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HL Paper 1

Three planes have equations:

$2x - y + z = 5$

$x + 3y - z = 4$ , where $a{\text{, }}b \in \mathbb{R}$ .

$3x - 5y + az = b$

Find the set of values of $a$ and $b$ such that the three planes have no points of intersection.

Two distinct lines, ${l_1}$ and ${l_2}$ , intersect at a point ${\text{P}}$ . In addition to ${\text{P}}$ , four distinct points are marked out on ${l_1}$ and three distinct points on ${l_2}$ . A mathematician decides to join some of these eight points to form polygons.

The line ${l_1}$ has vector equation r₁ $= \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 1 \end{array}} \right)$ , $\lambda \in \mathbb{R}$ and the line ${l_2}$ has vector equation r₂ $= \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 2 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 5 \\ 6 \\ 2 \end{array}} \right)$ , $\mu \in \mathbb{R}$ .

The point ${\text{P}}$ has coordinates (4, 6, 4).

The point ${\text{A}}$ has coordinates (3, 4, 3) and lies on ${l_1}$ .

The point ${\text{B}}$ has coordinates (−1, 0, 2) and lies on ${l_2}$ .

Find how many sets of four points can be selected which can form the vertices of a quadrilateral.

[2]

a.i.

Find how many sets of three points can be selected which can form the vertices of a triangle.

[4]

a.ii.

Verify that ${\text{P}}$ is the point of intersection of the two lines.

[3]

Write down the value of $\lambda$ corresponding to the point ${\text{A}}$ .

[1]

Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .

[2]

Let ${\text{C}}$ be the point on ${l_1}$ with coordinates (1, 0, 1) and ${\text{D}}$ be the point on ${l_2}$ with parameter $\mu = - 2$ .

Find the area of the quadrilateral ${\text{CDBA}}$ .

[8]

Let S be the sum of the roots found in part (a).

Find the roots of ${z^{24}} = 1$ which satisfy the condition $0 < {\text{arg}}\left( z \right) < \frac{\pi }{2}$ , expressing your answers in the form $r{e^{{\text{i}}\theta }}$ , where $r$ , $\theta \in {\mathbb{R}^ + }$ .

[5]

Show that Re S = Im S.

[4]

b.i.

By writing $\frac{\pi }{{12}}$ as $\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)$ , find the value of cos $\frac{\pi }{{12}}$ in the form $\frac{{\sqrt a + \sqrt b }}{c}$ , where $a$ , $b$ and $c$ are integers to be determined.

[3]

b.ii.

Hence, or otherwise, show that S = $\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)$ .

[4]

b.iii.

Let $z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi$ .

Solve $2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ$ .

[5]

Show that $\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}$ .

[3]

Find the modulus and argument of $z$ in terms of $\theta$ . Express each answer in its simplest form.

[9]

c.i.

Hence find the cube roots of $z$ in modulus-argument form.

[5]

c.ii.

Consider the three planes

$\prod_{1} : 2 x - y + z = 4$

$\prod_{2} : x - 2 y + 3 z = 5$

$\prod_{3} : - 9 x + 3 y - 2 z = 32$

Show that the three planes do not intersect.

[4]

Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .

[1]

b.i.

Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .

[4]

b.ii.

Find the distance between $L$ and $\prod_{3}$ .

[6]

In the following diagram, $\overrightarrow {{\text{OA}}}$ = a, $\overrightarrow {{\text{OB}}}$ = b. C is the midpoint of [OA] and $\overrightarrow {{\text{OF}}} = \frac{1}{6}\overrightarrow {{\text{FB}}}$ .

N17/5/MATHL/HP1/ENG/TZ0/09

It is given also that $\overrightarrow {{\text{AD}}} = \lambda \overrightarrow {{\text{AF}}}$ and $\overrightarrow {{\text{CD}}} = \mu \overrightarrow {{\text{CB}}}$ , where $\lambda ,{\text{ }}\mu \in \mathbb{R}$ .

Find, in terms of a and b $\overrightarrow {{\text{OF}}}$ .

[1]

a.i.

Find, in terms of a and b $\overrightarrow {{\text{AF}}}$ .

[2]

a.ii.

Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\lambda$ ;

[2]

b.i.

Find an expression for $\overrightarrow {{\text{OD}}}$ in terms of a, b and $\mu$ .

[2]

b.ii.

Show that $\mu = \frac{1}{{13}}$ , and find the value of $\lambda$ .

[4]

Deduce an expression for $\overrightarrow {{\text{CD}}}$ in terms of a and b only.

[2]

Given that area $\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})$ , find the value of $k$ .

[5]

Show that ${\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x$ .

[2]

Show that ${\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}$ .

[4]

Hence or otherwise find $\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x$ in the form ${\text{ln}}\left( {a + \sqrt b } \right)$ where $a$ , $b \in \mathbb{Z}$ .

[9]

The following figure shows a square based pyramid with vertices at O(0, 0, 0), A(1, 0, 0), B(1, 1, 0), C(0, 1, 0) and D(0, 0, 1).

The Cartesian equation of the plane ${\Pi _2}$ , passing through the points B , C and D , is $y + z = 1$ .

The plane ${\Pi _3}$ passes through O and is normal to the line BD.

${\Pi _3}$ cuts AD and BD at the points P and Q respectively.

Find the Cartesian equation of the plane ${\Pi _1}$ , passing through the points A , B and D.

[3]

Find the angle between the faces ABD and BCD.

[4]

Find the Cartesian equation of ${\Pi _3}$ .

[3]

Show that P is the midpoint of AD.

[4]

Find the area of the triangle OPQ.

[5]

Find the value of $\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}$ .

[2]

Show that $\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .

[2]

Use the principle of mathematical induction to prove that

$\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .

[9]

Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .

[6]

Use the binomial theorem to expand ${(\cos θ + i \sin θ)}^{4}$ . Give your answer in the form $a + b i$ where $a$ and $b$ are expressed in terms of $\sin θ$ and $\cos θ$ .

[3]

Use de Moivre’s theorem and the result from part (a) to show that $\cot 4 θ = \frac{\cot^{4} θ - 6 \cot^{2} θ + 1}{4 \cot^{3} θ - 4 \cot θ}$ .

[5]

Use the identity from part (b) to show that the quadratic equation $x^{2} - 6 x + 1 = 0$ has roots ${cot}^{2} \frac{π}{8}$ and ${cot}^{2} \frac{3 π}{8}$ .

[5]

Hence find the exact value of ${cot}^{2} \frac{3 π}{8}$ .

[4]

Deduce a quadratic equation with integer coefficients, having roots ${cosec}^{2} \frac{π}{8}$ and ${cosec}^{2} \frac{3 π}{8}$ .

[3]

In the following diagram, the points ${\text{A}}$ , ${\text{B}}$ , ${\text{C}}$ and ${\text{D}}$ are on the circumference of a circle with centre ${\text{O}}$ and radius $r$ . $\left[ {{\text{AC}}} \right]$ is a diameter of the circle. ${\text{BC}} = r$ , ${\text{AD}} = {\text{CD}}$ and ${\text{A}}\mathop {\text{B}}\limits^ \wedge {\text{C}} = {\text{A}}\mathop {\text{D}}\limits^ \wedge {\text{C}} = 90^\circ$ .

Given that ${\text{cos}}\,75^\circ = q$ , show that ${\text{cos}}\,105^\circ = - q$ .

[1]

Show that ${\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 75^\circ$ .

[3]

By considering triangle ${\text{ABD}}$ , show that ${\text{B}}{{\text{D}}^2} = 5{r^2} - 2{r^2}q\sqrt 6$ .

[4]

c.i.

By considering triangle ${\text{CBD}}$ , find another expression for ${\text{B}}{{\text{D}}^2}$ in terms of $r$ and $q$ .

[3]

c.ii.

Use your answers to part (c) to show that ${\text{cos}}\,75^\circ = \frac{1}{{\sqrt 6 + \sqrt 2 }}$ .

[3]

Given any two non-zero vectors, $a$ and $b$ , show that ${|a \times b|}^{2} = {|a|}^{2} {|b|}^{2} - {(a \cdot b)}^{2}$ .

The following diagram shows the graph of $y = arctan (2 x + 1) + \frac{π}{4}$ for $x \in ℝ$ , with asymptotes at $y = - \frac{π}{4}$ and $y = \frac{3 π}{4}$ .

Describe a sequence of transformations that transforms the graph of $y = arctan x$ to the graph of $y = arctan (2 x + 1) + \frac{π}{4}$ for $x \in ℝ$ .

[3]

Show that $arctan p + arctan q \equiv arctan (\frac{p + q}{1 - p q})$ where $p, q > 0$ and $p q < 1$ .

[4]

Verify that $arctan (2 x + 1) = arctan (\frac{x}{x + 1}) + \frac{π}{4}$ for $x \in ℝ, x > 0$ .

[3]

Using mathematical induction and the result from part (b), prove that $Σ_{r = 1}^{n} arctan (\frac{1}{2 r^{2}}) = arctan (\frac{n}{n + 1})$ for $n \in ℤ^{+}$ .

[9]

Consider the lines ${l_1}$ and ${l_2}$ defined by

${l_1}:$ r $= \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ a \end{array}} \right) + \beta \left( {\begin{array}{*{20}{c}} 1 \\ 4 \\ 2 \end{array}} \right)$ and ${l_2}:\frac{{6 - x}}{3} = \frac{{y - 2}}{4} = 1 - z$ where $a$ is a constant.

Given that the lines ${l_1}$ and ${l_2}$ intersect at a point P,

find the value of $a$ ;

[4]

determine the coordinates of the point of intersection P.

[2]

Consider a triangle OAB such that O has coordinates (0, 0, 0), A has coordinates (0, 1, 2) and B has coordinates (2 $b$ , 0, $b$ − 1) where $b$ < 0.

Let M be the midpoint of the line segment [OB].

Find, in terms of $b$ , a Cartesian equation of the plane Π containing this triangle.

[5]

Find, in terms of $b$ , the equation of the line L which passes through M and is perpendicular to the plane П.

[3]

Show that L does not intersect the $y$ -axis for any negative value of $b$ .

[7]

The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let θ be the angle between the two given sides. The triangle has an area of $\frac{{5\sqrt {15} }}{2}$ cm².

Show that ${\text{sin}}\,\theta = \frac{{\sqrt {15} }}{4}$ .

[1]

Find the two possible values for the length of the third side.

[6]

The points A and B are given by ${\text{A}}(0,{\text{ }}3,{\text{ }} - 6)$ and ${\text{B}}(6,{\text{ }} - 5,{\text{ }}11)$ .

The plane Π is defined by the equation $4x - 3y + 2z = 20$ .

Find a vector equation of the line L passing through the points A and B.

[3]

Find the coordinates of the point of intersection of the line L with the plane Π.

[3]

The function $f$ is defined by $f\left( x \right) = {{\text{e}}^x}\,{\text{cos}}{\,^2}x$ , where 0 ≤ $x$ ≤ 5. The curve $y = f\left( x \right)$ is shown on the following graph which has local maximum points at A and C and touches the $x$ -axis at B and D.

Use integration by parts to show that $\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c$ , $c \in \mathbb{R}$ .

[5]

Hence, show that $\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c$ , $c \in \mathbb{R}$ .

[3]

Find the $x$ -coordinates of A and of C , giving your answers in the form $a + {\text{arctan}}\,b$ , where $a$ , $b \in \mathbb{R}$ .

[6]

Find the area enclosed by the curve and the $x$ -axis between B and D, as shaded on the diagram.

[5]

The points A, B, C and D have position vectors a, b, c and d, relative to the origin O.

It is given that $\mathop {{\text{AB}}}\limits^ \to = \mathop {{\text{DC}}}\limits^ \to$ .

The position vectors $\mathop {{\text{OA}}}\limits^ \to$ , $\mathop {{\text{OB}}}\limits^ \to$ , $\mathop {{\text{OC}}}\limits^ \to$ and $\mathop {{\text{OD}}}\limits^ \to$ are given by

a = i + 2j − 3k

b = 3i − j + pk

c = qi + j + 2k

d = −i + rj − 2k

where p , q and r are constants.

The point where the diagonals of ABCD intersect is denoted by M.

The plane $\Pi$ cuts the x, y and z axes at X , Y and Z respectively.

Explain why ABCD is a parallelogram.

[1]

a.i.

Using vector algebra, show that $\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to$ .

[3]

a.ii.

Show that p = 1, q = 1 and r = 4.

[5]

Find the area of the parallelogram ABCD.

[4]

Find the vector equation of the straight line passing through M and normal to the plane $\Pi$ containing ABCD.

[4]

Find the Cartesian equation of $\Pi$ .

[3]

Find the coordinates of X, Y and Z.

[2]

f.i.

Find YZ.

[2]

f.ii.

A function $f$ is defined by $f (x) = \frac{1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x \neq - 1, x \neq 3$ .

A function $g$ is defined by $g (x) = \frac{1}{x^{2} - 2 x - 3}$ , where $x \in ℝ, x > 3$ .

The inverse of $g$ is $g^{- 1}$ .

A function $h$ is defined by $h (x) = arctan \frac{x}{2}$ , where $x \in ℝ$ .

Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.

[6]

Show that $g^{- 1} (x) = 1 + \frac{\sqrt{4 x^{2} + x}}{x}$ .

[6]

b.i.

State the domain of $g^{- 1}$ .

[1]

b.ii.

Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .

[7]

By using the substitution $u = sec x$ or otherwise, find an expression for $\int_{0}^{\frac{π}{3}} {sec}^{n} x \tan x d x$ in terms of $n$ , where $n$ is a non-zero real number.

A straight line, ${L_\theta }$ , has vector equation r $= \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 0 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right){\text{, }}\lambda {\text{, }}\theta \in \mathbb{R}$ .

The plane ${\Pi _p}$ , has equation $x = p{\text{, }}p \in \mathbb{R}$ .

Show that the angle between ${L_\theta }$ and ${\Pi _p}$ is independent of both $\theta$ and $p$ .

The lines $l_{1}$ and $l_{2}$ have the following vector equations where $λ, μ \in ℝ$ and $m \in ℝ$ .

$l_{1} : r_{1} = (\begin{matrix} 3 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 2 \\ 1 \\ m \end{matrix}) l_{2} : r_{2} = (\begin{matrix} - 1 \\ - 4 \\ - 2 m \end{matrix}) + μ (\begin{matrix} 2 \\ - 5 \\ - m \end{matrix})$

The plane $Π$ has Cartesian equation $x + 4 y - z = p$ where $p \in ℝ$ .

Given that $l_{1}$ and $Π$ have no points in common, find

Show that $l_{1}$ and $l_{2}$ are never perpendicular to each other.

[3]

the value of $m$ .

[2]

b.i.

the condition on the value of $p$ .

[2]

b.ii.

Consider the line $L_{1}$ defined by the Cartesian equation $\frac{x + 1}{2} = y = 3 - z$ .

Consider a second line $L_{2}$ defined by the vector equation $r = (\begin{matrix} 0 \\ 1 \\ 2 \end{matrix}) + t (\begin{matrix} a \\ 1 \\ - 1 \end{matrix})$ , where $t \in ℝ$ and $a \in ℝ$ .

Show that the point $(- 1, 0, 3)$ lies on $L_{1}$ .

[1]

a.i.

Find a vector equation of $L_{1}$ .

[3]

a.ii.

Find the possible values of $a$ when the acute angle between $L_{1}$ and $L_{2}$ is $45 °$ .

[8]

It is given that the lines $L_{1}$ and $L_{2}$ have a unique point of intersection, $A$ , when $a \neq k$ .

Find the value of $k$ , and find the coordinates of the point $A$ in terms of $a$ .

[7]

Points ${\text{A}}$ (0 , 0 , 10) , ${\text{B}}$ (0 , 10 , 0) , ${\text{C}}$ (10 , 0 , 0) , ${\text{V}}$ ( $p$ , $p$ , $p$ ) form the vertices of a tetrahedron.

Consider the case where the faces ${\text{ABV}}$ and ${\text{ACV}}$ are perpendicular.

The following diagram shows the graph of $\theta$ against $p$ . The maximum point is shown by ${\text{X}}$ .

Show that $\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)$ and find a similar expression for $\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}}$ .

[3]

a.i.

Hence, show that, if the angle between the faces ${\text{ABV}}$ and ${\text{ACV}}$ is $\theta$ , then ${\text{cos}}\,\theta = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}$ .

[5]

a.ii.

Find the two possible coordinates of ${\text{V}}$ .

[3]

b.i.

Comment on the positions of ${\text{V}}$ in relation to the plane ${\text{ABC}}$ .

[1]

b.ii.

At ${\text{X}}$ , find the value of $p$ and the value of $\theta$ .

[3]

c.i.

Find the equation of the horizontal asymptote of the graph.

[2]

c.ii.

Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3,{\text{ }}x - y + z = 5$ and $x + y + 2z = 6$ .

Consider the function $g\left( x \right) = 4\,{\text{cos}}\,x + 1$ , $a \leqslant x \leqslant \frac{\pi }{2}$ where $a < \frac{\pi }{2}$ .

For $a = - \frac{\pi }{2}$ , sketch the graph of $y = g\left( x \right)$ . Indicate clearly the maximum and minimum values of the function.

[3]

Write down the least value of $a$ such that $g$ has an inverse.

[1]

For the value of $a$ found in part (b), write down the domain of ${g^{ - 1}}$ .

[1]

c.i.

For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .

[2]

c.ii.

Consider the vectors a $=$ i $- {\text{ }}3$ j $- {\text{ }}2$ k, b $= - {\text{ }}3$ j $+ {\text{ }}2$ k.

Find a $\times$ b.

[2]

Hence find the Cartesian equation of the plane containing the vectors a and b, and passing through the point $(1,{\text{ }}0,{\text{ }} - 1)$ .

[3]

$A$ and $B$ are acute angles such that ${\text{cos}}\,A = \frac{2}{3}$ and ${\text{sin}}\,B = \frac{1}{3}$ .

Show that ${\text{cos}}\left( {2A + B} \right) = - \frac{{2\sqrt 2 }}{{27}} - \frac{{4\sqrt 5 }}{{27}}$ .

The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ.

Find cos θ.

ABCD is a parallelogram, where $\overrightarrow {{\text{AB}}}$ = –i + 2j + 3k and $\overrightarrow {{\text{AD}}}$ = 4i – j – 2k.

Find the area of the parallelogram ABCD.

[3]

By using a suitable scalar product of two vectors, determine whether ${\rm{A\hat BC}}$ is acute or obtuse.

[4]

It is given that $cosec θ = \frac{3}{2}$ , where $\frac{π}{2} < θ < \frac{3 π}{2}$ . Find the exact value of $cot θ$ .

Let a = $\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)$ and b = $\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)$ , $k \in \mathbb{R}$ .

Given that a and b are perpendicular, find the possible values of $k$ .

Three points in three-dimensional space have coordinates A(0, 0, 2), B(0, 2, 0) and C(3, 1, 0).

Find the vector $\overrightarrow {{\text{AB}}}$ .

[1]

a.i.

Find the vector $\overrightarrow {{\text{AC}}}$ .

[1]

a.ii.

Hence or otherwise, find the area of the triangle ABC.

[4]

Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$ .

The lines $l_{1}$ and $l_{2}$ have the following vector equations where $λ, μ \in ℝ$ .

$l_{1} : r_{1} = (\begin{matrix} 3 \\ 2 \\ - 1 \end{matrix}) + λ (\begin{matrix} 2 \\ - 2 \\ 2 \end{matrix})$

$l_{2} : r_{2} = (\begin{matrix} 2 \\ 0 \\ 4 \end{matrix}) + μ (\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix})$

Show that $l_{1}$ and $l_{2}$ do not intersect.

[3]

Find the minimum distance between $l_{1}$ and $l_{2}$ .

[5]

Let $a = {\text{sin}}\,b,\,\,0 < b < \frac{\pi }{2}$ .

Find, in terms of b, the solutions of ${\text{sin}}\,2x = - a,\,\,0 \leqslant x \leqslant \pi$ .

Consider the functions $f$ and $g$ defined on the domain $0 < x < 2\pi$ by $f\left( x \right) = 3\,{\text{cos}}\,2x$ and $g\left( x \right) = 4 - 11\,{\text{cos}}\,x$ .

The following diagram shows the graphs of $y = f\left( x \right)$ and $y = g\left( x \right)$

Find the $x$ -coordinates of the points of intersection of the two graphs.

[6]

Find the exact area of the shaded region, giving your answer in the form $p\pi + q\sqrt 3$ , where $p$ , $q \in \mathbb{Q}$ .

[5]

At the points A and B on the diagram, the gradients of the two graphs are equal.

Determine the $y$ -coordinate of A on the graph of $g$ .

[6]

Consider quadrilateral $PQRS$ where $[PQ]$ is parallel to $[SR]$ .

In $PQRS$ , $PQ = x$ , $SR = y$ , $R \hat{S} P = α$ and $Q \hat{R} S = β$ .

Find an expression for $PS$ in terms of $x, y, \sin β$ and $\sin (α + β)$ .

Consider the complex numbers $z_{1} = 1 + b i$ and $z_{2} = (1 - b^{2}) - 2 b i$ , where $b \in ℝ, b \neq 0$ .

Find an expression for $z_{1} z_{2}$ in terms of $b$ .

[3]

Hence, given that $arg (z_{1} z_{2}) = \frac{π}{4}$ , find the value of $b$ .

[3]

The vectors a and b are defined by a = $\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ t \end{array}} \right)$ , b = $\left( {\begin{array}{*{20}{c}} 0 \\ { - t} \\ {4t} \end{array}} \right)$ , where $t \in \mathbb{R}$ .

Find and simplify an expression for a • b in terms of $t$ .

[2]

Hence or otherwise, find the values of $t$ for which the angle between a and b is obtuse .

[4]

Let $f (x) = \sqrt{1 + x}$ for $x > - 1$ .

Show that $f'' (x) = - \frac{1}{4 \sqrt{{(1 + x)}^{3}}}$ .

[3]

Use mathematical induction to prove that $f^{(n)} (x) = {(- \frac{1}{4})}^{n - 1} \frac{(2 n - 3)!}{(n - 2)!} {(1 + x)}^{\frac{1}{2} - n}$ for $n \in ℤ, n \geq 2$ .

[9]

Let $g (x) = e^{m x}, m \in ℚ$ .

Consider the function $h$ defined by $h (x) = f (x) \times g (x)$ for $x > - 1$ .

It is given that the $x^{2}$ term in the Maclaurin series for $h (x)$ has a coefficient of $\frac{7}{4}$ .

Find the possible values of $m$ .

[8]

Solve the equation $2 \cos^{2} x + 5 \sin x = 4, 0 \leq x \leq 2 π$ .

A sector of a circle with radius $r$ cm , where $r$ > 0, is shown on the following diagram.
The sector has an angle of 1 radian at the centre.

Let the area of the sector be $A$ cm² and the perimeter be $P$ cm. Given that $A = P$ , find the value of $r$ .

The plane П has the Cartesian equation $2x + y + 2z = 3$

The line L has the vector equation r $= \left( {\begin{array}{*{20}{c}} 3 \\ { - 5} \\ 1 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right){\text{,}}\,\,\mu {\text{,}}\,p \in \mathbb{R}$ . The acute angle between the line L and the plane П is 30°.

Find the possible values of $p$ .