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<h2>SL Paper 2</h2><div class="specification">
<p>John purchases a new bicycle for 880 US dollars (USD) and pays for it with a Canadian credit card. There is a transaction fee of 4.2 % charged to John by the credit card company to convert this purchase into Canadian dollars (CAD).</p>
<p>The exchange rate is 1 USD = 1.25 CAD.</p>
</div>
<div class="specification">
<p>John insures his bicycle with a US company. The insurance company produces the following table for the bicycle’s value during each year.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATIAAACzCAYAAAAKXZAMAAAgAElEQVR4Ae2dDXRT15Xv/1J4lFJjGJq8cAUtrNjBIcUpE3uRNvG0MhA5mTSUByWkSWxTe03JC02Y8IIcM+FNGlpIsJeZps2sTLKkGvuFBoI0JJlMsIkcT8dpakYiH2YGyzUUWluiQ0r8oRC+pPPWuR/S1ZUEtpBkfWytZe7Vuedj79/Zd99z9jni6hhjDPQhAkSACGQwAX0Gy06iEwEiQAREAuTIyBCIABHIeAKTFA10Op1ySkciQASIQNoTUEfFgo6MJ3Jnpr6Y9pokQUBikASoVCURSDAB7cCLppYJBkzVEQEikHoC5MhSz5xaJAJEIMEEyJElGChVRwSIQOoJkCNLPXNqkQgQgQQTIEeWYKBUHREgAqknQI4s9cypRSJABBJMgBxZgoFSdUSACKSeADmy1DOnFokAEUgwAXJkCQZK1REBIpB6AuTIUs+cWiQCRCDBBBLkyIbgavyO+BMnneFHsA9eCBczcBL22mLpevlOuHyB8OsZ/e0TdNSVSrpVWNEXoVoAIx2bYdDpoNPdB2vfuStr67WjWsz/MOzeS1fOn9IcAfj62tBYLfenzoAKay8i1OYy+frQsa8RD+90IahFsnW75EJjIWddiGr7Hy9D5hK89oelfqu2w3uZnMm79EfYqwvHIOtYJQjA59qJcrlPLrgaUSjaEeeh+lP0DbKSrxU2wiV2FO/jDljrKuRyxahutMPlHUFfx28lVsF+VNWrasNQ3Yh9r7vg1RhGoM+KCp0B5Y2H4BurWmPJx/8/MuUD8J9axvkZ7WYNRoH/32ZMMDvYcLAaPxt1NjEjwIB7WIPz0+CVdDwZPwM/G3bUM0HUbzWzuD/XqHWaOcwlUbhosqm/emysSqxvHbN5LqqvpMF5SB/Oiv8VNDhZpJSjzNlgjLyebN0uOllDAZergFXZ/nAZXheZx7ZOlA9VNua5TM7kXfoDs1UVjEHWMUrgP8osJoFBqGeOYb9cSN1f0ezzPBuwmdk62wmmlPAP2FiNsJBVNXUxj5h4nnmcLczM7+8wVp8zt2W1xFDF2+/pYk1VC8V0oeoXrNtzXqWAIk80WVTZrnCqvU8TNCIDkLcI3390JQQA3h1NeMk1JPlRnxMvPtGATgBCTQ2+/5czxuJfMyiPHvmly1DJFUcX9nSdCB+djHyMtlYXgBJUVtyC/AzSLLqon2PoFO9bASbLUfgZQ/8TJQj+7wPRC6Vh6iQIK18U/5MEtkuy2zQUchwiBTDS2YKn2r0QKpehNF+5tadg+nXT5HpmYsY0bU9NxvVzi3Dz3C9DKuHDB7/6OazeO/Dd+2+DICZOhlBSie27d6F+1jUqmSZh2oyZqu/SqV64A4+/YEGDUYC35UdY8dS/YDA4MpuJ0goTBLyGpyy/wUhE6fgSFG3jKx1WajJmr9iEX9QsBPAWNm22oy9wAYPtLWjq9ALCevzime9gttiiduhqQHmdFR19GrV8fTjYWC1Py/gQVhniKlNXZWhejmft+7C53ADd5aY6YfIm8Ev+LaioLOEuHO17foP+YKcFMOJ8B6183iKYUFHKO107NdNBZ6hGoz1yGB6SMNY0yAdXY3nk1IRP6ax1KFeG+uV1sHb0jWEoz6cOe8KmjWH9Ik4nvopVLcckXWsX4BpdtOkv75dFKN3EH1/AsU2l+B+6cjS6NJOJgBeuXYqcFajbdUgzFeHyWFEn9ivv/wrUWTvQN9bQRGAQh3Yq9sNtp01VNhbTC/C67CoGOhiqd+KgaJvn0Ge9L3oYIdALawW3v1LUdXwidR3Xz96IaoMy/dLKEOrhsLO4+u8MnG3t8KIAd95241U8MM9h+PQogDfw8394Q8UL0At/hbUV14eJGvNL3i1YXlkmXvZat+H5TpkJ9MhfUIo7uQW1vgPnSPBmiVnVmC6oR3Da4Zr62ljPpWEpH9ovZFXbn2ZVgnReEzF0laYlyvREPAo1zHJUnpT6TzBbjTQ8DcvDp641NjYgDnmVobm6rhJmdpweq7gR+eJjEGt6qQyjQ9PtEB+1zBpGEdOvWNMgZfqmmkbF5LaQVVl62GiExkrCMDtqqZGnyFrZ5JBAUC719WjT32j9YmQNzlHGgnUYWVWVNPUM9a/ATJaj8hSHT3nWR5VHqGpmR0eViZAiv3wMTi3VMobOQ2WjMVWHQUJlJNtcz2wD5xkbdjCzaNPhUyO/28JMfKodnNZ9ypwN98jTrvC6Iu03Af0XQy7GFBvhMkTrK8YuOi3sZ7xvxI+Wu4mZLQ7mjspbxVA1tZQrYkEm2vCDMgVG/Peq9j4NC4ppLyoCje+oBaF2PLwm5eZW31ihm0iJrwUhGJuYU4SoqreggTnFoIzqhlHy+T9jo5/FMPIxKBI3g6AhqW7GYJoSG1RiCgIzNnRLTkXleIKxpuDNrhieymDCYhSKkSo3Qsihhm5YPxs92iw9UII3WSSIIG/+AFIcnn+AOepN0s2o8GUKc6XNyLqkFEU2TQwtqJvAjGabdIOo21H0U9ipH26jPcwixl4ucwOoHZnxaeYQ4zPnmcfxtBynVcpGYRq8wQRmrG+X4kMq2STbVOxX1c8s1K+KIw7xNLF6x4DonP2edlYvxpEVGbQsE9B/EX0c6oexOTLGmEpn0YmLsVATMzd3yzEzpc9VDKM4stBDCyw8thaNoVLn2I7a+zSBU0tlAKieYvIplXpKCeDSSRy28ZjREbTUFmOaOP2ZjgW1VnE1RBlu6ufXoI0x+Hd/CwPt+2G3bsFDq16QVkzOnsHwWfWQVICp8m78ZZ4e0E9F3tQkqKWoF+sYMb1UTStNK7FcjA1OwfyavWDsBHaXf4J2+z5Y63+IVdYjYq1nTw/jbKz6x5R+Fr87/L7EsWUtFky7BjrdNZi2YC1a+PTW244255koNZ1Df9cBtPMrpsfxd2sXIo+f62djyZN1MPP4X+cbeNd9ddKFN1yGytq/xnyxzwSULisR46tKnku/OwybKLMVtQumS9O5acWobeGsXGht+/gK8ZUCVD1agyXCZACTISxZjy1mPv13wXb4ZGgVVWmQH//0X/i3dt7ocjz6aLkUH+IMtrWJsTTPc0uQDyXGowoj+D7GG61dAMqwpmwe9LiEPx05JPGsWodHl8wW40964U5se9cDxpx4bsm16pbl83j7DwiMngGf8GPqTEy/WvsXdX4NbocFZqMY/AXQjh1rb0PJ39hV8a4oKowpSYnbeXHszGfhMeUxlY/MlJw7Xj8b37znDqm1O5fgm7O5Mcmf0yfRIxJXEjRH7xkMfRYAfEdgrS7GNYZSrFi1Cqtqd4gLBmLuiM4yYNG8a+Vgpaa+lH1VDJz3+QF09Q/KMQsBpjW3o1CJDfbuQrXhCzCU3oNVq1ajdofoPiS1rpuOqeOS91Oc7FFvMdB+11Y2hFNDn2sT+dMFo2dOi+kFd34dN6itYup0XCcK9Uf0nPw0Stl4k9SBZz2mTp+p0v0STp/sl27MGNV7Tw3hsxjXpOSvoHjuX6hyKDePKklzeslzHO9p0iK/6pG/eAU28htc7OezGDm0X4wDCzX3o6JwCoBz8Bx3Rxa9Ykr8/XclXldsOiJDPuYvqcFz7/aGObTweFdEobCEwOgQTskpBcVzcV3Y1cR+UZtsYmuOVduXZmCW6OSNaHCOSqtGjKmOL2KlcAl9e5+Rnr5GMyy2t+D0nMdFZwMKotY7DddN5wY0kR/N6qX9FbwlrlYqT2ke5+/D3g31aPEKMJpfgm2/Ex7/KJwNxgQJ/kXMmCWtChc0OHExjCtn3I9dK78Spa1JmDZTMrNjBz/CcfVg9+wwTosDMa1jiFJNwpL0+NKMmdIIraABzotq+5DPr7jS6MbBjwZUT3sliB1byEmGGyA9fs9gaDS48y2yQDCQzVepD6FbDLKXoPLBb8mLWVNguKFIKndqCKNqnpG1qVLi77/r5hbGuDdU1Uc9DeDs8BTMNSj3jw8fdLhUo13ZoTneh01cyIv1MNRWHoBvoB89YnIJVt06N6kr26l3ZMEpWCeanrehl69ABQbRsVnafGeo68AIfBhwHxcRCDfchooVf42S60/j17aDl31Ka1Gm/Hv+rbhv4z3Sil59PZr4LMV0F8rEpzTfIOqBu4cnfhk33GbCiu+W4Po/vQ/bW1d6eqtu7IMdeF/ccHwB3g4rfi6uICqahkaFx5r+ES29fBWY5/uxtIJp2IyOqKtEU1BYdhdMvJr2nfhp8xFphZP3y7PPYQcX2bgc5UXjGy8qUo3/qHooHNuF51skeQLeg/LKtGplMGblXrS3tmC/uNrIGbyArTukbTAxb6rrb8a3Tfwp24XW5n+XV1BH0GutlVbOgxueFV5etFt+jG38gRVcleYCTcL1CxfLPPeguXNQcqjyLCP2ynq8/Qfop82UHFlE2GUqbrz1m/K03Y3jHu2G7DNwdo5gznWhbRmBwy+Htk8pfIOzrBmYNeOLSmrsY+CPeOeVN8Uwh1CzGY8Z1VNp5aEioGDmlxIzk1KH1rQBNPW18Z2rgoBK8DZYgSr4LG+oDAYUg4FdVWBfned2IzPyFaNgQFMJlsorYsE24j+5WgahIC9fJVrI1Ku1TBXYD+oMgd1uvF1cnVMWOkJBUiXYz+OvfJNi+OoXUMKMRs2GymBAXJtXFcSPiif2Klv4RmaF+ZWC/UoAXJFDu2oZ0o2Lc9HZwAp4XwftJbQAFGIl1RVayIiiiDrYr7Yd+Ty0YhjNRi+zaolQ0F5sNawv1YF/RabL8AwuQkRhGW//KYsjCF9NFaVRb1ZXb1D1e1h3Uz3bIi9GSJIriwPhwX1lk2uIH8+t7mOVPYweZTaztEgUuSGWLybIG3fTe9WSKxjNSCRM0r9+Nup2MIusLDdUoaqJtbtDvwdgfg9zNpvllSa+wtXCnJ5+eSe0dtUnfRxZqJPUDjeku9/TzZqDesvGMiDv5FccdHBlT32zq3ZX85vSaGYWh5M5xN3zKiPiTY26mcOisONyVLGGdvdltl4o8g0zt+NV1iDvygY4dwtzqPtlzKuWjCnGLzki7RYOtW7RHBmXictjkXaUi45oIatqOBBjK4CsQ9CRFbCq195X2ZC2bCwb5ZxtKgbcNhuYzekJ7nyXWgqtMCKa8+CZuA3bGuQtSNwJcxlszBnc6R7FkfFycfWfshKo3BsyD+WgrVO856LpNcoO73ewAb+WfbhjCz1slQeV+sjt5iVmc8SwOcXpKvauyDiOI7cp9Sfsm/aiOmOunBODXOnpq9VTNWI0WZg7/h0/VyuIXD7kWIMj+wTVnNhqEiOn9j5NfYws9qyarhCBDCCg/JpE2TJUAnPdcsyf8DsptJqqbGFKT5jKLxBW4ye1t1/FLxDCtZtw/OHi0DcikO4E/gJzi5WVXxPMtv+HLVH3hE2AHnml+OGWtRBi7hecAJk0TQYGf41XWl0QzA/je/OVlVJNpji+6viwUSlHb9mGuPFShURBQ0ciQATSiIDWV9GILI06h0QhAkQgPgLkyOLjRqWIABFIIwLkyNKoM0gUIkAE4iNAjiw+blSKCBCBNCJAjiyNOoNEIQJEID4C5Mji40aliAARSCMCwV+K8uVM/lGOaSRjykUhBilHTg0SgasiEHRkfO+Udm/GVdWcoYWJQYZ2HImdUwS0gw2aWuZU95OyRCA7CZAjy85+Ja2IQE4RIEeWU91NyhKB7CRAjiw7+5W0IgI5RYAcWU51NylLBLKTADmy7OxX0ooI5BQBcmQ51d2kLBHITgLkyLKzX0krIpBTBMiR5VR3k7JEIDsJkCPLzn4lrYhAThFIrSPz9aHDvhuN1RWo6/gkp0CTspcjMIK+gztRbdCJP5PTGarReLBPeklwsFgAvr42NFYXS3l0xahubEMff8Gz+hPwwrWrTn4hcTV2HvKq3jauzkjn2UQgdY4s0Avr/U9hl207NrX8OS6GgcHfovOY9k3JcVVFhdKGwAUM2jfDWH0E3+4Y5q8nhN/1EM5s+wG2qh52gcH92GB8Aj3f/hVGGQPzH0D1mSYYt3aCv09d+gzB1fQInnAvw26/VM/pukfQ5BpSMtAxSwmkzpHpb0LNv+zFL//+celV8uMGOoQPfvXPOO4fd0EqkM4EAsfR9k92oPJBfO+mfFFSvfBXWFs5G61tH8tO6hz6216FFfei+ns3I4/n0s+Gce0aFLe+A+eINCoL9NmxuWkBtjy5FIIe0AtL8eSWBWjabEefZuCWzkhItvETSJ0jG79sqhIB+Fy/xBObfq9Ko9OsIKC/FvMWGeA99BF+F5wm+jDgPovKilvk9x5Oxqx5hRC8H+Pw75TxVwC+gX4cq1yG0nxuxufQ33UA7cWFmJOnmLUe+aXLUNlzAF39NJLPCnuJoYTS4zEupzjZ1wt7XQV0YvzDjkOdv8UHI5fgc/0M95ZuRCdeQ23RF6EzbEaH/BROsYTUXMIJzMTi+x6EsXMj7l3fil7fOXg7mtE2qx5/a7xWbk15+exhbLr3/8DaO4KA14HGtq9i99+WSc4ucAJde7ogLJqHWRFW3YU9XScoVpbwvkufCoP/H9nEi3QOfXsb8NZtL8HP5kKPCxh8/Z/xOyxGXsnjcLin4+6iA1jjbkFNAl/sOfF657oEeuSVrMfu7sl4csVaLJhWD5OlA28/fhPC/FHeYmzc/RquefJh1C6YjlqTBe63Hw694dvngbsHKF5jkKaeuY41x/QPs5UJ1V18on6ES8dd+MB7AcBkzP7uGhjFacOESkaNJ53AJEybMQfFG7fDbATaazfgqY7BiBGUftqXMbe4Gg1mE9D+FNY9dRBein0lvXcyoYH0cWT6G1Cx7hs4uGkVSkv+Bo37OiOX1jOBKMk4TgIB+HpbsaHJj+9vfBLPOf4Djnpg+9J14auNviOwbvgl8P3H8MRzb8LjWAdsr8YDTYekbRp5BhQVAz1uj2bbxjjFoewZSSB9HBkfga1sgsv5JiyVp7BpdTmK5j8G+yAfndEnawkE+rB3w1YMLL5ZXGnkq5FLtu2BswHYFFxt5GGHZ1A7UISFwmRxtC4s2YI3nZuATY3Y23cO0M9D2ZqyCEyBUyfwobcMa8rmhU9VI3JSQiYTSCNHxjFOhlDyHdQ81wa/pwtNd/4bfvR8l2qfUCajJtmjEhBjW2c1l/Jx4623QAim8lXM48Fv0okeeTd+HYuDmaagsOyusO0YgLSy2WO6C2WFUzTl6Ws2EUgjR/YJOu2/DTot/fVzMHfSF1BcpA7eHod7YAS+D9rQQSO17LDD/Ftx38Zb0f5UA5p75a0Vvv/Cvl3v4+51S1EoWqi8stm+Ez9tPiJPHUfQu+8V2O6+HxWyk9LPX4ltG49i67MOMXbGVzaf3XoUG7etDC0KZAc10kJLgKk+AN9YnazPaeYwlzDeRvDPZGFuv9Len9lhx1us3fYSMxsFBpiYubmbeZTr/gHmqDeJ6fWOAaYkK6UTdUwug0RJmWX1+D3M2WxmRsU2hCrW0O5mo2FqnmceZ4tsG9yGFrKqhgPMPaqxBL+HdTdVMYHXZTSzZqcnabYSJh59SSkB7X2q460rzo1ehSa911OFREFDRyJABNKIgNZXpdHUMo0okShEgAhkFAFyZBnVXSQsESAC0QiQI4tGhdKIABHIKALkyDKqu0hYIkAEohEgRxaNCqURASKQUQTIkWVUd5GwRIAIRCNAjiwaFUojAkQgowiQI8uo7iJhiQARiEaAHFk0KpRGBIhARhEgR5ZR3UXCEgEiEI1A8H+I5Vv++Uc5RsucK2nEIFd6mvTMFgJBR8Z/X6j9/VK2KDkePYjBeGhRXiIwMQS0gw2aWk5MP1CrRIAIJJAAObIEwqSqiAARmBgC5Mgmhju1SgSIQAIJkCNLIEyqiggQgYkhQI5sYrhTq0SACCSQADmyBMKkqogAEZgYAuTIJoY7tUoEiEACCZAjSyBMqooIEIGJIUCObGK4U6tEgAgkkEBmOrKAF67XX0Zd+YOw8rdM04cIEIGcJpBcR+brw8HGahh0Ouh0BpTXtcLlvXCVwM+hr3kDSlf8EDs6L15lXVR84gkEMNKxWbYRbifqv/vCH1T8AbarDuU8j6EaOw95EbiMAoFBO2oNmjouk58uZS6B5DmywEm8/nIHsPzn8DAGv+c1LD+1A6UPvACX73LmdyWYUzC/pgVuy+orZaTrGUHgDJxt7fBGk9V0F8rkt4gDQ3A1PYIn3Muw28/gdz2E03WPoMk1FK0kEDiJ/f/372GNWnH0IpSauQSS5sgCvx/A9PtqcOf8fJGOXrgDG/7ucZg6X8HeQ2cylxhJnlgCI0dw+Avb4fEz/pp7+c+PYccW3LPmdhTKFhros2Nz0wJseXIpBD2gF5biyS0L0LTZjr6I5yJ3eg1479pbICRWWqotTQkkzZHpC+6AcfbkMLX1s+ZhEVlWGJNc/xL47Cv4X2bJOQVZBPqw77ljWFk2D5KBnkN/1wG0FxdiTp5isnrkly5DZc8BdPWr46QB+FwteB4PYWPF3GCVdJLdBBSrSJ2WU7+B24qkURowgr6OPWisroW177/RZ98sxz9qYe0dCZMp4D2EXXUVYgzFUG3Bx2coPhYGKEO/6IUCFASdk6REoP83sM9ZhQplWhk4ga49XRAWzcOsCIvtwp6uE6FYmc+JF58HHnu4FNMylAmJPX4CEWYx/irGWiKAEWcnPny4CiZxpMaDvM/CuPR+bGo5jvfeeAdD33wa77LTcFR+hNoN+4JThoD3IJ564Bm4K6zwM4aBZ7+GP/b891gbpnwZRYCPvg7haw9+C7MV6/R54O4BiosMyLusLkNwvbgbeKwKJRrneNlidDHjCSimknxFfE68tOtabHu4VDZGPfKX/AS9jnoIuA53LF+JxQKfiuZhTtENQE8/BsRFgSF88MrP0Lx4I55cMlucauiFb+Ceb89OvszUQuoJ8NGX/X+ionTmONvmU8pXYbvhcWwsmTHOspQ90wmkyJENwfXy25hZv3b8T8qRw9jbdHgMT+NM7wqSnxMQp5VfM6I0X2WaeQYUFQM9bg98sTD5nHjZJmD9irlyXC1WRkrPRgIqa0mWehcw+PqrOHL346i5SYmNJastqjezCcjTyopbEGYp+nkoW1MWoVrg1Al86C3DmrKvwndoPxq2r8Cca5R9aNdg+tLt8OI11BZ9EboKazBUEVERJWQ8gSQ7sgvwdlixd9q9qAw6sRH07mpF50jEmnl0mFGfxj4MuI9Hz0+pmUsg5rRyCgrL7kJx6ztwBu0mAN9AP3rEvWZTkb9km7hfMXwLBw9brIbF/TlYWw3mJ9naMxd85kuexK4dQZ/9aTyw9H9j49I5uCa4Y3s6Fuy+CMNYg7H6G1CxbiWwYwMe3fkevIEAfL3v4fAnfE2KP22rwnd/Z36f5KwGUaeVMg39/JXYtvEotj7rgDcABLwOPLv1KDZuW0kOKmctRqU4U30Avh8xEZ/zbMC2ngkA43WG/wnMZDnK/Oxz5rasVl0rYWbHfzKHuUSVtppZ3J8zxoaZu72JVQm8roWsytLNnJYHmNH8Etvv9DB/IkSW60gcgwQKlRNVcXswswbnp7G19XtYd1OVZFdGM2u+bN/72bCjnglQbCh2tXQl8who71MdV0Hxa/QqNOm9niokCho6EgEikEYEtL4qiVPLNNKaRCECRCCrCZAjy+ruJeWIQG4QIEeWG/1MWhKBrCZAjiyru5eUIwK5QYAcWW70M2lJBLKaADmyrO5eUo4I5AYBcmS50c+kJRHIagLkyLK6e0k5IpAbBMiR5UY/k5ZEIKsJTFK04ztl+Uc5Kum5eCQGudjrpHMmEwg6Mv6zHO22/0xWLF7ZiUG85KgcEUgdAe1gg6aWqWNPLREBIpAkAuTIkgSWqiUCRCB1BMiRpY41tUQEiECSCJAjSxJYqpYIEIHUESBHljrW1BIRIAJJIkCOLElgqVoiQARSR4AcWepYU0tEgAgkiQA5siSBpWqJABFIHQFyZKljTS0RASKQJAIZ58gCXhdet9ahnF64miSToGqJQOYRSI0jC3hxaGc1DOK7LStQZ++FLx5WgV40r70XK2p3oDOe8lQmjQl8go66UvFncvznJzqdARXWXkR9jXPAC9fru9FYXQydrhR1HZ8E9Qp438NOMV0HXflm2PtGgtdinvj6cLBRsU8Dyuta4fJeiMzO291Vh3Iun6EaOw95o8sXWZJSkkwgBY5sCB/sfw+4/2V42Hl4upfj1I8ewlaV8Y1ZR/1NqHm7AxaTMOYilDEzCAQGf41XWl0qYcuwpmwetAYqOqofmHCv3YN51TaMMieeW3KtVM73Efa3X4P7f9kD5vege/kgfmR8Fh3Bt5OrqldOAyfx+ssdwPKfi28q93tew/JTO1D6wAtw+dRudAiupkfwhHsZdvsZ/K6HcLruETS5hpSa6DiRBNSv5tS+9FJ9Ld5zf38Xe3fgvKr4afElvILZwYZVqWM+9R9lFpPAYLIwdyLfzCsLkAwGY9YtZzN+ypwNVczsOH15AqPdrMFYwqqaupgnou8/Z/3vvs8G1OnDDmYW+IufY9cbaZ+M+d0WZkJ4OTFNqGeOYaUB+QXASbLDy4Ogq9r7VPvAS7hP1RfcAePsyaF6A5/gxIdfwcb7bkV+KJXOcpnAyGHsbWrBjq0NsNo70Rc2ElLADMH14jNoumEztm24A0KE5U5BgfEbmK1KD5w6gQ+LHsR9i2cqlUQcI+wTgH7WPCwKG/SfQ3/XAbQXF2JOntKAHvmly1DZcwBd/eci6qWE1BJQeiUFrQbg6+uAtf4f0F/3C2wsmSG3ydM7sa+xGoV1/4reQy+g2lCMausROY52AV5XK+rKDdDpilFt+Q+cSYG01ESqCJxD374XscMLoHMHaleVo+jepyJiW4E+OzZvuojKOz7Dr37AY2M6GKp34mDUGNgI+jqsqP+pF3W716Mk6HzGodPUb+C2IvlRGziBrj1dEBbNw6yIO6YLe7pOUKxsHGiTkYT7diIAAAZjSURBVDWiW5LRCMADuYsxrWgpane8gffbutGvPHVHOrHVWI7Vm1pw9tTvMfKV1XhsYyH+cOYsgAvwdmzHA08cR8XuE2DMhWdv/BQ9ca0UJEczqvVqCUzB/Jq9YDx+6nwTFrMJ6NyOVessqhiVPCIyLsLNX1+Gjbt6wEZ78BNYYQrLB2CkA3WG6ShaWosdLb9G2/vHx7mwFMCIsxMfPlwFkzKT8Hng7gGKiwzIu1p1qXxyCKhn29p5p/paQs79HuZsNjMjwIQaWyieIcYywCLiZmJM5PbwGAfFyBLSFelbyXnmcTzNjGExquhxVSmWJTCT5ShTIleKXn5PN2s2mxiwkNXYTkRcV/JFHLnNVf2MOUdVNYr2GaWdWOkRlVJCoglofVWKRmSyE9YLKKn+Mf7Jshret51wK6OyqD46gJFD+9HUORtFc+g5GBVRViZOhrBkPbaYgda2jyFunhDjqp4IbfWFt2ONCehxeyJGXXphMaq3/wwW05/xdvexiOsRlYkJQ3C9/DZm1q8Nn47mGVBUHL2d6PVQaqoJpNaRidpNQWHZXTClWlNqL4MI5GFOUVFoKqe/FvMWGeD98AROqXdEiBpNDeXTaqifh7I1ZdrUGN8vYPD1V3Hk7sdRc5NmGSpGPeJigjf6NpEYjVBykghMgCMLwDfQj2N3l6LoskFYPfLmFKIYx+EeUAXFxHgFjwzTJ3sJ+DDQX4y6782X95HNRGmFCULPYRxRb1QVbeHWqPvNJDY+DLiHcfdtBVeIbfFYrBV7p92LyqATG0HvrlZ0invQpIdvces7cAb3pEl23GO6C2WFU7K3KzJFM/XcVTvvVF+L7/w8G7CtZ4LRzJqdHjFO4fe0s3rTOmY5qtpFFitG5j/BbDULGYQq1tTNyw+zo7YGVmMUGJc1GXvJEs8gPnI5U4rHTfe/xZweea+h38O6m+rZFsdAeFxLtgWhqpkd5fErMV8NMzV0s1EOS7xewozmFrkuKdZmUvIrQJV8Sjk2zNy2ejFuK9oUtyvlL2yPGN/rtoIZ69vFPWyiHRtXsAbnp0rNdEwhAe19CnXb2ovqa/Gd+9no0WZWJSjGsZBVNdhCRsvtT9x8KF+/3ciM92g2uo66WXtDFRO4cQk1zPKf7zOL6R5mtrwZVk988kWWSjyDyDYoRUXAP8Ac9Twoz22A28erzOFWPeRUWZnaFmBi5uZu1cbYYXbUUiPZiWgrVazB5lRdlysKc2Tyg1ZxXGHHKMF90XnKtqh6OKtFpPPUENDepzrerDJ6pFehSe/1VCFR0NCRCBCBNCKg9VUTECNLIxokChEgAllBgBxZVnQjKUEEcpsAObLc7n/SnghkBQFyZFnRjaQEEchtAuTIcrv/SXsikBUEyJFlRTeSEkQgtwmQI8vt/iftiUBWECBHlhXdSEoQgdwmQI4st/uftCcCWUFgkqIF3ynLP8pRSc/FIzHIxV4nnTOZQNCR8Z/laLf9Z7Ji8cpODOIlR+WIQOoIaAcbNLVMHXtqiQgQgSQRIEeWJLBULREgAqkjQI4sdaypJSJABJJEgBxZksBStUSACKSOADmy1LGmlogAEUgSAXJkSQJL1RIBIpA6AuTIUseaWiICRCBJBMiRJQksVUsEiEDqCJAjSx1raokIEIEkESBHliSwVC0RIAKpI0COLHWsqSUiQASSRIAcWZLAUrVEgAikjgA5stSxppaIABFIEgFyZEkCS9USASKQOgLkyFLHmloiAkQgSQTIkSUJLFVLBIhA6giQI0sda2qJCBCBJBEgR5YksFQtESACqSNAjix1rKklIkAEkkSAHFmSwFK1RIAIpI5Ayh1ZYNCOWkMxau0nEVDpOd50VVE6zXUCgZOw1xbDUGvHYJhRjTM91zlmsP4pd2QZzIpEJwJEIE0J6Bh/D5z8oVehSe/1VCFR0NCRCBCBNCKg9VU0IkujziFRiAARiI8AObL4uFEpIkAE0ogAObI06gwShQgQgfgIkCOLjxuVIgJEII0IkCNLo84gUYgAEYiPADmy+LhRKSJABNKIADmyNOoMEoUIEIH4CJAji48blSICRCCNCJAjS6POIFGIABGIjwA5svi4USkiQATSiMAkRRa+5Z9/lKOSnotHYpCLvU46ZzKBoCOj3xdmcjeS7EQgtwnQ1DK3+5+0JwJZQYAcWVZ0IylBBHKbwP8HMdJX/9F+uz8AAAAASUVORK5CYII="></p>
<p>The values of the bicycle form a geometric sequence.</p>
</div>
<div class="specification">
<p>During the 1st year John pays 120 USD to insure his bicycle. Each year the amount he pays to insure his bicycle is reduced by 3.50 USD.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate, in CAD, the total amount John pays for the bicycle.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the bicycle during the 5th year. <strong>Give your answer to two decimal places</strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate, in years, when the bicycle value will be less than 50 USD.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total amount John has paid to insure his bicycle for the first 5 years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>John purchased the bicycle in 2008.</p>
<p>Justify why John should not insure his bicycle in 2019.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>1.042 × 880 × 1.25 <strong>OR</strong> (880 + 0.042 × 880) × 1.25 <em><strong> (M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for multiplying 880 by 1.042 and <em><strong>(M1)</strong></em> for multiplying 880 by 1.25.</p>
<p>1150 (CAD) (1146.20 (CAD)) <em><strong> (A1)(G2)</strong></em></p>
<p><strong>Note:</strong> Accept 1146.2 (CAD)</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{704}}{{880}}">
<mfrac>
<mrow>
<mn>704</mn>
</mrow>
<mrow>
<mn>880</mn>
</mrow>
</mfrac>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{563.20}}{{704}}">
<mfrac>
<mrow>
<mn>563.20</mn>
</mrow>
<mrow>
<mn>704</mn>
</mrow>
</mfrac>
</math></span> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correctly dividing sequential terms to find the common ratio, or 0.8 seen.</p>
<p>880(0.8)<sup>5−1</sup> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution into geometric sequence formula.</p>
<p>360.45 (USD) <em><strong> (A1)(G3)</strong></em></p>
<p><strong>Note:</strong> Do not award the final <em><strong>(A1)</strong></em> if the answer is not correct to 2 decimal places. Award at most <em><strong>(M0)(M1)(A0)</strong></em> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 1.25">
<mi>r</mi>
<mo>=</mo>
<mn>1.25</mn>
</math></span>.</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="880{\left( {0.8} \right)^{n - 1}} < 50">
<mn>880</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.8</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo><</mo>
<mn>50</mn>
</math></span> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution into geometric sequence formula and (in)equating to 50. Accept weak or strict inequalities. Accept an equation. Follow through from their common ratio in part (b). Accept a sketch of their GP with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 50">
<mi>y</mi>
<mo>=</mo>
<mn>50</mn>
</math></span> as a valid method.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{13}} = 60.473">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>13</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>60.473</mn>
</math></span> <strong>AND</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{14}} = 48.379">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>14</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>48.379</mn>
</math></span> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{13}}">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>13</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{14}}">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>14</mn>
</mrow>
</msub>
</mrow>
</math></span> <strong>both</strong> seen. If the student states <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{14}} = 48.379 < 50">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>14</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>48.379</mn>
<mo><</mo>
<mn>50</mn>
</math></span>, without <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{13}} = 60.473">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>13</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>60.473</mn>
</math></span> seen, this is not sufficient to award <em><strong>(M1)</strong></em>.</p>
<p>14 or “14th year” or “after the 13th year” <strong> <em>(A1)</em>(ft)<em>(G2)</em></strong></p>
<p><strong>Note:</strong> The context of the question requires the final answer to be an integer. Award at most <em><strong>(M1)(A0)</strong></em> for a final answer of 13.9 years. Follow through from their 0.8 in part (b).</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{2}\left( {\left( {2 \times 120} \right) + \left( { - 3.5\left( {5 - 1} \right)} \right)} \right)">
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>120</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>3.5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>5</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution into arithmetic series formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p>565 (USD) <strong> <em>(A1)</em><em>(G2)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2019 is the 12th year/term <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for 12 seen.</p>
<p>75.59 (value of bicycle) <strong>AND </strong> 81.5 (cost of insurance policy) <strong><em>(A1)</em>(ft)</strong></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for both sequences’ 12th term seen. The value of the bicycle will follow through from their common ratio in part (b). Do not award <em><strong>(M0)(A1)</strong></em>.</p>
<p>the cost of the insurance policy is greater than the value of the bicycle <strong><em>(R1)</em>(ft)</strong></p>
<p><strong>Note:</strong> Award <strong><em>(R1)</em>(ft)</strong> for a reason consistent with their cost of insurance policy and their value of the bicycle. Follow through within this part. Award <em><strong>(R0)</strong></em> if the correct values are not explicitly seen. Accept the following contextualized reasons: “the insurance is not worth it", "the values are too close", "insurance is as much as the value of the bike", but only if <strong>their</strong> cost of insurance is greater than the value of the bicycle.</p>
<p><strong>OR</strong></p>
<p>75.59 < 81.5 <em><strong>(R1)</strong></em><strong>(ft)</strong></p>
<p><strong>Note:</strong> Award <em><strong>(R1)</strong></em><strong>(ft)</strong> for a correct numerical comparison showing their cost of insurance policy is greater than their value of the bicycle. Follow through within this part.</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mo>)</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo> </mo></math>.</p>
<p>Given that the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mn>412</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>product of a binomial coefficient, a power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> (and a power of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math>) seen <em><strong>(M1)</strong></em></p>
<p>evidence of correct term chosen <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mmultiscripts><mrow></mrow><mprescripts></mprescripts><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><none></none></mmultiscripts></mmultiscripts><mo>×</mo><msup><mn>3</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>-</mo><mn>2</mn></mrow></msup><mo>×</mo><msup><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mn>2</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>×</mo><msup><mn>3</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></mfenced></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mi>r</mi><mo>=</mo><mn>1</mn></math></p>
<p>equating their coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20412</mn></math> or their term to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20412</mn><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mmultiscripts><mrow></mrow><mprescripts></mprescripts><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><none></none></mmultiscripts></mmultiscripts><mo>×</mo><msup><mn>3</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>20412</mn></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mrow><mi>r</mi><mo>+</mo><mn>2</mn></mrow></mmultiscripts><mo>×</mo><msup><mn>3</mn><mi>r</mi></msup><mo>=</mo><mn>20412</mn><mo>⇒</mo><mi>r</mi><mo>=</mo><mn>6</mn></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong><br>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></p>
<p>product of a binomial coefficient, and a power of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> seen <em><strong>(M1)</strong></em></p>
<p>evidence of correct term chosen <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>×</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>×</mo><msup><mfenced><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msup><mn>3</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mn>2</mn></mfrac><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mi>x</mi><mn>4</mn></msup></mrow></mfenced></math> </p>
<p>equating their coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20412</mn></math> or their term to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20412</mn><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>3</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>×</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>20412</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A large underground tank is constructed at Mills Airport to store fuel. The tank is in the shape of an isosceles trapezoidal prism, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABCDEFGH</mtext></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext><mo>=</mo><mn>70</mn><mo> </mo><mtext>m</mtext></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AF</mtext><mo>=</mo><mn>200</mn><mo> </mo><mtext>m</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AD</mtext><mo>=</mo><mn>40</mn><mo> </mo><mtext>m</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mn>40</mn><mo> </mo><mtext>m</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>CD</mtext><mo>=</mo><mn>110</mn><mo> </mo><mtext>m</mtext></math>. Angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ADC</mtext><mo>=</mo><mn>60</mn><mo>°</mo></math> and angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BCD</mtext><mo>=</mo><mn>60</mn><mo>°</mo></math>. The tank is illustrated below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Once construction was complete, a fuel pump was used to pump fuel <strong>into</strong> the empty tank. The amount of fuel pumped into the tank by this pump <strong>each hour</strong> decreases as an arithmetic sequence with terms <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>u</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>u</mi><mn>3</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msub><mi>u</mi><mi>n</mi></msub></math>.</p>
<p>Part of this sequence is shown in the table.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>At the end of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext>nd</mtext></math> hour, the total volume of fuel in the tank was <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>88</mn><mo> </mo><mn>200</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, the height of the tank.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the tank is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624</mn><mo> </mo><mn>000</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>, correct to three significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the common difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount of fuel pumped into the tank in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>13th</mtext></math> hour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the number of hours that the pump was pumping fuel into the tank.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total amount of fuel pumped into the tank in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the tank will never be completely filled using this pump.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>60</mn><mo>°</mo><mo>=</mo><mfrac><mi>h</mi><mn>40</mn></mfrac></math><strong> OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>60</mn><mo>°</mo><mo>=</mo><mfrac><mi>h</mi><mn>20</mn></mfrac></math><strong> </strong> <strong><em>(M1)</em></strong></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitutions in trig ratio.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>20</mn><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mn>40</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><msqrt><msup><mn>40</mn><mn>2</mn></msup><mo>-</mo><msup><mn>20</mn><mn>2</mn></msup></msqrt></mfenced></math><strong> </strong> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitutions in Pythagoras’ theorem.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>34</mn><mo>.</mo><mn>6</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><msqrt><mn>1200</mn></msqrt><mo>,</mo><mo> </mo><mn>20</mn><msqrt><mn>3</mn></msqrt><mo>,</mo><mo> </mo><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(A1)</em><em>(G2)</em></strong></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>70</mn><mo>+</mo><mn>110</mn></mrow></mfenced><mfenced><mrow><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo></mrow></mfenced><mo>×</mo><mn>200</mn></math><strong> </strong> <strong><em>(M1)(M1)</em></strong></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correctly substituted area of trapezium formula, provided all substitutions are positive. Award <em><strong>(M1)</strong></em> for multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn></math>. Follow through from part (a).</p>
<p><br><strong>OR</strong></p>
<p><strong><br></strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>20</mn><mo>×</mo><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo><mo>+</mo><mn>70</mn><mo>×</mo><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo></mrow></mfenced><mo>×</mo><mn>200</mn></math><strong> </strong> <strong><em>(M1)(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for the addition of correct areas for two triangles and one rectangle. Award <em><strong>(M1)</strong></em> for multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn></math>. Follow through from part (a).</p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>70</mn><mo>×</mo><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo><mo>×</mo><mn>200</mn><mo>+</mo><mn>2</mn><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>34</mn><mo>.</mo><mn>6410</mn><mo>…</mo><mo>×</mo><mn>20</mn><mo>×</mo><mn>200</mn></math><strong> </strong> <strong><em>(M1)(M1)</em></strong></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution in volume of cuboid formula. Award <em><strong>(M1)</strong></em> for correctly substituted volume of triangular prism(s). Follow through from part (a).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>623538</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong> (AG)</strong></em></p>
<p><strong><br>Note:</strong> Both an unrounded answer that rounds to the given answer and the rounded value must be seen for the <em><strong>(A1)</strong></em> to be awarded.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>d</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mo>-</mo><mn>1800</mn></math><strong> </strong> <strong><em>(A1)</em></strong></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>u</mi><mn>13</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mn>13</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></math><strong> </strong> <strong><em>(M1)</em></strong></p>
<p><strong><br>Note:</strong> Award <strong><em>(M1)</em></strong> for correct substitutions in arithmetic sequence formula.<br><strong>OR</strong><br>Award <strong><em>(M1)</em></strong> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>4</mtext><mtext>th</mtext></msup></math> term seen as part of list.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>23400</mn><mo> </mo><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(G2)</strong></em></p>
<p><strong><br>Note:</strong> Follow through from part (c) for their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></math><strong> </strong> <strong><em>(M1)</em></strong></p>
<p><strong><br>Note:</strong> Award <strong><em>(M1)</em></strong> for their correct substitution into arithmetic sequence formula, equated to zero.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mn>26</mn></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(G2)</strong></em></p>
<p><strong><br>Note:</strong> Follow through from part (c). Award at most <em><strong>(M1)(A0)</strong></em> if their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is not a positive integer.</p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn></math><strong> </strong> <em><strong>(A1)</strong></em><strong>(ft)</strong></p>
<p><strong><br>Note:</strong> Follow through from part (e)(i), but only if their final answer in (e)(i) is positive. If their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> in part (e)(i) is not an integer, award <em><strong>(A1)</strong></em><strong>(ft)</strong> for the nearest lower integer.</p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mn>8</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>8</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mn>8</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced></math><strong> </strong> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitutions in arithmetic series formula. If a list method is used, award <em><strong>(M1)</strong></em> for the addition of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> correct terms.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>310</mn><mo> </mo><mn>000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo> </mo><mfenced><mrow><mn>309</mn><mo> </mo><mn>600</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(G2)</strong></em></p>
<p><br><strong>Note:</strong> Follow through from part (c). Award at most <em><strong>(M1)(A0)</strong></em> if their final answer is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624</mn><mo> </mo><mn>000</mn></math>.</p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mn>25</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>25</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mn>25</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced><mo> </mo><mo> </mo><mo>,</mo><mo> </mo><mo> </mo><mfenced><mrow><msub><mi>S</mi><mn>25</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>25</mn><mn>2</mn></mfrac><mfenced><mrow><mn>45000</mn><mo>+</mo><mn>1800</mn></mrow></mfenced></math><strong> </strong> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitutions into arithmetic series formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>25</mn></msub><mo>=</mo><mn>585000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(G1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for correctly finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>26</mn></msub><mo>=</mo><mn>585000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math>, provided working is shown e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mn>26</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>26</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mn>26</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mn>26</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>26</mn><mn>2</mn></mfrac><mfenced><mrow><mn>45000</mn><mo>+</mo><mn>0</mn></mrow></mfenced></math>. Follow through from part (c) and either their (e)(i) or (e)(ii). If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo><</mo><mn>0</mn></math> and their final answer is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624</mn><mo> </mo><mn>000</mn></math>, award at most <em><strong>(M1)(A1)</strong></em><strong>(ft)</strong><em><strong>(R0)</strong></em>. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>></mo><mn>0</mn></math>, there is no maximum, award at most <em><strong>(M1)(A0)(R0)</strong></em>. Award no marks if their number of terms is not a positive integer.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo><</mo><mn>624000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>(R1)</strong></em></p>
<p>Hence it will never be filled <em><strong>(AG)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>(AG)</strong></em> line must be seen. If it is omitted do not award the final <em><strong>(R1)</strong></em>. Do not follow through within the part.<br>For unsupported <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>S</mi><mn>25</mn></msub></mfenced><mo>=</mo><mn>585000</mn></math> seen, award at most <em><strong>(G1)(R1)(AG)</strong></em>. Working must be seen to follow through from parts (c) and (e)(i) or (e)(ii).</p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced></math><strong> </strong> <em><strong>(M1)</strong></em></p>
<p><em><strong><br></strong></em><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution into arithmetic series formula, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p><br>Maximum of this function <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585225</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Follow through from part (c). Award at most <em><strong>(M1)(A1)</strong></em><strong>(ft)</strong><em><strong>(R0)</strong></em> if their final answer is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624</mn><mo> </mo><mn>000</mn></math>. Award at most <em><strong>(M1)(A0)(R0)</strong></em> if their common difference is not <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mn>1800</mn></math>. Award at most <em><strong>(M1)(A0)(R0)</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585</mn><mo> </mo><mn>225</mn></math> is not explicitly identified as the maximum of the function.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585225</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo><</mo><mn>624000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>(R1)</strong></em><br><br>Hence it will never be filled <em><strong>(AG)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>(AG)</strong></em> line must be seen. If it is omitted do not award the final <em><strong>(R1)</strong></em>. Do not follow through within the part.</p>
<p><br><strong>OR</strong></p>
<p><br>sketch with concave down curve <strong>and</strong> labelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624000</mn></math> horizontal line<strong> </strong> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Accept a label of “tank volume” instead of a numerical value. Award <em><strong>(M0)</strong></em> if the line and the curve intersect.</p>
<p><br>curve explicitly labelled as <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced></math> </strong>or equivalent <em><strong>(A1)<br></strong></em><strong><br>Note:</strong> Award <em><strong>(A1)</strong></em> for a written explanation interpreting the sketch. Accept a comparison of values, e.g <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585225</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo><</mo><mn>624000</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>585225</mn></math> is the graphical maximum. Award at most <em><strong>(M1)(A0)(R0)</strong></em> if their common difference is not <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>–</mo><mn>1800</mn></math>.</p>
<p><br>the line and the curve do not intersect <em><strong>(R1)</strong></em></p>
<p>hence it will never be filled <em><strong>(AG)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>(AG)</strong></em> line must be seen. If it is omitted do not award the final <em><strong>(R1)</strong></em>. Do not follow through within the part.</p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624000</mn><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>45000</mn><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mn>1800</mn></mrow></mfenced></mrow></mfenced></math><strong> </strong> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their correctly substituted arithmetic series formula equated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>624000</mn><mo> </mo><mo>(</mo><mn>623538</mn><mo>)</mo></math>.</p>
<p><br>Demonstrates there is no solution <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for a correct working that the discriminant is less than zero <strong>OR</strong> correct working indicating there is no real solution in the quadratic formula.</p>
<p><br>There is no (real) solution (to this equation) <em><strong>(</strong><strong>R1)</strong></em></p>
<p>hence it will never be filled <em><strong>(AG)</strong></em></p>
<p><br><strong>Note:</strong> At most <em><strong>(M1)(A0)(R0)</strong></em> for their correctly substituted arithmetic series formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>624000</mn><mo>,</mo><mo> </mo><mn>623538</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>622800</mn></math> with a statement "no solution". Follow through from their part (b).</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Two friends Amelia and Bill, each set themselves a target of saving <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>20</mn><mo> </mo><mn>000</mn></math>. They each have <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>9000</mn></math> to invest.</p>
</div>
<div class="specification">
<p>Amelia invests her <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>9000</mn></math> in an account that offers an interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>%</mo></math> per annum compounded <strong>annually</strong>.</p>
</div>
<div class="specification">
<p>A third friend Chris also wants to reach the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>20</mn><mo> </mo><mn>000</mn></math> target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Amelia’s investment after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years to the nearest hundred dollars.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of years required for Amelia’s investment to reach the target.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Bill invests his <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>9000</mn></math> in an account that offers an interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>%</mo></math> per annum compounded <strong>monthly</strong>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is set to two decimal places.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> needed for Bill to reach the target after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Chris will never reach the target if his initial deposit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>9000</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount Chris needs to deposit initially in order to reach the target after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9000</mn><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>7</mn><mn>100</mn></mfrac></mrow></mfenced><mn>5</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12622</mn><mo>.</mo><mn>965</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>I</mtext><mo>%</mo><mo>=</mo><mn>7</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PV</mtext><mo>=</mo><mo>∓</mo><mn>9000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P/Y</mtext><mo>=</mo><mn>1</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C/Y</mtext><mo>=</mo><mn>1</mn></math> <em><strong>(A1)</strong></em><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>12622</mn><mo>.</mo><mn>965</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>$</mo><mo>)</mo><mo> </mo><mn>12600</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9000</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>7</mn><mn>100</mn></mfrac></mrow></mfenced><mi>x</mi></msup><mo>=</mo><mn>20000</mn></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>I</mtext><mo>%</mo><mo>=</mo><mn>7</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PV</mtext><mo>=</mo><mo>∓</mo><mn>9000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>FV</mtext><mo>=</mo><mo>±</mo><mn>20000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P/Y</mtext><mo>=</mo><mn>1</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C/Y</mtext><mo>=</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn></math> (years) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to substitute into compound interest formula (condone absence of compounding periods) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9000</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>r</mi><mrow><mn>100</mn><mo>×</mo><mn>12</mn></mrow></mfrac></mrow></mfenced><mrow><mn>12</mn><mo>×</mo><mn>10</mn></mrow></msup><mo>=</mo><mn>20000</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>01170</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>02</mn><mo> </mo><mfenced><mo>%</mo></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>10</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PV</mtext><mo>=</mo><mo>±</mo><mn>9000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>FV</mtext><mo>=</mo><mo>∓</mo><mn>20000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P/Y</mtext><mo>=</mo><mn>1</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C/Y</mtext><mo>=</mo><mn>12</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>01170</mn><mo>…</mo></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to use a financial app in their technology, award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo> </mo><mi>r</mi><mo>=</mo><mo>)</mo><mo> </mo><mn>8</mn><mo>.</mo><mn>01170</mn><mo>…</mo></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>02</mn><mo> </mo><mfenced><mo>%</mo></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognising geometric series (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>4500</mn><mn>9000</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>considering <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>9000</mn><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mfenced><mrow><mo>=</mo><mn>18000</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>correct reasoning that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18000</mn><mo><</mo><mn>20000</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo><</mo><mn>20000</mn></math> only if <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub></math> has been calculated.</p>
<p> </p>
<p><strong>OR</strong></p>
<p>considering <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> for a large value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>80</mn></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> only if the candidate gives a valid reason for choosing a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn><mo>≤</mo><mi>n</mi><mo><</mo><mn>80</mn></math>.</p>
<p> </p>
<p>correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> for their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p>valid reason why Chris will not reach the target, which involves their choice of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> and Chris’ age OR using two large values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> to recognize asymptotic behaviour of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo></math>. <em><strong>R</strong><strong>1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the <em><strong>R</strong></em> mark without the preceding <em><strong>A</strong></em> mark.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>Therefore, Chris will never reach the target. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognising geometric sum <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><msup><mn>5</mn><mn>5</mn></msup></mrow></mfenced></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mn>20000</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10322</mn><mo>.</mo><mn>58</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>$</mo><mo>)</mo><mo> </mo><mn>10323</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Tommaso plans to compete in a regional bicycle race after he graduates, however he needs to buy a racing bicycle. He finds a bicycle that costs 1100 euro (EUR). Tommaso has 950 EUR and invests this money in an account that pays 5 % interest per year, <strong>compounded monthly</strong>.</p>
</div>
<div class="specification">
<p>The cost of the bicycle, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>, can be modelled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 20x + 1100">
<mi>C</mi>
<mo>=</mo>
<mn>20</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1100</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is the number of years since Tommaso invested his money.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> complete <strong>months</strong> Tommaso will, for the first time, have enough money in his account to buy the bicycle.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="950 \times {\left( {1 + \frac{5}{{12 \times 100}}} \right)^{12 \times 3}}">
<mn>950</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mn>100</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong><strong><em>(A1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for substitution in the compound interest formula: <em><strong>(A1)</strong></em> for correct substitution.</p>
<p><strong>OR</strong></p>
<p>N = 3<br>I% = 5<br>PV = 950<br>P/Y = 1<br>C/Y = 12 <strong><em>(A1)</em></strong><strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for C/Y = 12 seen, <em><strong>(M1)</strong></em> for other correct entries.</p>
<p><strong>OR</strong></p>
<p>N = 36<br>I% = 5<br>PV = 950<br>P/Y = 12<br>C/Y = 12 <strong><em>(A1)</em></strong><strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for C/Y = 12 seen, <em><strong>(M1)</strong></em> for other correct entries.</p>
<p>1103.40 (EUR) <strong><em>(A1)</em></strong><strong><em>(G3)</em></strong></p>
<p><strong>Note:</strong> Answer must be given to 2 decimal places.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(20 × 3 + 1100) − 1103.40 <strong><em>(M1)</em></strong><strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for correct substitution into cost of bike function, <strong><em>(M1)</em></strong> for subtracting their answer to part (a). This subtraction may be implied by their final answer (follow through from their part (a) for this implied subtraction).</p>
<p>55.60 (EUR) <strong><em>(A1)</em>(ft)</strong><strong><em>(G3)</em></strong></p>
<p><strong>Note:</strong> Follow through from part (a). The answer must be two decimal places.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="950 \times {\left( {1 + \frac{5}{{12 \times 100}}} \right)^{12x}} = 20x + 1100">
<mn>950</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mn>100</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mn>12</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>20</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1100</mn>
</math></span> <strong><em>(M1)</em></strong><strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for their correct substitution in the compound interest formula with a variable in the exponent; <em><strong>(M1)</strong></em> for comparing their expressions provided variables are the same (not an expression with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for years and another with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> representing months). Award at most <em><strong>(M0)(M1)(A0)(M1)(A0)</strong></em> for substitution of an integer in both expressions and comparison of the results. Accept inequality.</p>
<p>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> =) 4.52157… (years) <strong><em>(A1)</em>(ft)</strong></p>
<p>4.52157… × 12 (= 54.2588…) <strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for multiplying <strong>their</strong> value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> by 12. This may be implied.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> = 55 (months) <strong><em>(A1)</em>(ft)</strong><strong><em>(G4)</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="950 \times {\left( {1 + \frac{5}{{12 \times 100}}} \right)^m} = 20 \times \frac{m}{{12}} + 1100">
<mn>950</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mn>100</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>m</mi>
</msup>
</mrow>
<mo>=</mo>
<mn>20</mn>
<mo>×</mo>
<mfrac>
<mi>m</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mn>1100</mn>
</math></span> <strong><em>(M1)</em></strong><strong><em>(M1)</em></strong><strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for their correct substitution in the compound interest formula with a variable in the exponent to solve; <strong>(M1)</strong> for comparing their expressions provided variables are the same; <em><strong>(M1)</strong></em> for converting years to months in these expressions. Award at most <em><strong>(M0)(M1)(A0)(M1)(A0)</strong></em> for substitution of an integer in both expressions and comparison of the results. Accept inequality.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> = 54.2588… (months) <strong><em>(A1)</em>(ft)</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> = 55 (months) <strong><em>(A1)</em>(ft)</strong><strong><em>(G4)</em></strong></p>
<p> </p>
<p><em><strong>METHOD 3</strong></em></p>
<p><em><strong><img src="data:image/png;base64,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"> (M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for each graph drawn.</p>
<p>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> =) 4.52157… (years) <strong><em>(A1)</em>(ft)</strong></p>
<p>4.52157… × 12 (= 54.2588…) <strong><em>(M1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for multiplying <strong>their</strong> value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> by 12. This may be implied.</p>
<p> If the graphs drawn are in terms of months, leading to a value of 54.2588…, award <em><strong>(M1)(M1)(M1)(A1)</strong></em>, consistent with METHOD 2.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> = 55 (months) <strong><em>(A1)</em>(ft)</strong><strong><em>(G4)</em></strong></p>
<p><strong>Note: </strong>Follow through for a compound interest formula consistent with their part (a). The final <strong><em>(A1)</em>(ft)</strong> can only be awarded for correct answer, or their correct answer following through from previous parts and only if value is rounded up. For example, do not award <strong><em>(M0)(M0)(A0)(M1)(A1)</em>(ft)</strong> for an unsupported “5 years × 12 = 60” or similar.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence has first term <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> and common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>th term of the sequence is zero, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> denote the sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the sequence.</p>
<p>Find the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>25</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempting to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p>use of a graph or a table to attempt to find the maximum sum <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>750</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong><br>EITHER</strong></p>
<p>recognizing maximum occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>25</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>25</mn></msub><mo>=</mo><mfrac><mn>25</mn><mn>2</mn></mfrac><mfenced><mrow><mn>60</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mo>,</mo><mo> </mo><msub><mi>S</mi><mn>25</mn></msub><mo>=</mo><mfrac><mn>25</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>60</mn><mo>+</mo><mn>24</mn><mo>×</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempting to calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>24</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>24</mn></msub><mo>=</mo><mfrac><mn>24</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>60</mn><mo>+</mo><mn>23</mn><mo>×</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>750</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A new café opened and during the first week their profit was $60.</p>
<p>The café’s profit increases by $10 every week.</p>
</div>
<div class="specification">
<p>A new tea-shop opened at the same time as the café. During the first week their profit was also $60.</p>
<p>The tea-shop’s profit increases by 10 % every week.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the café’s <strong>total</strong> profit for the first 12 weeks.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the tea-shop’s <strong>total</strong> profit for the first 12 weeks.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{12}}{2}\left( {2 \times 60 + 11 \times 10} \right)">
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>60</mn>
<mo>+</mo>
<mn>11</mn>
<mo>×</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em><strong>(ft)</strong></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting the arithmetic series formula, <em><strong>(A1)</strong></em><strong>(ft)</strong> for correct substitution. Follow through from their first term and common difference in part (a).</p>
<p>= ($) 1380 <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(G2)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{60\left( {{{1.1}^{12}} - 1} \right)}}{{1.1 - 1}}">
<mfrac>
<mrow>
<mn>60</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mn>1.1</mn>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1.1</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)(A1)</em>(ft)</strong></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting the geometric series formula, <strong><em>(A1)</em>(ft)</strong> for correct substitution. Follow through from part (c) for their first term and common ratio.</p>
<p>= ($)1280 (1283.05…) <strong><em>(A1)</em>(ft)<em>(G2)</em></strong></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, give all answers correct to two decimal places.</strong></p>
<p>Sam invests <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>1700</mn></math> in a savings account that pays a nominal annual rate of interest of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>74</mn><mo>%</mo></math>, compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account.</p>
</div>
<div class="specification">
<p>David also invests <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>1700</mn></math> in a savings account that pays an annual rate of interest of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>%</mo></math>, compounded yearly. David makes no further payments or withdrawals from this account.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount that Sam will have in his account after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> required so that the amount in David’s account after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years will be equal to the amount in Sam’s account.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the interest David will earn over the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> The first time an answer is not given to two decimal places, the final <em><strong>A1</strong></em> in that part is not awarded.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>20</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>2</mn><mo>.</mo><mn>74</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>2</mn><mo>.</mo><mn>74</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mfenced><mo>∓</mo></mfenced><mn>1700</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mfenced><mo>∓</mo></mfenced><mn>1700</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> <strong> <em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt to use a financial app in their technology with at least two entries seen, and award <em><strong>(A1)</strong></em> for all entries correct. Accept a positive or negative value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1700</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0274</mn></mrow><mn>2</mn></mfrac></mrow></mfenced><mrow><mn>2</mn><mo>×</mo><mn>10</mn></mrow></msup></math> <strong> <em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution into compound interest formula.<br>Award <em><strong>(A1)</strong></em> for correct substitution.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>2231</mn><mo>.</mo><mn>71</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> The first time an answer is not given to two decimal places, the final <em><strong>A1</strong></em> in that part is not awarded.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mo>∓</mo><mn>1700</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mi>V</mi><mo>=</mo><mo>±</mo><mn>2231</mn><mo>.</mo><mn>71</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math> <strong> <em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt to use a financial app in their technology with at least two entries seen.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1700</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>r</mi><mn>100</mn></mfrac></mrow></mfenced><mn>10</mn></msup><mo>=</mo><mn>2231</mn><mo>.</mo><mn>71</mn><mo>…</mo></math> <strong> <em>(M1)</em></strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>75876</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>76</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Ignore omission of opposite signs for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mi>V</mi></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>76</mn></math> is obtained.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> The first time an answer is not given to two decimal places, the final <em><strong>A1</strong></em> in that part is not awarded.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>531</mn><mo>.</mo><mn>71</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was the only short question which was not common to both HL and SL papers.</p>
<p>Candidates found this question a straightforward start, with the majority being able to attempt all three parts. While most candidates initially gave answers to two decimal places as required, many subsequently rounded their answers to three significant figures. In parts (a) and (b), most used the compound interest formula rather than the finance app on their GDC. Difficulty using the formula arose in determining the value of <em>r</em> (0.0274 was often seen, rather than 2.74) and the value of <em>k</em>. Many candidates struggled to correctly interpret 'half-yearly', and attempted to use either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>6</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the formula.</p>
<p>In part (b), a correct equation was usually seen, but lengthy analytical methods to solving it were favoured, with varying degrees of success, over an approach using the GDC. A common error was an attempt to use logarithms rather than the tenth root. It was not uncommon to see a final answer of either 2.76% or 0.0276, rather than 2.76. Many did not know what was meant by 'interest' in part (c). Those that did often recalculated David's amount using their value of <em>r</em>, rather than subtract $1700 from Sam's amount.</p>
<p>The candidates who attempted to use a finance app on their GDC in this question were generally able to give accurate and concise answers. In the case where they obtained an incorrect answer, most gave sufficient detail of the values they were using in the app, to be awarded the method mark. The most common error seen was the use of an incorrect value for the number of payments per year or the number of compounding periods per year.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows values of ln <em>x</em> and ln <em>y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between ln <em>x</em> and ln <em>y</em> can be modelled by the regression equation ln <em>y</em> = <em>a</em> ln <em>x</em> + <em>b</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the regression equation to estimate the value of <em>y</em> when<em> x</em> = 3.57.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The relationship between <em>x</em> and <em>y</em> can be modelled using the formula <em>y</em> = <em>kx<sup>n</sup></em>, where <em>k</em> ≠ 0 , <em>n</em> ≠ 0 , <em>n</em> ≠ 1.</p>
<p>By expressing ln <em>y</em> in terms of ln <em>x</em>, find the value of <em>n</em> and of <em>k</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> one correct value</p>
<p>−0.453620, 6.14210</p>
<p><em>a</em> = −0.454, <em>b</em> = 6.14 <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong> (A1)</strong></em></p>
<p><em>eg </em>−0.454 ln 3.57 + 6.14</p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> ln <em>y</em> = 5.56484</p>
<p>261.083 (260.409 from 3 sf)</p>
<p><em>y</em> = 261, (<em>y</em> = 260 from 3sf) <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> If no working shown, award <em><strong>N1</strong></em> for 5.56484.<br>If no working shown, award <em><strong>N2</strong> </em>for ln <em>y</em> = 5.56484.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</math></span></p>
<p>correct application of addition rule for logs <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,k + {\text{ln}}\,\left( {{x^n}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct application of exponent rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,k + n\,{\text{ln}}\,x">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>+</mo>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span></p>
<p>comparing one term with regression equation (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = a,\,\,b = {\text{ln}}\,k">
<mi>n</mi>
<mo>=</mo>
<mi>a</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</math></span></p>
<p>correct working for <em>k</em> <strong>(A1)</strong></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,k = 6.14210,\,\,\,k = {e^{6.14210}}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>=</mo>
<mn>6.14210</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>6.14210</mn>
</mrow>
</msup>
</mrow>
</math></span></p>
<p>465.030</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 0.454,\,\,k = 465">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.454</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>=</mo>
<mn>465</mn>
</math></span> (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{e^{{\text{ln}}\,y}} = {e^{a\,{\text{ln}}\,x + b}}">
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
<p>correct use of exponent laws for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{e^{a\,{\text{ln}}\,x + b}}">
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{e^{a\,{\text{ln}}\,x}} \times {e^b}">
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
</math></span></p>
<p>correct application of exponent rule for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\,{\text{ln}}\,x">
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,{x^a}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
</mrow>
</math></span></p>
<p>correct equation in<em> y</em> <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {x^a} \times {e^b}">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
</math></span></p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = {e^b},\,\,n = a">
<mi>k</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>=</mo>
<mi>a</mi>
</math></span></p>
<p>465.030</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 0.454,\,\,k = 465">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.454</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>=</mo>
<mn>465</mn>
</math></span> (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
</math></span></p>
<p>correct application of exponent rule for logs (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,\left( {{x^a}} \right) + b">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>b</mi>
</math></span></p>
<p>correct working for <em>b</em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = {\text{ln}}\,\left( {{e^b}} \right)">
<mi>b</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct application of addition rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,\left( {{e^b}{x^a}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = {e^b},\,\,n = a">
<mi>k</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mi>b</mi>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>=</mo>
<mi>a</mi>
</math></span></p>
<p>465.030</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 0.454,\,\,k = 465">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.454</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>=</mo>
<mn>465</mn>
</math></span> (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of a geometric sequence is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mfenced><mfrac><mn>7</mn><mn>8</mn></mfrac></mfenced><mi>r</mi></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first term of the sequence, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>S</mi><mi>n</mi></msub><mo><</mo><mn>0</mn><mo>.</mo><mn>001</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msub><mi>S</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>×</mo><mfrac><mn>7</mn><mn>8</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>14</mn><mn>24</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>7</mn><mn>12</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>583333</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>7</mn><mn>8</mn></mfrac><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>875</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>substituting their values for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>14</mn><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>4</mn><mo>.</mo><mn>66666</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute their values into the inequality or formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>14</mn><mn>3</mn></mfrac><mo>-</mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mfenced><mfrac><mn>7</mn><mn>8</mn></mfrac></mfenced><mi>r</mi></msup><mo><</mo><mn>0</mn><mo>.</mo><mn>001</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mn>7</mn><mn>12</mn></mfrac></mstyle><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mfenced><mstyle displaystyle="true"><mfrac><mn>7</mn><mn>8</mn></mfrac></mstyle></mfenced><mi>n</mi></msup></mrow></mfenced></mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>7</mn><mn>8</mn></mfrac></mstyle></mrow></mfenced></mfrac></math></p>
<p>attempt to solve their inequality using a table, graph or logarithms</p>
<p>(must be exponential) <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M0)</strong></em> if the candidate attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>u</mi><mi>n</mi></msub><mo><</mo><mn>0</mn><mo>.</mo><mn>001</mn></math>.</p>
<p><br>correct critical value or at least one correct crossover value <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn><mo>.</mo><mn>2675</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>S</mi><mn>63</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>001036</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>S</mi><mn>64</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>000906</mn><mo>…</mo></math></p>
<p>OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>S</mi><mn>63</mn></msub><mo>-</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>=</mo><mn>0</mn><mo>.</mo><mn>0000363683</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>-</mo><msub><mi>S</mi><mn>64</mn></msub><mo>-</mo><mn>0</mn><mo>.</mo><mn>001</mn><mo>=</mo><mn>0</mn><mo>.</mo><mn>0000931777</mn><mo>…</mo></math></p>
<p>least value is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>64</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.</p>
</div>
<div class="specification">
<p>A marathon is 42.195 kilometres.</p>
<p>In the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>th training session Rosa will run further than a marathon for the first time.</p>
</div>
<div class="specification">
<p>Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each <strong>month</strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the distance Rosa runs in the third training session;</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the distance Rosa runs in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>th training session.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the total distance, in <strong>kilometres</strong>, Rosa runs in the first 50 training sessions.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance Carlos runs in the fifth month of training.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the total distance Carlos runs in the first year.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>3800 m <strong><em>(A1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 + (n - 1)400{\text{ m}}">
<mn>3000</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mn>400</mn>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2600 + 400n{\text{ m}}">
<mn>2600</mn>
<mo>+</mo>
<mn>400</mn>
<mi>n</mi>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into arithmetic sequence formula, <strong><em>(A1) </em></strong>for correct substitution.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 + (k - 1)400 > 42195">
<mn>3000</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mn>400</mn>
<mo>></mo>
<mn>42195</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>(M1) </em></strong>for their correct inequality. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + (k - 1)0.4 > 42.195">
<mn>3</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mn>0.4</mn>
<mo>></mo>
<mn>42.195</mn>
</math></span>.</p>
<p>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong>OR </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \geqslant ">
<mo>⩾</mo>
</math></span>. Award <strong><em>(M0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3000 + (k - 1)400 > 42.195">
<mn>3000</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mn>400</mn>
<mo>></mo>
<mn>42.195</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(k = ){\text{ }}99">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>99</mn>
</math></span> <strong><em>(A1)</em>(ft)<em>(G2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a)(ii), but only if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is a positive integer.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{50}}{2}\left( {2 \times 3000 + (50 - 1)(400)} \right)">
<mfrac>
<mrow>
<mn>50</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>3000</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>50</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>400</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into sum of an arithmetic series formula, <strong><em>(A1)</em>(ft) </strong>for correct substitution.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="640\,000{\text{ m}}">
<mn>640</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="640\,000">
<mn>640</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> seen.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 640{\text{ km}}">
<mo>=</mo>
<mn>640</mn>
<mrow>
<mtext> km</mtext>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)<em>(G3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1)</em>(ft) </strong>for correctly converting their answer in metres to km; this can be awarded independently from previous marks.</p>
<p> </p>
<p><strong><em>OR</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{50}}{2}\left( {2 \times 3 + (50 - 1)(0.4)} \right)">
<mfrac>
<mrow>
<mn>50</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>50</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>0.4</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)(A1)</em>(ft)<em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into sum of an arithmetic series formula, <strong><em>(A1)</em>(ft) </strong>for correct substitution, <strong><em>(A1) </em></strong>for correctly converting 3000 m and 400 m into km.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 640{\text{ km}}">
<mo>=</mo>
<mn>640</mn>
<mrow>
<mtext> km</mtext>
</mrow>
</math></span> <strong><em>(A1)(G3)</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7500 \times {1.2^{5 - 1}}">
<mn>7500</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>1.2</mn>
<mrow>
<mn>5</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into geometric series formula, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 15\,600{\text{ m }}(15\,552{\text{ m}})">
<mo>=</mo>
<mn>15</mn>
<mspace width="thinmathspace"></mspace>
<mn>600</mn>
<mrow>
<mtext> m </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mspace width="thinmathspace"></mspace>
<mn>552</mn>
<mrow>
<mtext> m</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)(G3)</em></strong></p>
<p><strong><em>OR</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7.5 \times {1.2^{5 - 1}}">
<mn>7.5</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>1.2</mn>
<mrow>
<mn>5</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into geometric series formula, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 15.6{\text{ km}}">
<mo>=</mo>
<mn>15.6</mn>
<mrow>
<mtext> km</mtext>
</mrow>
</math></span> <strong><em>(A1)(G3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7500({{1.2}^{12}} - 1)}}{{1.2 - 1}}">
<mfrac>
<mrow>
<mn>7500</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mn>1.2</mn>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>1.2</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>(M1) </em></strong>for substitution into sum of a geometric series formula, <strong><em>(A1) </em></strong>for correct substitutions. Follow through from their ratio (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>) in part (d). If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r < 1">
<mi>r</mi>
<mo><</mo>
<mn>1</mn>
</math></span> (distance does not increase) or the final answer is unrealistic (<em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 20">
<mi>r</mi>
<mo>=</mo>
<mn>20</mn>
</math></span>), do not award the final <em><strong>(A1)</strong></em>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 297\,000{\text{ m }}(296\,853 \ldots {\text{ m}},{\text{ }}297{\text{ km}})">
<mo>=</mo>
<mn>297</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mrow>
<mtext> m </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>296</mn>
<mspace width="thinmathspace"></mspace>
<mn>853</mn>
<mo>…</mo>
<mrow>
<mtext> m</mtext>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>297</mn>
<mrow>
<mtext> km</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)(G2)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Helen and Jane both commence new jobs each starting on an annual salary of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>70</mn><mo>,</mo><mn>000</mn></math>. At the start of each new year, Helen receives an annual salary increase of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>2400</mn></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><msub><mi>H</mi><mi>n</mi></msub></math> represent Helen’s annual salary at the start of her <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th year of employment.</p>
</div>
<div class="specification">
<p>At the start of each new year, Jane receives an annual salary increase of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>%</mo></math> of her previous year’s annual salary.</p>
<p>Jane’s annual salary, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><msub><mi>J</mi><mi>n</mi></msub></math>, at the start of her <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th year of employment is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mi>n</mi></msub><mo>=</mo><mn>70</mn><mo> </mo><mn>000</mn><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>03</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>At the start of year <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>, Jane’s annual salary exceeds Helen’s annual salary for the first time.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mi>n</mi></msub><mo>=</mo><mn>2400</mn><mi>n</mi><mo>+</mo><mn>67</mn><mo> </mo><mn>600</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mi>n</mi></msub></math> follows a geometric sequence, state the value of the common ratio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find Jane’s total earnings at the start of her <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>th year of employment. Give your answer correct to the nearest dollar.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mi>n</mi></msub><mo>=</mo><msub><mi>H</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>1</mn></msub><mo>=</mo><mn>70</mn><mo> </mo><mn>000</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>2400</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mi>n</mi></msub><mo>=</mo><mn>70</mn><mo> </mo><mn>000</mn><mo>+</mo><mn>2400</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mi>n</mi></msub><mo>=</mo><mn>2400</mn><mi>n</mi><mo>+</mo><mn>67</mn><mo> </mo><mn>600</mn></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[2</strong><strong> marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>03</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1</strong><strong> mark]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of use of an appropriate table or graph or GDC numerical solve feature to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mi>n</mi></msub><mo>></mo><msub><mi>H</mi><mi>n</mi></msub></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>for example, an excerpt from an appropriate table</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAABACAYAAAB7qyKHAAAOXElEQVR4Ae2dX4gd1R3H5+HCPgR8CNyHC3koBCIsNg/bSgn4cOVCLBYqShdc2iwhUtuGagiaYEtksWJiLREUdivRUGllg2u2LYjVTW4Q6p8sER9SYrbCQsX9Q4kt7uYGrCzDt8yfM3fmzO/Mnbvun5k530C4M3POzJzzPZ893zPnnJnjgP+oABWgAlSACpRQAaeEaWaSqQAVoAJUgAqABkYIqAAVoAJUoJQK0MBKWWxMNBWgAlSACiQMzHESu1SHCvRUgMz0lIgRBAXIjSAKD2UqIDGTcCwvAv9TAzJABsgAGSgiA7rDpQxMj8B9KpClgAc5/1GBfhUgN/0qxvgSM4naR4pA2ahAlgJkJksdhpkUIDcmZXjcpIDEDA3MpBaP51JAgirXiYxktQLkxuriX1fmJWZoYOuSkicpBSSoVBh/qYBJAXJjUobHTQpIzNDATGrxeC4FJKhynchIVitAbqwu/nVlXmKGBrYuKXmSUkCCSoXxlwqYFCA3JmV43KSAxAwNzKQWj+dSQIIq14mViPQ/LL8+lH71ZHQa17z8rYzjeEObjt18HlMdtxK5/yaZsJsbSblVfPrSbo2l76A584UU2cpjEjM0MCtR2LhMS1Bt3NXLcSV37jD2e+9QiuY0j0uP3QbH+QGGZ78sR4a2IJXkRhbZXRjDoYaD2qFpXGU7JyGSxAwNLCGRt6O3hIZxeO6rRKyowope/La3pSRBlRDLgp2AhwYaZ+awlsrvxzj/4wE4+8/iXVZIkTrkJpIisRGwlK5zEpEs3ZGYoYEZYVjD6oUWGo6hYnLbeOXuAdQfn8E1iysmCSqjpJUMUA0eQ6WzMo5jtZrB3CopSK5MkRtJppAlNnYkcfzuVT2ABqYrEu3P4+Iv9uLBH+2Qu4a8imngCYyvWOxegAhVJKENG24bZ1s1wxNWD3OzQR9DHmlggjAhS7Vjl7AsBNt+SGKGBmaiYmUcrdbb+Nx/CtO7CIOnswGCZr2BZXYfKnNr/Mr6ho7+ZyZVRnoc2/YDlvS6xjYVzPmVmKGBiXqtYXXmfrQufAGIraJ5tB/ZxRlC1j+BqScsbaZhNDYaHGeLOv1HJlVG6Vg2HQlZYmPHWOgSMzQwUa5VzD39s7DVLIDF7sNINQmqKLDqG5lPWMrc2KKWMLCaG0kQsaEsRbT3mMQMDUziwW3j9MiFqB862U3E7sO4ZBJU8fAqbwdcOBCfsDLNrcqq5MubzdxICrH7UFIleUxihgaW1Mjfc68/gZG3b3RD3CuYHt0RDtT/m92HXWUsHgPLfsLKNLeYfrZuSpWRrVpEr+6w+zATAYkZGlhKsnj3oQpUU+qHcfj9Sc4+VLLYPAaW+YSVbW4x+azdlCojO8RYwEfP7ErMWnUXX8AT+wZQPzWLWLPZDjn6yKXEDA1MF9BtY2LoTPql0/B9np17doKzD7uiSVB1Q6u6dQtLM/ei6U3WUJ+NimXVr5DuqsFxDO+GxeLaumknN/AnhXnvj3ovvbu4haUPj+LYXTXUDryK9k27X8np9bcgMUMD01Rz5x7F0IT0RQX1SSAOysclk6CKh1due20Kv9utzzpUTNzAR6fqfreqp4v6L46RVU6Y/jJkHTeRPGvoXB/zTcvnozGK4T/NWv0xhEiaHhsSMzSwuGjuVVz85e3G6fHeuEarxpeX45JJUMXDua0roEyuieF3PsQbRxpwLBz7IDc6F8K+aiztPoW/XA1Mz+bGkMQMDcznRo1Z9Gg1a7MTBeSsOyRBZZ0I/WZ4aQyjThODD5/Gm28ftLJRRG7yQBOudtAcRfPJP6M9sVue8ZrnUhWIIzFDA6tAwW5nFiSotjM9xb+3mhDkfZ3+P/73Nm0cUyU3eUgNhy38VQ6WrZ/9LDFDA8vDEeMYFZCgMkZmQLTagdcV9F/Mo320jsGpz61ThtzkKHK3jZe/uzMY0vAnkT2MsaX0egc5rlSJKBIzNLBKFO32ZUKCavtSU4Y7f4zzD9SDSikc47BxXIPc5GB1aQwHwjH3tdlh7HbUZKEc51YwisQMDayCBb2VWZKg2sr7817lVIDclLPctjPVEjM0sO0skQrcW4KqAtliFjZZAXKzyQJX8PISMzSwChb0VmZJgmor7897lVMBclPOctvOVEvM0MC2s0QqcG8Jqgpki1nYZAXIzSYLXMHLS8zQwCpY0FuZJQmqrbw/71VOBchNOcttO1MtMZMyMC8S/1MDMkAGyAAZKBoDuoGmDEyPwH0qkKWABzj/UYF+FSA3/SrG+BIzidpHikDZqECWAmQmSx2GmRQgNyZleNykgMQMDcykFo/nUkCCKteJjGS1AuTG6uJfV+YlZmhg65KSJykFJKhUGH+pgEkBcmNShsdNCkjM0MBMavF4LgUkqHKdyEhWK0BurC7+dWVeYoYGti4peZJSQIJKhRX+172GKxPNYGVl5w4Mnnyvx8KCt7D4+p2otc6mV+wWMusuTuKst96X00D96KuYWvg6HSuehuZxPHp5CanPteaJk75yoY+UmZtuuTpwGqMY+WA5XWYx9d2FMRyqPYDDc1/Fjpo2u6s0O85+NCcua0yqhXWFGZL7TVz2x60pZdt9XGKGBrbdpVLy+0tQlSNLC7jym9u7ptV5Cy8/eBvqp2Zxw5ABvyJqOHCMFUXsxM5beKm1G60Li8HS8TP3ol5/DlOd+LLxy5ibGELjsQt+JeUuvoDH600Mz34Zu1CeOLHoJdksLTedSZxsDIemtYbOJ4/iQO37WpnFCsG9gunRHXCc4RwGFlzvvsavMb74NeBeRfvoriSTbhtnWzXhVacGGmekleSBvriNJb1omxIzNLB4KbnX8MHECZw50shYOO4Wlt57DudebOIurs7s/yHFJSzLtrS6dnZLed5fj2nfvoEcBha0eJPrfM2jfaSBvVOfRa11734HB+IrfAdrhQ0cPI+roc/liVMWzePplCqjeHgxt4OFb5OrB2Q93axhdeYe7NmzE408BuZewRvDjeSK8N4yKgM/x1j49O5efxojf/xEa2TN4+LBIYNB9sNtMVVXqZKYoYEpdTCPiw/diSOHvS4fx2Bga1h554fY+9B9QbeThUvBR3KFGxJUepzi7YcrcOtPUm4br9w9ILRk19CZ/QkGn5rG9GO39TYw8Tr6PfX9QCXPWPdHlV2eOMVTN0+KSsmNWK5Assxiue9M4qnBk/jbuWYuAxOvk7jnGm5e+0wzLwAr43hk1xmhW7tPbmNJL+KmxAwNTC+plXEcb5gMTEUO+6FpYCV9AgvLTzcwBMeTLWwAbhsTe5/FVGcZl/IYmL/44LeTLWkET1e16KndcK/EuXniKCbL9StVRoXPgV82tXQDJ1FmKher+PRFrzs4WHW79xOYzoe6joEBFexxNXMPdkndh/1yG12zmBsSMzQwvaxoYLoimfsSVJknFCIwfLKJnnRUooLKIjnGdQuLr92PEX9cSgpX53Z/g5a0vvhgUEFFFVk4lpEyy1gl6eaIk5rw0U1GobdKyY0af9IbPrEyU+XhLjyN1pMf4gZcrF5o5XgCC5lMNYp7MWfqPuyf20IDA4iNZRqYXmo0MF2RzP1SVkRejlQ5H3gV7ZvegFN39lfcVLwxqKGfXsSyr0KvysSLpBlVpJ46HhqbUOn5UVW6jl3CSo44Qbqim5Rmo5zcqDK8A4O//0fQlRfNEI01WNwreH3oOMZXPK7UOb0mcZjYCo+njC0sakP3Yf/cFh8diRkamF5usQrEXDn0gEq/ZoX3JajKkV1vxtfzePZBb4aYg9qB03jj7ydwrBbv+lvA7NH7owF0hF2MySe0dG7FsQy9IkuMbcSuETMt7wlMHJOLxVEt/tgVSrFZXm6W8c83RzDqzUb1Xr146k1cPtdEt2vYG3d6CK1z/won6+Q1MHm8M5s5U/fh+rgtOjgSMzQwvdRoYLoimfsSVJknFDYwnE0WjVHpFZGXcFMrWcuUbzBxI/TCg4qsW9EF14o/7flXSZybJ45275LsVoabcJp8VI6dSTzZmopmkapyj7qOjeWj86EiGhjwg6Xuw2/ArbplQX8lZmhgemHRwHRFMvclqDJPKGpgZxK/3RefgRialXF5IfN7N96kj/STk97C1vcDYZJPb3niFFXQ7HRVgxvPLEbQjMZS1dOW8JKx4kgfP4vJlCz7MEBkKQwTuw+/AbextBRxU2KGBqaXFA1MVyRzX4Iq84QiBvovMe9A7dB0rOUsJTSsHDIqoeAsvgcmqRc/Vn5u1EvMexLv9sXzGGwrU+s1BubNdu39Hlj3+qbuw26M7lZebrtnFHFLYoYGppcUDUxXJHNfgirzhEIFLmPurcM4tm8A9aPn8a4/mSMrgX1UBJ1JnKp/r8eXOBbw0TPfQv3xmYwvceSJk5XmYoaVmpub72LmRe8TZPvRPDeXfi8rIXkfBobgia5eP2H+Ekd0ban7MArUNvrgVjuzSLsSMzQwvYRoYLoimfsSVJknFCJQVSre5I3ncOKv13tUQirR/VUE7uIfcNqfJJL9LcTLJwfR8LqYMr6F2DOOSmJJfsvJTVj+/uSN1zB+bSWH2oq1HE9g/tVuYen9g+EkEelbiOEtvbHSxFdcspLSH7dZV9rOMIkZGlhUIuF4g+qr9n/T0AX91PE+7oyxkOja1d2QoKpubpmzjVKA3GyUkvZcR2KGBmZP+W9KTiWoNuVGvGilFCA3lSrOLcmMxAwNbEukr+5NJKiqm1vmbKMUIDcbpaQ915GYoYHZU/6bklMJqk25ES9aKQXITaWKc0syIzFDA9sS6at7Ewmq6uaWOdsoBcjNRilpz3UkZlIG5kXif2pABsgAGSADRWNAt+uEgemB3KcCVIAKUAEqUFQFaGBFLRmmiwpQASpABTIVoIFlysNAKkAFqAAVKKoCNLCilgzTRQWoABWgApkK/B9EOunWCkgtgAAAAABJRU5ErkJggg=="> <strong>(A1)</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>for example, use of a GDC numerical solve feature to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>800</mn><mo>…</mo></math> <strong> (A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for an appropriate graph. Condone use of a continuous graph.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>11</mn></math> <strong> A1</strong></p>
<p> </p>
<p><strong>[3</strong><strong> marks]</strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>11</mn></msub><mo>=</mo><mn>94</mn><mo> </mo><mn>000</mn><mo> </mo><mfenced><mo>$</mo></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mn>11</mn></msub><mo>=</mo><mn>94</mn><mo> </mo><mn>074</mn><mo> </mo><mfenced><mo>$</mo></mfenced></math> <strong>A1</strong></p>
<p>Helen’s annual salary is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>94</mn><mo> </mo><mn>000</mn></math> and Jane’s annual salary is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>94</mn><mo> </mo><mn>074</mn></math></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mn>11</mn></msub></math> value and <strong>A1</strong> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mn>11</mn></msub></math> value seen in part (c) (i).</p>
<p> </p>
<p><strong>[2</strong><strong> marks]</strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at the start of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>th year, Jane will have worked for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> years so the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>9</mn></msub></math> is required <strong>R1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>R1</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>9</mn></msub></math> is seen anywhere.</p>
<p> </p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>J</mi><mn>1</mn></msub><mfenced><mrow><msup><mi>r</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>J</mi><mn>1</mn></msub><mo>=</mo><mn>70</mn><mo> </mo><mn>000</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>03</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>10</mn></math> is used.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>9</mn></msub><mo>=</mo><mfrac><mrow><mn>70</mn><mo> </mo><mn>000</mn><mfenced><mrow><msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>03</mn></mrow></mfenced><mn>9</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo>.</mo><mn>03</mn><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>711</mn><mo> </mo><mn>137</mn><mo>.</mo><mn>42</mn><mo>…</mo></math> <strong>(A1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>711</mn><mo> </mo><mn>137</mn><mo> </mo><mfenced><mo>$</mo></mfenced></math></p>
<p>Jane’s total earnings are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>711</mn><mo> </mo><mn>137</mn></math> (correct to the nearest dollar)</p>
<p> </p>
<p><strong>[4</strong><strong> marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>An infinite geometric series has first term <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mi>a</mi></math> and second term <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>a</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> for which the sum to infinity of the series exists.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>76</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of dividing terms (in any order) <em><strong>(M1)</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>u</mi><mn>1</mn></msub><msub><mi>u</mi><mn>2</mn></msub></mfrac><mo>,</mo><mo> </mo><mfrac><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>a</mi></mstyle><mi>a</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mo>-</mo><mn>3</mn></math> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo> </mo><mi>r</mi><mo> </mo></mrow></mfenced><mo><</mo><mn>1</mn></math> (must be in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>) <em><strong>(M1)</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo><</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><mo>-</mo><mn>1</mn><mo>≤</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mo>-</mo><mn>3</mn><mo>≤</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><mo>-</mo><mn>4</mn><mo><</mo><mi>a</mi><mo>-</mo><mn>12</mn><mo><</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo><</mo><mi>a</mi><mo><</mo><mn>16</mn></math> <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct equation <em><strong>(A1)</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mn>76</mn><mo> </mo><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>76</mn><mfenced><mrow><mn>4</mn><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>76</mn><mn>5</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>15</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math> (exact) <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {({x^2} + 3)^7}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>7</mn>
</msup>
</mrow>
</math></span>. Find the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5}">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</math></span> in the expansion of the derivative, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1 </strong></p>
<p>derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A2</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7{({x^2} + 3)^6}(x2)">
<mn>7</mn>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>6</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p>recognizing need to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({x^2} + 3)^6}">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>6</mn>
</msup>
</mrow>
</math></span> (seen anywhere) <strong><em>R1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="14x{\text{ (term in }}{x^4})">
<mn>14</mn>
<mi>x</mi>
<mrow>
<mtext> (term in </mtext>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p>valid approach to find the terms in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({x^2} + 3)^6}">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>6</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ r \end{array}} \right){({x^2})^{6 - r}}{(3)^r},{\text{ }}{({x^2})^6}{(3)^0} + {({x^2})^5}{(3)^1} + \ldots ">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>r</mi>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>6</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>0</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>5</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>1</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span>, Pascal’s triangle to 6th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{5th term, }}r = 2,{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right),{\text{ }}{({x^2})^2}{(3)^4}">
<mrow>
<mtext>5th term, </mtext>
</mrow>
<mi>r</mi>
<mo>=</mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
</math></span></p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right){({x^2})^2}{(3)^4},{\text{ }}15 \times {3^4},{\text{ }}14x \times 15 \times 81{({x^2})^2}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>15</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>3</mn>
<mn>4</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>14</mn>
<mi>x</mi>
<mo>×</mo>
<mn>15</mn>
<mo>×</mo>
<mn>81</mn>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="17010{x^5}">
<mn>17010</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>recognition of need to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^6}">
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
</math></span> in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({x^2} + 3)^7}">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>7</mn>
</msup>
</mrow>
</math></span> (seen anywhere) <strong><em>R1 </em></strong></p>
<p>valid approach to find the terms in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({x^2} + 3)^7}">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>7</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7 \\ r \end{array}} \right){({x^2})^{7 - r}}{(3)^r},{\text{ }}{({x^2})^7}{(3)^0} + {({x^2})^6}{(3)^1} + \ldots ">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mn>7</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>r</mi>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>7</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>0</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>6</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>1</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span>, Pascal’s triangle to 7th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>6th term, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 3,{\text{ }}\left( {\begin{array}{*{20}{c}} 7 \\ 3 \end{array}} \right),{\text{ (}}{{\text{x}}^2}{)^3}{(3)^4}">
<mi>r</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> (</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>x</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
</math></span></p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7 \\ 4 \end{array}} \right){{\text{(}}{{\text{x}}^2})^3}{(3)^4},{\text{ }}35 \times {3^4}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mtext>(</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>x</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>4</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>35</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>3</mn>
<mn>4</mn>
</msup>
</mrow>
</math></span></p>
<p>correct term <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2835{x^6}">
<mn>2835</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
</math></span></p>
<p>differentiating their term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^6}">
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(2835{x^6})',{\text{ (6)(2835}}{{\text{x}}^5})">
<mo stretchy="false">(</mo>
<mn>2835</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>6</mn>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> (6)(2835</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>x</mtext>
</mrow>
<mn>5</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="17010{x^5}">
<mn>17010</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, for <em>x</em> > 0.</p>
<p>The <em>k </em>th maximum point on the graph of <em>f</em> has <em>x</em>-coordinate <em>x<sub>k</sub></em> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>x<sub>k</sub></em><sub> + 1</sub> = <em>x<sub>k</sub></em> + <em>a</em>, find <em>a</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <em>n</em> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 1}^n {{x_k} = 861} "> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mrow> <msub> <mi>x</mi> <mi>k</mi> </msub> </mrow> <mo>=</mo> <mn>861</mn> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>valid approach to find maxima <em><strong>(M1)</strong></em></p>
<p><em>eg</em> one correct value of <em>x<sub>k</sub></em>, sketch of <em>f</em></p>
<p>any two correct consecutive values of <em>x<sub>k</sub></em> <em><strong>(A1)(A1)</strong></em></p>
<p><em>eg x</em><sub>1</sub> = 1, <em>x</em><sub>2</sub> = 5</p>
<p><em>a</em> = 4 <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing the sequence <em>x</em><sub>1,<sup> </sup></sub> <em>x</em><sub>2</sub><sub>,<sup> </sup></sub> <em>x</em><sub>3, …,</sub> <em>x</em><sub>n</sub> is arithmetic <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em>d</em> = 4</p>
<p>correct expression for sum<em> <strong>(A1)</strong><br></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{n}{2}\left( {2\left( 1 \right) + 4\left( {n - 1} \right)} \right)"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>valid attempt to solve for <em>n</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> graph, 2<em>n</em><sup>2</sup> − <em>n</em> − 861 = 0</p>
<p><em>n</em> = 21 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The first terms of an infinite geometric sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>, are 2, 6, 18, 54, …</p>
<p>The first terms of a second infinite geometric sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{v_n}">
<mrow>
<msub>
<mi>v</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>, are 2, −6, 18, −54, …</p>
<p>The terms of a third sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_n}">
<mrow>
<msub>
<mi>w</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>, are defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_n} = {u_n} + {v_n}">
<mrow>
<msub>
<mi>w</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>v</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The finite series, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 1}^{225} {{w_k}} ">
<munderover>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>225</mn>
</mrow>
</munderover>
<mrow>
<mrow>
<msub>
<mi>w</mi>
<mi>k</mi>
</msub>
</mrow>
</mrow>
</math></span> , can also be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 0}^m {4{r^k}} ">
<munderover>
<mo movablelimits="false">∑<!-- ∑ --></mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>m</mi>
</munderover>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three <strong>non-zero</strong> terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_n}"> <mrow> <msub> <mi>w</mi> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to add corresponding terms <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + 2{\text{,}}\,\,6 + \left( { - 6} \right){\text{,}}\,\,2{\left( 3 \right)^{n - 1}} + \,2{\left( { - 3} \right)^{n - 1}}"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>6</mn> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span></p>
<p>correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_5}"> <mrow> <msub> <mi>w</mi> <mn>5</mn> </msub> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 324</p>
<p>4, 36, 324 (accept 4 + 36 + 324) <em><strong> A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 \times {r^1} = 36"> <mn>4</mn> <mo>×</mo> <mrow> <msup> <mi>r</mi> <mn>1</mn> </msup> </mrow> <mo>=</mo> <mn>36</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 \times {9^{n - 1}}"> <mn>4</mn> <mo>×</mo> <mrow> <msup> <mn>9</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 9"> <mi>r</mi> <mo>=</mo> <mn>9</mn> </math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 0}^m {4 \times {9^k}} "> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mn>4</mn> <mo>×</mo> <mrow> <msup> <mn>9</mn> <mi>k</mi> </msup> </mrow> </mrow> </math></span>; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m"> <mi>m</mi> </math></span> may be incorrect) <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that 225 terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_n}">
<mrow>
<msub>
<mi>w</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> consists of 113 non-zero terms <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_1^{113} {} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mn>1</mn>
<mrow>
<mn>113</mn>
</mrow>
</munderover>
<mrow>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_0^{112} {} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mn>0</mn>
<mrow>
<mn>112</mn>
</mrow>
</munderover>
<mrow>
</mrow>
</math></span>, 113</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 112">
<mi>m</mi>
<mo>=</mo>
<mn>112</mn>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{k = 0}^112 {4 \times {r^k}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mn>1</mn>
</munderover>
<mn>12</mn>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>k</mi>
</msup>
</mrow>
</mrow>
</math></span>; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> may be incorrect) <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></mfenced><mn>9</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The coefficient of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>6</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6048</mn></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid approach for expansion (must have correct substitution for parameters, but accept an incorrect value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>). <strong><em>(M1)</em></strong></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable></mfenced><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mn>9</mn><mo>-</mo><mi>r</mi></mrow></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></mfenced><mi>r</mi></msup><mo> </mo><mo>,</mo><mo> </mo><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>9</mn></msup><mo>+</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>8</mn></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></mfenced><mn>1</mn></msup><mo>+</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>7</mn></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p>valid attempt to identify correct term <strong><em>(M1)</em></strong></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mn>9</mn><mo>-</mo><mi>r</mi></mrow></mfenced><mo>-</mo><mi>r</mi><mo>=</mo><mn>6</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mi>r</mi></msup><msup><mfenced><msup><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mrow><mn>9</mn><mo>-</mo><mi>r</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mn>6</mn></msup></math></p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>5</mn></math></p>
<p>correct term or coefficient in binominal expansion <strong><em>(A1)</em></strong></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>5</mn></msup><msup><mfenced><mrow><mo>-</mo><mfrac><mi>k</mi><mi>x</mi></mfrac></mrow></mfenced><mn>4</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mn>126</mn><mfenced><mrow><mn>243</mn><msup><mi>x</mi><mn>10</mn></msup></mrow></mfenced><mfenced><mfrac><msup><mi>k</mi><mn>4</mn></msup><msup><mi>x</mi><mn>4</mn></msup></mfrac></mfenced><mo>,</mo><mo> </mo><mn>30618</mn><msup><mi>k</mi><mn>4</mn></msup></math></p>
<p>correct equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> <strong><em>(A1)</em></strong></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mfenced><mn>243</mn></mfenced><mfenced><msup><mi>k</mi><mn>4</mn></msup></mfenced><msup><mi>x</mi><mn>6</mn></msup><mo>=</mo><mn>6048</mn><msup><mi>x</mi><mn>6</mn></msup><mo> </mo><mo>,</mo><mo> </mo><mn>30618</mn><msup><mi>k</mi><mn>4</mn></msup><mo>=</mo><mn>6048</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> (exact) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>667</mn></math> <em><strong>A1 N3</strong></em> </p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if additional answers given.</p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first 8 terms.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <em>n</em> for which <em>S</em><sub><em>n</em></sub> > 163.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>correct substitution into infinite sum <em><strong>(A1)</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="200 = \frac{4}{{1 - r}}"> <mn>200</mn> <mo>=</mo> <mfrac> <mn>4</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </math></span></p>
<p><em>r </em>= 0.98 (exact) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{4\left( {1 - {{0.98}^8}} \right)}}{{1 - 0.98}}"> <mfrac> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mn>0.98</mn> </mrow> <mn>8</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.98</mn> </mrow> </mfrac> </math></span></p>
<p>29.8473</p>
<p>29.8 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to set up inequality (accept equation) <em><strong>(M1)</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{4\left( {1 - {{0.98}^n}} \right)}}{{1 - 0.98}} > 163,\,\,\frac{{4\left( {1 - {{0.98}^n}} \right)}}{{1 - 0.98}} = 163"> <mfrac> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mn>0.98</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.98</mn> </mrow> </mfrac> <mo>></mo> <mn>163</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mn>0.98</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.98</mn> </mrow> </mfrac> <mo>=</mo> <mn>163</mn> </math></span></p>
<p>correct inequality for <em>n</em> (accept equation) or crossover values <em><strong>(A1)</strong></em><br><em>eg n</em> > 83.5234, <em>n</em> = 83.5234, <em>S</em><sub><em>83</em></sub> = 162.606 <strong>and </strong><em>S</em><sub><em>84</em></sub> = 163.354</p>
<p><em>n</em> = 84 <em><strong>A1 N1</strong></em></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The first two terms of a geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 2.1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2.1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} = 2.226">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2.226</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}}"> <mrow> <msub> <mi>u</mi> <mrow> <mn>10</mn> </mrow> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_n} > 5543"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </mrow> <mo>></mo> <mn>5543</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u_1}}}{{{u_2}}}"> <mfrac> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2.226}}{{2.1}}"> <mfrac> <mrow> <mn>2.226</mn> </mrow> <mrow> <mn>2.1</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.226 = 2.1r"> <mn>2.226</mn> <mo>=</mo> <mn>2.1</mn> <mi>r</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 1.06"> <mi>r</mi> <mo>=</mo> <mn>1.06</mn> </math></span> (exact) <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.1 \times {1.06^9}"> <mn>2.1</mn> <mo>×</mo> <mrow> <msup> <mn>1.06</mn> <mn>9</mn> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3.54790"> <mn>3.54790</mn> </math></span> <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}} = 3.55"> <mrow> <msub> <mi>u</mi> <mrow> <mn>10</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>3.55</mn> </math></span></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_n}"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </mrow> </math></span> formula <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2.1\left( {{{1.06}^n} - 1} \right)}}{{1.06 - 1}}"> <mfrac> <mrow> <mn>2.1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>1.06</mn> </mrow> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1.06</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2.1\left( {{{1.06}^n} - 1} \right)}}{{1.06 - 1}} > 5543"> <mfrac> <mrow> <mn>2.1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>1.06</mn> </mrow> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1.06</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>></mo> <mn>5543</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.1\left( {{{1.06}^n} - 1} \right) = 332.58"> <mn>2.1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mn>1.06</mn> </mrow> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>332.58</mn> </math></span>, sketch of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_n}"> <mrow> <msub> <mi>S</mi> <mi>n</mi> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 5543"> <mi>y</mi> <mo>=</mo> <mn>5543</mn> </math></span></p>
<p>correct inequality for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> <strong>or</strong> crossover values <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 87.0316"> <mi>n</mi> <mo>></mo> <mn>87.0316</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{87}} = 5532.73"> <mrow> <msub> <mi>S</mi> <mrow> <mn>87</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>5532.73</mn> </math></span> <strong>and </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{88}} = 5866.79"> <mrow> <msub> <mi>S</mi> <mrow> <mn>88</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>5866.79</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 88"> <mi>n</mi> <mo>=</mo> <mn>88</mn> </math></span> <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Consider a geometric sequence where the first term is 768 and the second term is 576.</p>
<p>Find the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> such that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>th term of the sequence is less than 7.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{576}}{{768}},{\text{ }}\frac{{768}}{{576}},{\text{ }}0.75"> <mfrac> <mrow> <mn>576</mn> </mrow> <mrow> <mn>768</mn> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>768</mn> </mrow> <mrow> <mn>576</mn> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.75</mn> </math></span></p>
<p>correct expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}"> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="768{(0.75)^{n - 1}}"> <mn>768</mn> <mrow> <mo stretchy="false">(</mo> <mn>0.75</mn> <msup> <mo stretchy="false">)</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span></p>
<p><strong>EITHER (solving inequality)</strong></p>
<p>valid approach (accept equation) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} < 7"> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> </mrow> <mo><</mo> <mn>7</mn> </math></span></p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> <strong><em>M1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="768{(0.75)^{n - 1}} = 7,{\text{ }}n - 1 > {\log _{0.75}}\left( {\frac{7}{{768}}} \right)"> <mn>768</mn> <mrow> <mo stretchy="false">(</mo> <mn>0.75</mn> <msup> <mo stretchy="false">)</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>7</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>></mo> <mrow> <msub> <mi>log</mi> <mrow> <mn>0.75</mn> </mrow> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>7</mn> <mrow> <mn>768</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, sketch</p>
<p>correct value</p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 17.3301"> <mi>n</mi> <mo>=</mo> <mn>17.3301</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 18"> <mi>n</mi> <mo>=</mo> <mn>18</mn> </math></span> (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (table of values)</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} > 7"> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> </mrow> <mo>></mo> <mn>7</mn> </math></span>, one correct crossover value</p>
<p>both crossover values, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{17}} = 7.69735"> <mrow> <msub> <mi>u</mi> <mrow> <mn>17</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>7.69735</mn> </math></span> <strong>and</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{18}} = 5.77301"> <mrow> <msub> <mi>u</mi> <mrow> <mn>18</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>5.77301</mn> </math></span> <strong><em>A2</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 18"> <mi>n</mi> <mo>=</mo> <mn>18</mn> </math></span> (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (sketch of functions)</strong></p>
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>sketch of appropriate functions</p>
<p>valid approach <strong><em>(M1) </em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>finding intersections or roots (depending on function sketched)</p>
<p>correct value</p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 17.3301"> <mi>n</mi> <mo>=</mo> <mn>17.3301</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 18"> <mi>n</mi> <mo>=</mo> <mn>18</mn> </math></span> (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>On 1st January 2020, Laurie invests $<em>P</em> in an account that pays a nominal annual interest rate of 5.5 %, compounded <strong>quarterly</strong>.</p>
<p>The amount of money in Laurie’s account <strong>at the end of each year</strong> follows a geometric sequence with common ratio, <em>r</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>r</em>, giving your answer to four significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Laurie makes no further deposits to or withdrawals from the account.</p>
<p>Find the year in which the amount of money in Laurie’s account will become double the amount she invested.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + \frac{{5.5}}{{4 \times 100}}} \right)^4}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>5.5</mn>
</mrow>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>100</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong> (M1)(A1)</strong></em></p>
<p>1.056 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2P = P \times {\left( {1 + \frac{{5.5}}{{100 \times 4}}} \right)^{4n}}">
<mn>2</mn>
<mi>P</mi>
<mo>=</mo>
<mi>P</mi>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>5.5</mn>
</mrow>
<mrow>
<mn>100</mn>
<mo>×</mo>
<mn>4</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mn>4</mn>
<mi>n</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong>OR </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2P = P \times {\left( {{\text{their}}\,\left( a \right)} \right)^m}">
<mn>2</mn>
<mi>P</mi>
<mo>=</mo>
<mi>P</mi>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>their</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mi>a</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mi>m</mi>
</msup>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution into loan payment formula. Award <em><strong>(A1)</strong></em> for correct substitution.</p>
<p><strong>OR</strong></p>
<p>PV = ±1<br>FV = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \mp ">
<mo>∓</mo>
</math></span>1<br>I% = 5.5<br>P/Y = 4<br>C/Y = 4<br><em>n</em> = 50.756… <em><strong>(M1)(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p>PV = ±1<br>FV = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \mp ">
<mo>∓</mo>
</math></span>2<br>I% = 100(their (<em>a</em>) − 1)<br>P/Y = 1<br>C/Y = 1 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>⇒ 12.7 years</p>
<p>Laurie will have double the amount she invested during 2032 <em><strong>A1</strong></em><br><br></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1.3">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.3</mn>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} = 1.4">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.4</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_k} = 31.2">
<mrow>
<msub>
<mi>u</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>31.2</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the terms, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>, of this sequence such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="F">
<mi>F</mi>
</math></span> be the sum of the terms for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> is not a multiple of 3.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k}">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="F = 3240">
<mi>F</mi>
<mo>=</mo>
<mn>3240</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>An infinite geometric series is given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots ">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mfrac>
<mi>a</mi>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>a</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in {\mathbb{Z}^ + }">
<mi>a</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<p>Find the largest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } < F">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo><</mo>
<mi>F</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{300}}{2}\left( {1.3 + 31.2} \right)">
<mfrac>
<mrow>
<mn>300</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.3</mn>
<mo>+</mo>
<mn>31.2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{300}}{2}\left[ {2\left( {1.3} \right) + \left( {300 - 1} \right)\left( {0.1} \right)} \right]">
<mfrac>
<mrow>
<mn>300</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>300</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{300}}{2}\left[ {2.6 + 299\left( {0.1} \right)} \right]">
<mfrac>
<mrow>
<mn>300</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mn>2.6</mn>
<mo>+</mo>
<mn>299</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k} = 4875">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>4875</mn>
</math></span> <em><strong>A1 N2</strong></em></p>
<p style="text-align: start;"><em><strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing need to find the sequence of multiples of 3 (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em> first term is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_3}">
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> (= 1.5) (accept notation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1.5">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1.5</mn>
</math></span>) ,</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 0.1 \times 3">
<mi>d</mi>
<mo>=</mo>
<mn>0.1</mn>
<mo>×</mo>
<mn>3</mn>
</math></span> (= 0.3) , 100 terms (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 100">
<mi>n</mi>
<mo>=</mo>
<mn>100</mn>
</math></span>), last term is 31.2</p>
<p>(accept notation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{100}} = 31.2">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>100</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>31.2</mn>
</math></span>) , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_3} + {u_6} + {u_9} + \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>9</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="F = {u_3} + {u_6} + {u_9} + \ldots ">
<mi>F</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>9</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span>)</p>
<p>correct working for sum of sequence where n is a multiple of 3 <em><strong>A2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{100}}{2}\left( {1.5 + 31.2} \right)">
<mfrac>
<mrow>
<mn>100</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.5</mn>
<mo>+</mo>
<mn>31.2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="50\left( {2 \times 1.5 + 99 \times 0.3} \right)">
<mn>50</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>1.5</mn>
<mo>+</mo>
<mn>99</mn>
<mo>×</mo>
<mn>0.3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , 1635</p>
<p>valid approach (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k} - \left( {{u_3} + {u_6} + \ldots } \right)">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k} - \frac{{100}}{2}\left( {1.5 + 31.2} \right)">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>100</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.5</mn>
<mo>+</mo>
<mn>31.2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k} - ">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>−</mo>
</math></span> (their sum for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{u_3} + {u_6} + \ldots } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span>)</p>
<p>correct working (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_k} - 1635">
<mrow>
<msub>
<mi>S</mi>
<mi>k</mi>
</msub>
</mrow>
<mo>−</mo>
<mn>1635</mn>
</math></span> , 4875 − 1635</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="F = 3240">
<mi>F</mi>
<mo>=</mo>
<mn>3240</mn>
</math></span> <em><strong>AG N0</strong></em></p>
<p style="text-align: start;"><em><strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> <em><strong>(M1)</strong></em><br></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><em>eg</em> dividing consecutive terms<br></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">correct value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> (seen anywhere, including in formula)<br></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{\sqrt 2 }}">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> , 0.707106… , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{{0.293 \ldots }}">
<mfrac>
<mi>a</mi>
<mrow>
<mn>0.293</mn>
<mo>…</mo>
</mrow>
</mfrac>
</math></span></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">correct working (accept equation) <em><strong> (A1)</strong></em><br></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{{1 - \frac{1}{{\sqrt 2 }}}} < 3240">
<mfrac>
<mi>a</mi>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mfrac>
<mo><</mo>
<mn>3240</mn>
</math></span></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">correct working <em><strong>A1</strong></em><br></span></p>
<p style="text-align: start;"> </p>
<p style="text-align: start;"><span style="font-size: 10.5pt;"><strong><span style="font-family: Verdana, sans-serif;color: black;">METHOD 1 (analytical)</span></strong></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3240 \times \left( {1 - \frac{1}{{\sqrt 2 }}} \right)">
<mn>3240</mn>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a < 948.974">
<mi>a</mi>
<mo><</mo>
<mn>948.974</mn>
</math></span> , 948.974</span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><strong>METHOD 2 (using table, must find both <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty }">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
</math></span> values)</strong><br></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><em>eg</em> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 948">
<mi>a</mi>
<mo>=</mo>
<mn>948</mn>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = 3236.67 \ldots ">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3236.67</mn>
<mo>…</mo>
</math></span> <strong>AND </strong> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 949">
<mi>a</mi>
<mo>=</mo>
<mn>949</mn>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = 3240.08 \ldots ">
<mrow>
<msub>
<mi>S</mi>
<mi mathvariant="normal">∞</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3240.08</mn>
<mo>…</mo>
</math></span></span></p>
<p style="text-align: start;"><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 948">
<mi>a</mi>
<mo>=</mo>
<mn>948</mn>
</math></span> <em><strong>A1 N2</strong></em></span></p>
<p style="text-align: start;"><em><strong><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">[5 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>All answers in this question should be given to four significant figures.</strong></p>
<p><br>In a local weekly lottery, tickets cost <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>2</mn></math> each.</p>
<p>In the first week of the lottery, a player will receive <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mi>D</mi></math> for each ticket, with the probability distribution shown in the following table. For example, the probability of a player receiving <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>10</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>03</mn></math>. The grand prize in the first week of the lottery is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>1000</mn></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>If nobody wins the grand prize in the first week, the probabilities will remain the same, but the value of the grand prize will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>2000</mn></math> in the second week, and the value of the grand prize will continue to double each week until it is won. All other prize amounts will remain the same.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether this lottery is a fair game in the first week. Justify your answer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> for the value of the grand prize in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> week of the lottery.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mtext>th</mtext></math> week is the first week in which the player is expected to make a profit. Ryan knows that if he buys a lottery ticket in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mtext>th</mtext></math> week, his expected profit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mi>p</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>considering that sum of probabilities is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>85</mn><mo>+</mo><mi>c</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>03</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>002</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0001</mn><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1179</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced><mo>=</mo><mfenced><mrow><mn>0</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>85</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>1179</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>10</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>03</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>50</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>002</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1000</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>0001</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>7358</mn></math> <em><strong>A1</strong></em></p>
<p>No, not a fair game <em><strong>A1</strong></em></p>
<p>for a fair game, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced></math> would be <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>2</mn></math> OR players expected winnings are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>264</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition of GP with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn><mo>×</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>500</mn><mfenced><msup><mn>2</mn><mi>n</mi></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced><mo>></mo><mn>2</mn></math> <em><strong> (M1)</strong></em></p>
<p>correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>w</mi><mtext>th</mtext></msup></math> week (or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mtext>th</mtext></msup></math> week) <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>85</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>1179</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>10</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>03</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>50</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>002</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1000</mn><mo>×</mo><msup><mn>2</mn><mrow><mi>w</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>0001</mn></mrow></mfenced></math></p>
<p>correct inequality (accept equation) <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>6358</mn><mo>+</mo><mfenced><mrow><mn>1000</mn><mo>×</mo><msup><mn>2</mn><mrow><mi>w</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>0001</mn></mrow></mfenced><mo>></mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>></mo><mn>13</mn><mo>.</mo><mn>642</mn></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn><mo>></mo><mn>3</mn><mo>.</mo><mn>76998</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>76998</mn><mo>…</mo></math> <em><strong> (A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced><mo>=</mo><mn>1</mn><mo>.</mo><mn>4358</mn></math> in week <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mfenced><mi>D</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>2358</mn></math> in week <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> <em><strong> (A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>5</mn></math> <em><strong> A1</strong></em></p>
<p>expected profit per ticket <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>their E</mtext><mfenced><mi>D</mi></mfenced><mo>-</mo><mn>2</mn></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>2358</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Gemma and Kaia started working for different companies on January 1st 2011.</p>
<p>Gemma’s starting annual salary was <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>45</mn><mo> </mo><mn>000</mn></math>, and her annual salary increases <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>%</mo></math> on January 1st each year after 2011.</p>
</div>
<div class="specification">
<p>Kaia’s annual salary is based on a yearly performance review. Her salary for the years 2011, 2013, 2014, 2018, and 2022 is shown in the following table.</p>
<p style="text-align: left;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find Gemma’s annual salary for the year 2021, to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming Kaia’s annual salary can be approximately modelled by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, show that Kaia had a higher salary than Gemma in the year 2021, according to the model.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>using geometric sequence with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>02</mn></math> <em><strong>(M1)</strong></em></p>
<p>correct expression or listing terms correctly <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45000</mn><mo>×</mo><mn>1</mn><mo>.</mo><msup><mn>02</mn><mn>10</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45000</mn><mo>×</mo><mn>1</mn><mo>.</mo><msup><mn>02</mn><mrow><mn>11</mn><mo>-</mo><mn>1</mn></mrow></msup></math> OR listing terms</p>
<p>Gemma’s salary is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>54855</mn></math> (must be to the nearest dollar) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>PV</mtext><mo>=</mo><mo>∓</mo><mn>45000</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>I</mtext><mo>%</mo><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P/Y</mtext><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C/Y</mtext><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mi>V</mi><mo>=</mo><mo>±</mo><mn>54854</mn><mo>.</mo><mn>7489</mn><mo>…</mo></math> <em><strong>(M1)(A1)</strong></em></p>
<p>Gemma’s salary is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>54855</mn></math> (must be to the nearest dollar) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1096</mn><mo>.</mo><mn>89</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>2160753</mn><mo>.</mo><mn>8</mn><mo>…</mo></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>16</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math>) <em><strong>(A1)(A1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(A1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mn>1096</mn><mo>.</mo><mn>89</mn><mo>…</mo><mi>x</mi><mo>+</mo><mn>33028</mn><mo>.</mo><mn>49</mn><mo>…</mo></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mn>1096</mn><mo>.</mo><mn>89</mn><mo>…</mo><mi>x</mi><mo>+</mo><mn>43997</mn><mo>.</mo><mn>4</mn><mo>…</mo></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mn>1096</mn><mo>.</mo><mn>89</mn><mo>…</mo><mi>x</mi><mo>+</mo><mn>45094</mn><mo>.</mo><mn>3</mn><mo>…</mo></math></p>
<p><br>Kaia’s salary in 2021 is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>56063</mn><mo>.</mo><mn>21</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>$</mo><mn>56817</mn><mo>.</mo><mn>09</mn></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>16</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math>) <em><strong>A1</strong></em></p>
<p>Kaia had a higher salary than Gemma in 2021 <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many errors were seen in part (a). Some candidates used the incorrect formula <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>02</mn></mrow><mn>100</mn></mfrac></mrow></mfenced><mn>10</mn></msup></math> or used an incorrect value for the exponent e.g. 9 was often seen. Others lost the final mark for not answering to the nearest dollar.<br>Very few tried to make a table of values.</p>
<p>In part (b) students often let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> represent the number of years since a given year, rather than the year itself. Despite this, most were able to find the correct amount with their equation and were awarded marks as appropriate. Some students did not realise regression on GDC was expected and tried to work with a few given data points, others had difficulty dealing with the constant in the regression equation if it was reported using scientific notation.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>