HL Paper 2
A function is defined by \(f(x) = A\sin (Bx) + C,{\text{ }} - \pi \le x \le \pi \), where \(A,{\text{ }}B,{\text{ }}C \in \mathbb{Z}\). The following diagram represents the graph of \(y = f(x)\).
Find the value of
(i) \(A\);
(ii) \(B\);
(iii) \(C\).
Solve \(f(x) = 3\) for \(0 \le x \le \pi \).
Consider the function \(f\) defined by \(f(x) = 3x\arccos (x)\) where \( - 1 \leqslant x \leqslant 1\).
Sketch the graph of \(f\) indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
State the range of \(f\).
Solve the inequality \(\left| {3x\arccos (x)} \right| > 1\).
Given \(\Delta \)ABC, with lengths shown in the diagram below, find the length of the line segment [CD].
Consider triangle ABC with \({\rm{B}}\hat {\rm{A}}{\rm{C}} = 37.8^\circ \) , AB = 8.75 and BC = 6 .
Find AC.
The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that \({\rm{A\hat OB}} = \theta \), where \(\theta \) is in radians.
Show that the shaded area can be expressed as \(50\theta - 50\sin \theta \).
Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures.
In triangle \({\text{PQR, PR}} = 12{\text{ cm, QR}} = p{\text{ cm, PQ}} = r{\text{ cm}}\) and \({\rm{Q\hat PR}} = 30^\circ \).
Consider the possible triangles with \({\text{QR}} = 8{\text{ cm}}\).
Consider the case where \(p\), the length of QR is not fixed at 8 cm.
Use the cosine rule to show that \({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\).
Calculate the two corresponding values of PQ.
Hence, find the area of the smaller triangle.
Determine the range of values of \(p\) for which it is possible to form two triangles.
Consider the following diagram.
The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].
Find AM.
Find \({\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}\) in radians.
Find the area of the shaded region.
The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B.
Find:
(a) \({\rm{B\hat OC}}\);
(b) the area of the shaded region.
Triangle ABC has AB = 5 cm, BC = 6 cm and area 10 \({\text{c}}{{\text{m}}^2}\).
(a) Find \(\sin \hat B\).
(b) Hence, find the two possible values of AC, giving your answers correct to two decimal places.
In a triangle ABC, \(\hat A = 35^\circ \), BC = 4 cm and AC = 6.5 cm. Find the possible values of \(\hat B\) and the corresponding values of AB.
Given that \(\arctan \frac{1}{2} - \arctan \frac{1}{3} = \arctan a,{\text{ }}a \in {\mathbb{Q}^ + }\), find the value of a.
Hence, or otherwise, solve the equation \(\arcsin x = \arctan a\).
Consider the triangle \({\text{PQR}}\) where \({\rm{Q\hat PR = 30^\circ }}\), \({\text{PQ}} = (x + 2){\text{ cm}}\) and \({\text{PR}} = {(5 - x)^2}{\text{ cm}}\), where \( - 2 < x < 5\).
Show that the area, \(A\;{\text{c}}{{\text{m}}^2}\), of the triangle is given by \(A = \frac{1}{4}({x^3} - 8{x^2} + 5x + 50)\).
(i) State \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\).
(ii) Verify that \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\).
(i) Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that \(x = \frac{1}{3}\) gives the maximum area of triangle \(PQR\).
(ii) State the maximum area of triangle \(PQR\).
(iii) Find \(QR\) when the area of triangle \(PQR\) is a maximum.
The function \(f(x) = 3\sin x + 4\cos x\) is defined for \(0 < x < 2\pi \) .
Write down the coordinates of the minimum point on the graph of f .
The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .
Find p and q .
Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.
Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.
A circle of radius 4 cm , centre O , is cut by a chord [AB] of length 6 cm.
Find \({\rm{A\hat OB}}\), expressing your answer in radians correct to four significant figures.
Determine the area of the shaded region.
Consider the triangle ABC where \({\rm{B\hat AC}} = 70^\circ \), AB = 8 cm and AC = 7 cm. The point D on the side BC is such that \(\frac{{{\text{BD}}}}{{{\text{DC}}}} = 2\).
Determine the length of AD.
The depth, h(t) metres, of water at the entrance to a harbour at t hours after midnight on a particular day is given by
\[h(t) = 8 + 4\sin \left( {\frac{{\pi t}}{6}} \right),{\text{ }}0 \leqslant t \leqslant 24.\]
(a) Find the maximum depth and the minimum depth of the water.
(b) Find the values of t for which \(h(t) \geqslant 8\).
Farmer Bill owns a rectangular field, 10 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and attaches the other end to his goat Gruff.
Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to graze. Give your answer correct to the nearest integer.
Bill replaces Gruff’s rope with another, this time of length \(a,{\text{ }}4 < a < 10\), so that Gruff can now graze exactly one half of Bill’s field.
Show that \(a\) satisfies the equation
\[{a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40.\]
Find the value of \(a\).
In triangle \({\text{ABC}}\), \({\text{AB}} = 5{\text{ cm}}\), \({\text{BC}} = 12{\text{ cm}}\) and \({\rm{A\hat BC}} = 100^\circ \).
Find the area of the triangle.
Find \(AC\).
Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.
“Seaview” \((S)\) is at an angle of depression of 25°.
“Nauti Buoy” \((N)\) is at an angle of depression of 35°.
The following three dimensional diagram shows Barry and the two yachts at S and N.
X lies at the foot of the cliff and angle \({\text{SXN}} = \) 70°.
Find, to 3 significant figures, the distance between the two yachts.
An electricity station is on the edge of a straight coastline. A lighthouse is located in the sea 200 m from the electricity station. The angle between the coastline and the line joining the lighthouse with the electricity station is 60°. A cable needs to be laid connecting the lighthouse to the electricity station. It is decided to lay the cable in a straight line to the coast and then along the coast to the electricity station. The length of cable laid along the coastline is x metres. This information is illustrated in the diagram below.
The cost of laying the cable along the sea bed is US$80 per metre, and the cost of laying it on land is US$20 per metre.
Find, in terms of x, an expression for the cost of laying the cable.
Find the value of x, to the nearest metre, such that this cost is minimized.
The vertices of an equilateral triangle, with perimeter P and area A , lie on a circle with radius r . Find an expression for \(\frac{P}{A}\) in the form \(\frac{k}{r}\), where \(k \in {\mathbb{Z}^ + }\).
Consider the function \(f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}\).
Let \(u = \tan x\).
Determine an expression for \(f’(x)\) in terms of \(x\).
Sketch a graph of \(y = f’(x)\) for \(0 \leqslant x < \frac{\pi }{2}\).
Find the \(x\)-coordinate(s) of the point(s) of inflexion of the graph of \(y = f(x)\), labelling these clearly on the graph of \(y = f’(x)\).
Express \(\sin x\) in terms of \(\mu \).
Express \(\sin 2x\) in terms of \(u\).
Hence show that \(f(x) = 0\) can be expressed as \({u^3} - 7{u^2} + 15u - 9 = 0\).
Solve the equation \(f(x) = 0\), giving your answers in the form \(\arctan k\) where \(k \in \mathbb{Z}\).
The diagram shows two circles with centres at the points A and B and radii \(2r\) and \(r\), respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.
Let \(\alpha \) be the measure of the angle CAD and \(\theta \) be the measure of the angle CBD in radians.
Find an expression for the shaded area in terms of \(\alpha \), \(\theta \) and \(r\).
Show that \(\alpha = 4\arcsin \frac{1}{4}\).
Hence find the value of \(r\) given that the shaded area is equal to 4.
The shaded region S is enclosed between the curve \(y = x + 2\cos x\), for \(0 \leqslant x \leqslant 2\pi \), and the line \(y = x\), as shown in the diagram below.
Find the coordinates of the points where the line meets the curve.
The region \(S\) is rotated by \(2\pi \) about the \(x\)-axis to generate a solid.
(i) Write down an integral that represents the volume \(V\) of the solid.
(ii) Find the volume \(V\).
In a triangle \({\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}\) and \({\rm{B\hat AC}} = \frac{\pi }{9}\).
Use the cosine rule to find the two possible values for AC.
Find the difference between the areas of the two possible triangles ABC.
Let \(z = r(\cos \alpha + {\text{i}}\sin \alpha )\), where \(\alpha \) is measured in degrees, be the solution of \({z^5} - 1 = 0\) which has the smallest positive argument.
(i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\).
(ii) Hence use De Moivre’s theorem to prove
\[\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta .\]
(iii) State a similar expression for \(\cos 5\theta \) in terms of \(\cos \theta \) and \(\sin \theta \).
Find the value of \(r\) and the value of \(\alpha \).
Using (a) (ii) and your answer from (b) show that \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\).
Hence express \(\sin 72^\circ \) in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\).
In triangle ABC, BC = a , AC = b , AB = c and [BD] is perpendicular to [AC].
(a) Show that \({\text{CD}} = b - c\cos A\).
(b) Hence, by using Pythagoras’ Theorem in the triangle BCD, prove the cosine rule for the triangle ABC.
(c) If \({\rm{A\hat BC}} = 60^\circ \) , use the cosine rule to show that \(c = \frac{1}{2}a \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) .
The above three dimensional diagram shows the points P and Q which are respectively west and south-west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation of the top S of the flagpole from P and Q are respectively 25° and 40° , and PQ = 20 m .
Determine the height of the flagpole.
Let \(f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)\) where \(0 < x < \frac{\pi }{2}\).
Express \(f\left( x \right)\) in terms of sin \(x\) and cos \(x\).
ABCD is a quadrilateral where \({\text{AB}} = 6.5,{\text{ BC}} = 9.1,{\text{ CD}} = 10.4,{\text{ DA}} = 7.8\) and \({\rm{C\hat DA}} = 90^\circ \). Find \({\rm{A\hat BC}}\), giving your answer correct to the nearest degree.
A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is \(\theta \) radians.
The volume of water is increasing at a constant rate of \(0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).
Find an expression for the volume of water \(V{\text{ }}({{\text{m}}^3})\) in the trough in terms of \(\theta \).
Calculate \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) when \(\theta = \frac{\pi }{3}\).
Triangle \(ABC\) has area \({\text{21 c}}{{\text{m}}^{\text{2}}}\). The sides \(AB\) and \(AC\) have lengths \(6\) cm and \(11\) cm respectively. Find the two possible lengths of the side \(BC\).
Consider the planes \({\pi _1}:x - 2y - 3z = 2{\text{ and }}{\pi _2}:2x - y - z = k\) .
Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .
The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector equation of
\({L_1}\) is \(r = \left( {\begin{array}{*{20}{c}}
0\\
{2 - 3k}\\
{2k - 2}
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
1\\
5\\
{ - 3}
\end{array}} \right)\)
The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and \({L_2}\) intersect at a point X. Find the coordinates of X.
Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and \({L_2}\) .
Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ and the area of the triangle XYZ is 3. Find the perimeter of the triangle XYZ.
In triangle \(ABC\),
\(3\sin B + 4\cos C = 6\) and
\(4\sin C + 3\cos B = 1\).
Show that \(\sin (B + C) = \frac{1}{2}\).
Robert conjectures that \({\rm{C\hat AB}}\) can have two possible values.
Show that Robert’s conjecture is incorrect by proving that \({\rm{C\hat AB}}\) has only one possible value.
Find the set of values of \(k\) that satisfy the inequality \({k^2} - k - 12 < 0\).
The triangle ABC is shown in the following diagram. Given that \(\cos B < \frac{1}{4}\), find the range of possible values for AB.
A triangle \(ABC\) has \(\hat A = 50^\circ \), \({\text{AB}} = 7{\text{ cm}}\) and \({\text{BC}} = 6{\text{ cm}}\). Find the area of the triangle given that it is smaller than \(10{\text{ c}}{{\text{m}}^2}\).
A rectangle is drawn around a sector of a circle as shown. If the angle of the sector is 1 radian and the area of the sector is \(7{\text{ c}}{{\text{m}}^2}\), find the dimensions of the rectangle, giving your answers to the nearest millimetre.
The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.
The points P and Q lie on the larger circle and \({\rm{P}}\hat {\text{O}}{\text{Q}} = x\) , where \(0 < x < \frac{\pi }{2}\) .
(a) Show that the area of the shaded region is \(8\sin x - 2x\) .
(b) Find the maximum area of the shaded region.
In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO of the cone is 8 cm and the angle \({\rm{B\hat OC}}\) is 60°.
Calculate the size of the angle \({\rm{B\hat AC}}\).
Points A , B and T lie on a line on an indoor soccer field. The goal, [AB] , is 2 metres wide. A player situated at point P kicks a ball at the goal. [PT] is perpendicular to (AB) and is 6 metres from a parallel line through the centre of [AB] . Let PT be \(x\) metros and let \(\alpha = {\rm{A\hat PB}}\) measured in degrees. Assume that the ball travels along the floor.
The maximum for \(\tan \alpha \) gives the maximum for \(\alpha \).
Find the value of \(\alpha \) when \(x = 10\).
Show that \(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\).
(i) Find \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\).
(ii) Hence or otherwise find the value of \(\alpha \) such that \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = 0\).
(iii) Find \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) and hence show that the value of \(\alpha \) never exceeds 10°.
Find the set of values of \(x\) for which \(\alpha \geqslant 7^\circ \).
Solve the equation \(3{\cos ^2}x - 8\cos x + 4 = 0\), where \(0 \leqslant x \leqslant 180^\circ \), expressing your answer(s) to the nearest degree.
Find the exact values of \(\sec x\) satisfying the equation \(3{\sec ^4}x - 8{\sec ^2}x + 4 = 0\).
The points P and Q lie on a circle, with centre O and radius 8 cm, such that \({\rm{P\hat OQ}} = 59^\circ \) .
Find the area of the shaded segment of the circle contained between the arc PQ and the chord [PQ].
The graph below shows \(y = a\cos (bx) + c\).
Find the value of a, the value of b and the value of c.
A system of equations is given by
\[\cos x + \cos y = 1.2\]
\[\sin x + \sin y = 1.4{\text{ .}}\]
(a) For each equation express y in terms of x.
(b) Hence solve the system for \(0 < x < \pi ,{\text{ }}0 < y < \pi \) .
This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius \({\text{OB}} = 9{\text{ cm}}\) and four equal sectors of a smaller circle of radius \({\text{OA}} = 3{\text{ cm}}\).
The angle \({\text{BOC}} = \) 20°.
Find the area of the pendant.
A ship, S, is 10 km north of a motorboat, M, at 12.00pm. The ship is travelling northeast with a constant velocity of \(20{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\). The motorboat wishes to intercept the ship and it moves with a constant velocity of \(30{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\) in a direction \(\theta \) degrees east of north. In order for the interception to take place, determine
the value of \(\theta \).
the time at which the interception occurs, correct to the nearest minute.
A straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. The intensity of light at point P at ground level on the street is proportional to the angle \(\theta \) where \(\theta = {\rm{A\hat PB}}\), as shown in the diagram.
Find an expression for \(\theta \) in terms of x, where x is the distance of P from the base of the wall of height 8 m.
(i) Calculate the value of \(\theta \) when x = 0.
(ii) Calculate the value of \(\theta \) when x = 20.
Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).
Show that \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\).
Using the result in part (d), or otherwise, determine the value of x corresponding to the maximum light intensity at P. Give your answer to four significant figures.
The point P moves across the street with speed \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Determine the rate of change of \(\theta \) with respect to time when P is at the midpoint of the street.
The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas of these sectors form a geometric sequence with common ratio k. The angle of the first sector is \(\theta \) radians.
(a) Show that \(\theta = 2\pi (1 - k)\).
(b) The perimeter of the third sector is half the perimeter of the first sector.
Find the value of k and of \(\theta \).
The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The points A and C are 3 m apart. A goat \(G\) is tied by a 5 m length of rope at point A on the outside edge of the enclosure.
Given that the corner of the enclosure at C forms an angle of \(\theta \) radians and the area of field that can be reached by the goat is \({\text{44 }}{{\text{m}}^{\text{2}}}\), find the value of \(\theta \).
The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB.
Compactness is a measure of how compact an enclosed region is.
The compactness, \(C\) , of an enclosed region can be defined by \(C = \frac{{4A}}{{\pi {d^2}}}\), where \(A\) is the area of the region and \(d\) is the maximum distance between any two points in the region.
For a circular region, \(C = 1\).
Consider a regular polygon of \(n\) sides constructed such that its vertices lie on the circumference of a circle of diameter \(x\) units.
If \(n > 2\) and even, show that \(C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}\).
If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).
Find the regular polygon with the least number of sides for which the compactness is more than \(0.99\).
If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).
Comment briefly on whether C is a good measure of compactness.
Two non-intersecting circles C1 , containing points M and S , and C2 , containing points N and R, have centres P and Q where PQ \( = 50\) . The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is \( \alpha\), the size of the obtuse angle NQR is \( \beta\) , and the size of the angle MPQ is \( \theta\) . The arc length MS is \({l_1}\) and the arc length NR is \({l_2}\) . This information is represented in the diagram below.
The radius of C1 is \(x\) , where \(x \geqslant 10\) and the radius of C2 is \(10\).
(a) Explain why \(x < 40\) .
(b) Show that cosθ = x −10 50.
(c) (i) Find an expression for MN in terms of \(x\) .
(ii) Find the value of \(x\) that maximises MN.
(d) Find an expression in terms of \(x\) for
(i) \( \alpha\) ;
(ii) \( \beta\) .
(e) The length of the perimeter is given by \({l_1} + {l_2} + {\text{MN}} + {\text{SR}}\).
(i) Find an expression, \(b (x)\) , for the length of the perimeter in terms of \(x\) .
(ii) Find the maximum value of the length of the perimeter.
(iii) Find the value of \(x\) that gives a perimeter of length \(200\).
Consider a triangle ABC with \({\rm{B\hat AC}} = 45.7^\circ \) , AB = 9.63 cm and BC = 7.5 cm .
By drawing a diagram, show why there are two triangles consistent with this information.
Find the possible values of AC .
Points A, B and C are on the circumference of a circle, centre O and radius \(r\) . A trapezium OABC is formed such that AB is parallel to OC, and the angle \({\rm{A}}\hat {\text{O}}{\text{C}}\) is \(\theta\) , \(\frac{\pi }{2} \leqslant \theta \leqslant \pi \) .
(a) Show that angle \({\rm{B\hat OC}}\) is \(\pi - \theta \).
(b) Show that the area, T, of the trapezium can be expressed as
\[T = \frac{1}{2}{r^2}\sin \theta - \frac{1}{2}{r^2}\sin 2\theta .\]
(c) (i) Show that when the area is maximum, the value of \(\theta \) satisfies
\[\cos \theta = 2\cos 2\theta .\]
(ii) Hence determine the maximum area of the trapezium when r = 1.
(Note: It is not required to prove that it is a maximum.)
The diagram shows the plan of an art gallery a metres wide. [AB] represents a doorway, leading to an exit corridor b metres wide. In order to remove a painting from the art gallery, CD (denoted by L ) is measured for various values of \(\alpha \) , as represented in the diagram.
If \(\alpha \) is the angle between [CD] and the wall, show that \(L = \frac{a }{{\sin \alpha }} + \frac{b}{{\cos \alpha }}{\text{, }}0 < \alpha < \frac{\pi }{2}\).
If a = 5 and b = 1, find the maximum length of a painting that can be removed through this doorway.
Let a = 3k and b = k .
Find \(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}\).
Let a = 3k and b = k .
Find, in terms of k , the maximum length of a painting that can be removed from the gallery through this doorway.
Let a = 3k and b = k .
Find the minimum value of k if a painting 8 metres long is to be removed through this doorway.
Two discs, one of radius 8 cm and one of radius 5 cm, are placed such that they touch each other. A piece of string is wrapped around the discs. This is shown in the diagram below.
Calculate the length of string needed to go around the discs.