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AQA C4 2014 June Q5

15 marks Standard +0.3

1. Express $3 \sin x + 4 \cos x$ in the form $R \sin ( x + \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $3 \sin 2 \theta + 4 \cos 2 \theta = 5$ in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Show that the equation $\tan 2 \theta \tan \theta = 2$ can be written as $2 \tan ^ { 2 } \theta = 1$.
2. Hence solve the equation $\tan 2 \theta \tan \theta = 2$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Use the Factor Theorem to show that $2 x - 1$ is a factor of $8 x ^ { 3 } - 4 x + 1$.
2. Show that $4 \cos 2 \theta \cos \theta + 1$ can be written as $8 x ^ { 3 } - 4 x + 1$ where $x = \cos \theta$.
3. Given that $\theta = 72 ^ { \circ }$ is a solution of $4 \cos 2 \theta \cos \theta + 1 = 0$, use the results from parts (c)(i) and (c)(ii) to show that the exact value of $\cos 72 ^ { \circ }$ is $\frac { ( \sqrt { 5 } - 1 ) } { p }$ where $p$ is an integer.
  [0pt] [3 marks]

AQA C4 2014 June Q6

10 marks Moderate -0.3

6 The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 5 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } - 1 \\ 3 \\ 1 \end{array} \right]$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 7 \\ - 8 \\ 6 \end{array} \right] + \mu \left[ \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right]$.
The point $P$ lies on $l _ { 1 }$ where $\lambda = - 1$. The point $Q$ lies on $l _ { 2 }$ where $\mu = 2$.

Show that the vector $\overrightarrow { P Q }$ is parallel to $\left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]$.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R ( 3 , b , c )$.
1. Show that $b = - 2$ and find the value of $c$.
2. The point $S$ lies on a line through $P$ that is parallel to $l _ { 2 }$. The line $R S$ is perpendicular to the line $P Q$. \includegraphics[max width=\textwidth, alt={}, center]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-16_887_1159_1320_443} Find the coordinates of $S$. $7 \quad$ A curve has equation $\cos 2 y + y \mathrm { e } ^ { 3 x } = 2 \pi$.
  The point $A \left( \ln 2 , \frac { \pi } { 4 } \right)$ lies on this curve.

AQA C4 2014 June Q8

11 marks Standard +0.3

Express $\frac { 16 x } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$ in the form $\frac { A } { 1 - 3 x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$.
Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 x \mathrm { e } ^ { 2 y } } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$$ where $y = 0$ when $x = 0$.
Give your answer in the form $\mathrm { f } ( y ) = \mathrm { g } ( x )$.
[0pt] [7 marks]

AQA C4 2015 June Q1

9 marks Moderate -0.8

1 It is given that $\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }$ can be expressed as $\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }$, where $A$ and $B$ are integers.

Find the values of $A$ and $B$.
Hence show that $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5$, where $k$ is a rational number.
[0pt] [6 marks]

AQA C4 2015 June Q2

8 marks Standard +0.3

Express $2 \cos x - 5 \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$, giving your value of $\alpha$, in radians, to three significant figures.
1. Hence find the value of $x$ in the interval $0 < x < 2 \pi$ for which $2 \cos x - 5 \sin x$ has its maximum value. Give your value of $x$ to three significant figures.
2. Use your answer to part (a) to solve the equation $2 \cos x - 5 \sin x + 1 = 0$ in the interval $0 < x < 2 \pi$, giving your solutions to three significant figures.
  [0pt] [3 marks]

AQA C4 2015 June Q3

9 marks Moderate -0.3

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d$, where $d$ is a constant. When $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ), the remainder is - 2 . Use the Remainder Theorem to find the value of $d$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3$.
1. Given that $x = - \frac { 1 } { 2 }$ is a solution of the equation $\mathrm { g } ( x ) = 0$, write $\mathrm { g } ( x )$ as a product of three linear factors.
2. The function h is defined by $\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }$ for $x > 2$. Simplify $\mathrm { h } ( x )$, and hence show that h is a decreasing function.
  [0pt] [4 marks]

AQA C4 2015 June Q4

7 marks Standard +0.3

Find the binomial expansion of $( 1 + 5 x ) ^ { \frac { 1 } { 5 } }$ up to and including the term in $x ^ { 2 }$.
1. Find the binomial expansion of $( 8 + 3 x ) ^ { - \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
2. Use your expansion from part (b)(i) to find an estimate for $\sqrt [ 3 ] { \frac { 1 } { 81 } }$, giving your answer to four decimal places.
  [0pt] [2 marks]

AQA C4 2015 June Q5

11 marks Standard +0.8

5 A curve is defined by the parametric equations $x = \cos 2 t , y = \sin t$.
The point $P$ on the curve is where $t = \frac { \pi } { 6 }$.

Find the gradient at $P$.
Find the equation of the normal to the curve at $P$ in the form $y = m x + c$.
The normal at $P$ intersects the curve again at the point $Q ( \cos 2 q , \sin q )$. Use the equation of the normal to form a quadratic equation in $\sin q$ and hence find the $x$-coordinate of $Q$.
[0pt] [5 marks]

AQA C4 2015 June Q6

12 marks Challenging +1.2

6 The points $A$ and $B$ have coordinates $( 3,2,10 )$ and $( 5 , - 2,4 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]$.

Find the acute angle between $l$ and the line $A B$.
The point $C$ lies on $l$ such that angle $A B C$ is $90 ^ { \circ }$. \includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of $C$.
The point $D$ is such that $B D$ is parallel to $A C$ and angle $B C D$ is $90 ^ { \circ }$. The point $E$ lies on the line through $B$ and $D$ and is such that the length of $D E$ is half that of $A C$. Find the coordinates of the two possible positions of $E$.
[0pt] [4 marks]

AQA C4 2015 June Q7

7 marks Standard +0.3

7 A curve has equation $y ^ { 3 } + 2 \mathrm { e } ^ { - 3 x } y - x = k$, where $k$ is a constant.
The point $P \left( \ln 2 , \frac { 1 } { 2 } \right)$ lies on this curve.

Show that the exact value of $k$ is $q - \ln 2$, where $q$ is a rational number.
Find the gradient of the curve at $P$.

AQA C4 2015 June Q8

12 marks Standard +0.3

A pond is initially empty and is then filled gradually with water. After $t$ minutes, the depth of the water, $x$ metres, satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { \sqrt { 4 + 5 x } } { 5 ( 1 + t ) ^ { 2 } }$$ Solve this differential equation to find $x$ in terms of $t$.
Another pond is gradually filling with water. After $t$ minutes, the surface of the water forms a circle of radius $r$ metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
1. Write down a differential equation, in the variables $r$ and $t$ and a constant of proportionality, which represents how the radius of the surface of the water is changing with time.
  (You are not required to solve your differential equation.)
2. When the radius of the pond is 1 metre, the radius is increasing at a rate of 4.5 metres per second. Find the radius of the pond when the radius is increasing at a rate of 0.5 metres per second.
  [0pt] [2 marks]
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-18_1277_1709_1430_153}
  
  \includegraphics[max width=\textwidth, alt={}]{fdd3905e-11f7-4b20-adfe-4c686018a221-20_2288_1707_221_153}

Edexcel C4 Q1

6 marks Standard +0.3

The function $f$ is given by

$$f ( x ) = \frac { 3 ( x + 1 ) } { ( x + 2 ) ( x - 1 ) } , x \in \mathbb { R } , x \neq - 2 , x \neq 1$$

Express $\mathrm { f } ( x )$ in partial fractions.
Hence, or otherwise, prove that $\mathrm { f } ^ { \prime } ( x ) < 0$ for all values of $x$ in the domain.

Edexcel C4 Q2

4 marks Moderate -0.3

2. The curve $C$ is described by the parametric equations $$x = 3 \cos t , \quad y = \cos 2 t , \quad 0 \leq t \leq \pi .$$

Find a cartesian equation of the curve $C$.
Draw a sketch of the curve $C$.

Edexcel C4 Q3

6 marks Standard +0.3

3. Use the substitution $x = \sin \theta$ to show that, for $| x | \leq 1$, $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { x } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } } + c \text {, where } c \text { is an arbitrary constant. }$$

Edexcel C4 Q4

6 marks Moderate -0.3

A measure of the effective voltage, $M$ volts, in an electrical circuit is given by

$$M ^ { 2 } = \int _ { 0 } ^ { 1 } V ^ { 2 } \mathrm {~d} t$$ where $V$ volts is the voltage at time $t$ seconds. Pairs of values of $V$ and $t$ are given in the following table.

$t$	0	0.25	0.5	0.75	1
$V$	- 48	207	37	- 161	- 29
$V ^ { 2 }$

Use the trapezium rule with five values of $V ^ { 2 }$ to estimate the value of $M$.
(6)

Edexcel C4 Q5

8 marks Standard +0.3

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{964070ca-a2c0-4935-8a5b-f1f656495f2e-3_771_1049_251_477}

\end{figure} Figure 1 shows part of the curve with equation $y = 1 + \frac { 1 } { 2 \sqrt { x } }$. The shaded region $R$, bounded by the curve, that $x$-axis and the lines $x = 1$ and $x = 4$, is rotated through $360 ^ { \circ }$ about the $x$-axis. Using integration, show that the volume of the solid generated is $\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)$.
(8)

Edexcel C4 Q6

11 marks Moderate -0.3

6. Liquid is poured into a container at a constant rate of $30 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$. At time $t$ seconds liquid is leaking from the container at a rate of $\frac { 2 } { 15 } V \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$, where $V \mathrm {~cm} ^ { 3 }$ is the volume of liquid in the container at that time.

Show that $$- 15 \frac { \mathrm {~d} V } { \mathrm {~d} t } = 2 V - 450$$ Given that $V = 1000$ when $t = 0$,
find the solution of the differential equation, in the form $V = \mathrm { f } ( t )$.
Find the limiting value of $V$ as $t \rightarrow \infty$.

Edexcel C4 Q7

11 marks Standard +0.3

7. The curve $C$ has equation $y = \frac { x } { 4 + x ^ { 2 } }$.

Use calculus to find the coordinates of the turning points of $C$. Using the result $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 x \left( x ^ { 2 } - 12 \right) } { \left( 4 + x ^ { 2 } \right) ^ { 3 } }$, or otherwise,
determine the nature of each of the turning points.
Sketch the curve $C$.

Edexcel C4 Q8

13 marks Standard +0.3

Given that $\cos ( x + 30 ) ^ { \circ } = 3 \cos ( x - 30 ) ^ { \circ }$, prove that tan $x ^ { \circ } = - \frac { \sqrt { 3 } } { 2 }$.
(a) Prove that $\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta$.
(b) Verify that $\theta = 180 ^ { \circ }$ is a solution of the equation $\sin 2 \theta = 2 - 2 \cos 2 \theta$.
(c) Using the result in part (a), or otherwise, find the other two solutions, $0 < \theta < 360 ^ { \circ }$, of the equation using $\sin 2 \theta = 2 - 2 \cos 2 \theta$.

Edexcel C4 Q9

14 marks Standard +0.8

9. The equations of the lines $l _ { 1 }$ and $l _ { 2 }$ are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where $\lambda$ and $\mu$ are parameters.

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect and find the coordinates of $Q$, their point of intersection.
Show that $l _ { 1 }$ is perpendicular to $l _ { 2 }$. The point $P$ with $x$-coordinate 3 lies on the line $l _ { 1 }$ and the point $R$ with $x$-coordinate 4 lies on the line $l _ { 2 }$.
Find, in its simplest form, the exact area of the triangle $P Q R$. END

Edexcel C4 Q1

6 marks Moderate -0.8

The following is a table of values for $y = \sqrt { } ( 1 + \sin x )$, where $x$ is in radians.

$x$	0	0.5	1	1.5	2
$y$	1	1.216	$p$	1.413	$q$

Find the value of $p$ and the value of $q$.
(2)
Use the trapezium rule and all the values of $y$ in the completed table to obtain an estimate of $I$, where $$I = \int _ { 0 } ^ { 2 } \sqrt { } ( 1 + \sin x ) \mathrm { d } x$$ (4)

Edexcel C4 Q2

7 marks Moderate -0.3

Use integration by parts to find $$\int x \cos 2 x d x$$
Prove that the answer to part (a) may be expressed as $$\frac { 1 } { 2 } \sin x ( 2 x \cos x - \sin x ) + C ,$$ where $C$ is an arbitrary constant.

Edexcel C4 Q3

8 marks Standard +0.3

Expand $( 1 + 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each term.
Hence, or otherwise, find the first three terms in the expansion of $\frac { x + 4 } { ( 1 + 3 x ) ^ { 2 } }$ as a series in ascending powers of $x$.

Edexcel C4 Q4

12 marks Standard +0.3

4. Relative to a fixed origin $O$, the point $A$ has position vector $4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }$, and the point $B$ has position vector $7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }$.

Find the vector $\overrightarrow { A B }$.
Calculate the cosine of $\angle O A B$.
Show that, for all values of $\lambda$, the point P with position vector $\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }$ lies on the line through $A$ and $B$.
Find the value of $\lambda$ for which $O P$ is perpendicular to $A B$.
Hence find the coordinates of the foot of the perpendicular from $O$ to $A B$.

Edexcel C4 Q5

11 marks Challenging +1.2

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-3_668_1172_1231_354}

\end{figure} Figure 1 shows a graph of $y = x \sqrt { } \sin x , 0 < x < \pi$. The maximum point on the curve is $A$.

Show that the $x$-coordinate of the point $A$ satisfies the equation $2 \tan x + x = 0$. The finite region enclosed by the curve and the $x$-axis is shaded as shown in Fig. 1.
A solid body $S$ is generated by rotating this region through $2 \pi$ radians about the $x$-axis.
Find the exact value of the volume of $S$.
(7)

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\(x\)	0	0.5	1	1.5	2
\(y\)	1	1.216	\(p\)	1.413	\(q\)

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AQA C4 2014 June Q5

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