Questions - AQA - A-Level Maths

Find the range of f.
The inverse of f is $\mathrm { f } ^ { - 1 }$.
1. Find $\mathrm { f } ^ { - 1 } ( x )$.
2. State the domain of $\mathrm { f } ^ { - 1 }$.
The composite function fg is denoted by h .
1. Find an expression for $\mathrm { h } ( x )$.
2. Solve the equation $\mathrm { h } ( x ) = 3$.

AQA C3 2009 June Q3

8 marks Moderate -0.3

Solve the equation $\tan x = - \frac { 1 } { 3 }$, giving all the values of $x$ in the interval $0 < x < 2 \pi$ in radians to two decimal places.
Show that the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ can be written in the form $3 \tan ^ { 2 } x - 5 \tan x - 2 = 0$.
Hence, or otherwise, solve the equation $$3 \sec ^ { 2 } x = 5 ( \tan x + 1 )$$ giving all the values of $x$ in the interval $0 < x < 2 \pi$ in radians to two decimal places.
(4 marks)

AQA C3 2009 June Q4

12 marks Standard +0.3

Sketch the graph of $y = \left| 50 - x ^ { 2 } \right|$, indicating the coordinates of the point where the graph crosses the $y$-axis.
Solve the equation $\left| 50 - x ^ { 2 } \right| = 14$.
Hence, or otherwise, solve the inequality $\left| 50 - x ^ { 2 } \right| > 14$.
Describe a sequence of two geometrical transformations that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = 50 - x ^ { 2 }$.

AQA C3 2009 June Q5

6 marks Standard +0.3

Given that $2 \ln x = 5$, find the exact value of $x$.
Solve the equation $$2 \ln x + \frac { 15 } { \ln x } = 11$$ giving your answers as exact values of $x$.

AQA C3 2009 June Q6

19 marks Standard +0.3

6 The diagram shows the curve with equation $y = \sqrt { 100 - 4 x ^ { 2 } }$, where $x \geqslant 0$.
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}

Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis, giving your answer in terms of $\pi$.
Use the mid-ordinate rule with five strips of equal width to find an estimate for $\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x$, giving your answer to three significant figures.
The point $P$ on the curve has coordinates $( 3,8 )$.
1. Find the gradient of the curve $y = \sqrt { 100 - 4 x ^ { 2 } }$ at the point $P$.
2. Hence show that the equation of the tangent to the curve at the point $P$ can be written as $2 y + 3 x = 25$.
The shaded regions on the diagram below are bounded by the curve, the tangent at $P$ and the coordinate axes.
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.

AQA C3 2009 June Q7

10 marks Standard +0.3

Use integration by parts to find $\int ( t - 1 ) \ln t \mathrm {~d} t$.
Use the substitution $t = 2 x + 1$ to show that $\int 4 x \ln ( 2 x + 1 ) \mathrm { d } x$ can be written as $\int ( t - 1 ) \ln t \mathrm {~d} t$.
Hence find the exact value of $\int _ { 0 } ^ { 1 } 4 x \ln ( 2 x + 1 ) \mathrm { d } x$.

AQA C3 2010 June Q1

5 marks Standard +0.3

1 The curve $y = 3 ^ { x }$ intersects the curve $y = 10 - x ^ { 3 }$ at the point where $x = \alpha$.

Show that $\alpha$ lies between 1 and 2 .
1. Show that the equation $3 ^ { x } = 10 - x ^ { 3 }$ can be rearranged into the form $$x = \sqrt [ 3 ] { 10 - 3 ^ { x } }$$
2. Use the iteration $x _ { n + 1 } = \sqrt [ 3 ] { 10 - 3 ^ { x _ { n } } }$ with $x _ { 1 } = 1$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.

AQA C3 2010 June Q2

10 marks Moderate -0.3

The diagram shows the graph of $y = \sec x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-2_816_1447_1087_287}
1. The point $A$ on the curve is where $x = 0$. State the $y$-coordinate of $A$.
2. Sketch, on the axes given on page 3, the graph of $y = | \sec 2 x |$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Solve the equation $\sec x = 2$, giving all values of $x$ in degrees in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
Solve the equation $\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2$, giving all values of $x$ in degrees in the interval $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-3_839_1475_317_351}

AQA C3 2010 June Q3

12 marks Moderate -0.3

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when:
1. $\quad y = \ln ( 5 x - 2 )$;
2. $y = \sin 2 x$.
The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \ln ( 5 x - 2 ) , & \text { for real values of } x \text { such that } x \geqslant \frac { 1 } { 2 } \\ \mathrm {~g} ( x ) = \sin 2 x , & \text { for real values of } x \text { in the interval } - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } \end{array}$$
1. Find the range of f .
2. Find an expression for $\operatorname { gf } ( x )$.
3. Solve the equation $\operatorname { gf } ( x ) = 0$.
4. The inverse of g is $\mathrm { g } ^ { - 1 }$. Find $\mathrm { g } ^ { - 1 } ( x )$.

AQA C3 2010 June Q4

8 marks Standard +0.3

Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x$, giving your answer to three significant figures.
Find the exact value of $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x$.

AQA C3 2010 June Q5

5 marks Moderate -0.3

Show that the equation $$10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x$$ can be written in the form $$10 \cot ^ { 2 } x + 11 \cot x - 6 = 0$$
Hence, given that $10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x$, find the possible values of $\tan x$.

AQA C3 2010 June Q6

9 marks Standard +0.3

6 The diagram shows the curve $y = \frac { \ln x } { x }$.
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-4_586_1034_1612_513} The curve crosses the $x$-axis at $A$ and has a stationary point at $B$.

State the coordinates of $A$.
Find the coordinates of the stationary point, $B$, of the curve, giving your answer in an exact form.
Find the exact value of the gradient of the normal to the curve at the point where $x = \mathrm { e } ^ { 3 }$.

AQA C3 2010 June Q7

11 marks Standard +0.3

Use integration by parts to find:
1. $\quad \int x \cos 4 x \mathrm {~d} x$;
  (4 marks)
2. $\int x ^ { 2 } \sin 4 x d x$.
  (4 marks)
The region bounded by the curve $y = 8 x \sqrt { ( \sin 4 x ) }$ and the lines $x = 0$ and $x = 0.2$ is rotated through $2 \pi$ radians about the $x$-axis. Find the value of the volume of the solid generated, giving your answer to three significant figures.
(3 marks)

AQA C3 2010 June Q8

15 marks Standard +0.3

8 The diagram shows the curves $y = \mathrm { e } ^ { 2 x } - 1$ and $y = 4 \mathrm { e } ^ { - 2 x } + 2$.
\includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-6_958_1492_372_242} The curve $y = 4 \mathrm { e } ^ { - 2 x } + 2$ crosses the $y$-axis at the point $A$ and the curves intersect at the point $B$.

Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { e } ^ { x }$ onto the graph of $y = \mathrm { e } ^ { 2 x } - 1$.
Write down the coordinates of the point $A$.
1. Show that the $x$-coordinate of the point $B$ satisfies the equation $$\left( \mathrm { e } ^ { 2 x } \right) ^ { 2 } - 3 \mathrm { e } ^ { 2 x } - 4 = 0$$
2. Hence find the exact value of the $x$-coordinate of the point $B$.
Find the exact value of the area of the shaded region bounded by the curves $y = \mathrm { e } ^ { 2 x } - 1$ and $y = 4 \mathrm { e } ^ { - 2 x } + 2$ and the $y$-axis.

AQA C3 2011 June Q1

7 marks Moderate -0.3

1 The diagram shows the curve with equation $y = \ln ( 6 x )$.
\includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-2_448_501_370_790}

State the $x$-coordinate of the point of intersection of the curve with the $x$-axis.
(1 mark)
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2 marks)
Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for $\int _ { 1 } ^ { 7 } \ln ( 6 x ) \mathrm { d } x$, giving your answer to three significant figures.

AQA C3 2011 June Q2

9 marks Moderate -0.3

1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $y = x \mathrm { e } ^ { 2 x }$.
2. Find an equation of the tangent to the curve $y = x \mathrm { e } ^ { 2 x }$ at the point $\left( 1 , \mathrm { e } ^ { 2 } \right)$.
Given that $y = \frac { 2 \sin 3 x } { 1 + \cos 3 x }$, use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 + \cos 3 x }$$ where $k$ is an integer.

AQA C3 2011 June Q3

5 marks Standard +0.3

3 The curve $y = \cos ^ { - 1 } ( 2 x - 1 )$ intersects the curve $y = \mathrm { e } ^ { x }$ at a single point where $x = \alpha$.

Show that $\alpha$ lies between 0.4 and 0.5 .
Show that the equation $\cos ^ { - 1 } ( 2 x - 1 ) = \mathrm { e } ^ { x }$ can be written as $x = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x } \right)$.
Use the iteration $x _ { n + 1 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x _ { n } } \right)$ with $x _ { 1 } = 0.4$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.

AQA C3 2011 June Q4

12 marks Standard +0.3

1. Solve the equation $\operatorname { cosec } \theta = - 4$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$, giving your answers to the nearest $0.1 ^ { \circ }$.
2. Solve the equation $$2 \cot ^ { 2 } \left( 2 x + 30 ^ { \circ } \right) = 2 - 7 \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)$$ for $0 ^ { \circ } < x < 180 ^ { \circ }$, giving your answers to the nearest $0.1 ^ { \circ }$.
Describe a sequence of two geometrical transformations that maps the graph of $y = \operatorname { cosec } x$ onto the graph of $y = \operatorname { cosec } \left( 2 x + 30 ^ { \circ } \right)$.

AQA C3 2011 June Q5

8 marks Moderate -0.8

5 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { 2 x + 1 } & \text { for real values of } x , \quad x \neq - 0.5 \end{array}$$

Explain why f does not have an inverse.
The inverse of g is $\mathrm { g } ^ { - 1 }$. Find $\mathrm { g } ^ { - 1 } ( x )$.
State the range of $\mathrm { g } ^ { - 1 }$.
Solve the equation $\mathrm { fg } ( x ) = \mathrm { g } ( x )$.

AQA C3 2011 June Q6

6 marks Standard +0.3

Given that $3 \ln x = 4$, find the exact value of $x$.
By forming a quadratic equation in $\ln x$, solve $3 \ln x + \frac { 20 } { \ln x } = 19$, giving your answers for $x$ in an exact form.

AQA C3 2011 June Q7

12 marks Standard +0.3

On separate diagrams:
1. sketch the curve with equation $y = | 3 x + 3 |$;
2. sketch the curve with equation $y = \left| x ^ { 2 } - 1 \right|$.
1. Solve the equation $| 3 x + 3 | = \left| x ^ { 2 } - 1 \right|$.
2. Hence solve the inequality $| 3 x + 3 | < \left| x ^ { 2 } - 1 \right|$.
  $8 \quad$ Use the substitution $u = 1 + 2 \tan x$ to find $$\int \frac { 1 } { ( 1 + 2 \tan x ) ^ { 2 } \cos ^ { 2 } x } d x$$

AQA C3 2011 June Q9

11 marks Standard +0.8

Use integration by parts to find $\int x \ln x \mathrm {~d} x$.
Given that $y = ( \ln x ) ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2 marks)
The diagram shows part of the curve with equation $y = \sqrt { x } \ln x$.
\includegraphics[max width=\textwidth, alt={}, center]{7148f43d-dc7d-43e2-b96e-ed1fb94073bf-5_406_645_696_719} The shaded region $R$ is bounded by the curve $y = \sqrt { x } \ln x$, the line $x = \mathrm { e }$ and the $x$-axis from $x = 1$ to $x = \mathrm { e }$. Find the volume of the solid generated when the region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis, giving your answer in an exact form.
(6 marks)

AQA C3 2012 June Q1

4 marks Moderate -0.3

1 Use the mid-ordinate rule with four strips to find an estimate for $\int _ { 0.4 } ^ { 1.2 } \cot \left( x ^ { 2 } \right) \mathrm { d } x$, giving your answer to three decimal places.

AQA C3 2012 June Q2

7 marks Standard +0.3

2 For $0 < x \leqslant 2$, the curves with equations $y = 4 \ln x$ and $y = \sqrt { x }$ intersect at a single point where $x = \alpha$.

Show that $\alpha$ lies between 0.5 and 1.5.
Show that the equation $4 \ln x = \sqrt { x }$ can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$$
Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \sqrt { x _ { n } } } { 4 } \right) }$$ with $x _ { 1 } = 0.5$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
Figure 1, on the page 3, shows a sketch of parts of the graphs of $y = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$ and $y = x$, and the position of $x _ { 1 }$. On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-3_1285_1543_356_296}
\end{figure}

AQA C3 2012 June Q3

7 marks Moderate -0.3

3 A curve has equation $y = x ^ { 3 } \ln x$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
1. Find an equation of the tangent to the curve $y = x ^ { 3 } \ln x$ at the point on the curve where $x = \mathrm { e }$.
2. This tangent intersects the $x$-axis at the point $A$. Find the exact value of the $x$-coordinate of the point $A$.

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