Questions - AQA C3 - A-Level Maths

The area of the shaded region, bounded by the curve, the $x$-axis and the lines $x = 0$ and $x = 4$, is given by $\int _ { 0 } ^ { 4 } \sqrt { 27 + x ^ { 3 } } \mathrm {~d} x$. Use the mid-ordinate rule with five strips to find an estimate for this area. Give your answer to three significant figures.
With the aid of a diagram, explain whether the mid-ordinate rule applied in part (a) gives an estimate which is smaller than or greater than the area of the shaded region.
(2 marks)

AQA C3 2013 June Q6

Sketch the graph of $y = \cos ^ { - 1 } x$, where $y$ is in radians. State the coordinates of the end points of the graph.
Sketch the graph of $y = \pi - \cos ^ { - 1 } x$, where $y$ is in radians. State the coordinates of the end points of the graph.
(2 marks)
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_759_1258_678_431}
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_751_1241_1564_443}

AQA C3 2013 June Q7

7 The diagram shows a sketch of the curve with equation $y = \mathrm { f } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

On Figure 1, below, sketch the curve with equation $y = - \mathrm { f } ( 3 x )$, indicating the values where the curve cuts the coordinate axes.
On Figure 2, on page 7, sketch the curve with equation $y = \mathrm { f } ( | x | )$, indicating the values where the curve cuts the coordinate axes.
Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
\end{figure}

AQA C3 2013 June Q8

8 The curve with equation $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }$, is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-08_630_1173_424_443}

The inverse of f is $\mathrm { f } ^ { - 1 }$.
1. Find $\mathrm { f } ^ { - 1 } ( x )$.
2. State the range of $\mathrm { f } ^ { - 1 }$.
3. Sketch, on the axes given on page 9 , the curve with equation $y = \mathrm { f } ^ { - 1 } ( x )$, indicating the value of the $y$-coordinate of the point where the curve intersects the $y$-axis.
  (2 marks)
The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } - 4 , \text { for all real values of } x$$
1. Find $\mathrm { gf } ( x )$, giving your answer in the form $( a x - b ) ^ { 2 } - c$, where $a , b$ and $c$ are integers.
2. Write down an expression for $\mathrm { fg } ( x )$, and hence find the exact solution of the equation $\operatorname { fg } ( x ) = \ln 5$.
  \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_1079_1422_233_358}

AQA C3 2013 June Q9

9 The shape of a vase can be modelled by rotating the curve with equation $16 x ^ { 2 } - ( y - 8 ) ^ { 2 } = 32$ between $y = 0$ and $y = 16$ completely about the $\boldsymbol { y }$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-09_890_1210_1555_424} The vase has a base.
Find the volume of water needed to fill the vase, giving your answer as an exact value.

AQA C3 2013 June Q10

1. By writing $\ln x$ as $( \ln x ) \times 1$, use integration by parts to find $\int \ln x \mathrm {~d} x$.
2. Find $\int ( \ln x ) ^ { 2 } \mathrm {~d} x$.
  (4 marks)
Use the substitution $u = \sqrt { x }$ to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)

AQA C3 2014 June Q1

4 marks

1 Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for $$\int _ { 0 } ^ { \pi } x ^ { \frac { 1 } { 2 } } \sin x d x$$ Give your answer to four significant figures.
[0pt] [4 marks]

AQA C3 2014 June Q2

6 marks

2 A curve has equation $y = 2 \ln ( 2 \mathrm { e } - x )$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find an equation of the normal to the curve $y = 2 \ln ( 2 \mathrm { e } - x )$ at the point on the curve where $x = \mathrm { e }$.
[0pt] [4 marks]
The curve $y = 2 \ln ( 2 \mathrm { e } - x )$ intersects the line $y = x$ at a single point, where $x = \alpha$.
1. Show that $\alpha$ lies between 1 and 3 .
2. Use the recurrence relation $$x _ { n + 1 } = 2 \ln \left( 2 \mathrm { e } - x _ { n } \right)$$ with $x _ { 1 } = 1$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
3. Figure 1, on the opposite page, shows a sketch of parts of the graphs of $y = 2 \ln ( 2 \mathrm { e } - x )$ 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.
  [0pt] [2 marks] \section*{(c)(iii)} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-05_864_1284_1802_386}
  \end{figure}

AQA C3 2014 June Q3

5 marks

1. Differentiate $\left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }$ with respect to $x$.
2. Given that $y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } + 1 \right) ^ { \frac { 5 } { 2 } }$, find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 0$.
A curve has equation $y = \frac { 4 x - 3 } { x ^ { 2 } + 1 }$. Use the quotient rule to find the $x$-coordinates of the stationary points of the curve.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-06_1855_1709_852_153}

AQA C3 2014 June Q4

4 The sketch shows part of the curve with equation $y = \mathrm { f } ( x )$.
\includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}

On Figure 2 below, sketch the curve with equation $y = - | \mathrm { f } ( x ) |$.
On Figure 3 on the page opposite, sketch the curve with equation $y = \mathrm { f } ( | 2 x | )$.
1. Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } ( 2 x + 2 )$.
2. Find the coordinates of the image of the point $P ( 4 , - 3 )$ under the sequence of transformations given in part (c)(i). \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
  \end{figure}

AQA C3 2014 June Q5

5 The functions $f$ and $g$ are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } - 6 x + 5 , & \text { for } x \geqslant 3
\mathrm {~g} ( x ) = | x - 6 | , & \text { for all real values of } x \end{array}$$

Find the range of f.
The inverse of f is $\mathrm { f } ^ { - 1 }$. Find $\mathrm { f } ^ { - 1 } ( x )$. Give your answer in its simplest form.
1. Find $\mathrm { gf } ( x )$.
2. Solve the equation $\operatorname { gf } ( x ) = 6$.

AQA C3 2014 June Q6

3 marks

By using integration by parts twice, find $$\int x ^ { 2 } \sin 2 x d x$$
A curve has equation $y = x \sqrt { \sin 2 x }$, for $0 \leqslant x \leqslant \frac { \pi } { 2 }$. The region bounded by the curve and the $x$-axis is rotated through $2 \pi$ radians about the $x$-axis to generate a solid. Find the exact value of the volume of the solid generated.
[0pt] [3 marks]

AQA C3 2014 June Q7

6 marks

7 Use the substitution $u = 3 - x ^ { 3 }$ to find the exact value of $\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 3 - x ^ { 3 } } \mathrm {~d} x$.
[0pt] [6 marks]

AQA C3 2014 June Q8

12 marks

Show that the expression $\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x }$ can be written as $2 \sec x$.
[0pt] [4 marks]
Hence solve the equation $$\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x } = \tan ^ { 2 } x - 2$$ giving the values of $x$ to the nearest degree in the interval $0 ^ { \circ } \leqslant x < 360 ^ { \circ }$.
[0pt] [6 marks]
Hence solve the equation $$\frac { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } { \cos \left( 2 \theta - 30 ^ { \circ } \right) } + \frac { \cos \left( 2 \theta - 30 ^ { \circ } \right) } { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } = \tan ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) - 2$$ giving the values of $\theta$ to the nearest degree in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-16_1517_1709_1190_153}

\includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-20_2489_1730_221_139}

AQA C3 2016 June Q1

Given that $y = ( 4 x + 1 ) ^ { 3 } \sin 2 x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Given that $y = \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 }$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p x } { \left( 3 x ^ { 2 } + 4 \right) ^ { 2 } }$, where $p$ is a constant.
Given that $y = \ln \left( \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 } \right)$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.

AQA C3 2016 June Q2

2 The curve with equation $y = x ^ { x }$, where $x > 0$, intersects the line $y = 5$ at a single point, where $x = \alpha$.

Show that $\alpha$ lies between 2 and 3 .
Show that the equation $x ^ { x } = 5$ can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \ln 5 } { x } \right) }$$
Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \ln 5 } { x _ { n } } \right) }$$ with $x _ { 1 } = 2$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $$\int _ { 0.5 } ^ { 1.7 } \left( 5 - x ^ { x } \right) \mathrm { d } x$$ giving your answer to three significant figures.
2. Hence find an approximation to $\int _ { 0.5 } ^ { 1.7 } x ^ { x } \mathrm {~d} x$.

AQA C3 2016 June Q3

5 marks

3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]

AQA C3 2016 June Q4

6 marks

Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { e } ^ { x }$ onto the graph of $y = \mathrm { e } ^ { 2 x - 5 }$.
The normal to the curve $y = \mathrm { e } ^ { 2 x - 5 }$ at the point $P \left( 2 , \mathrm { e } ^ { - 1 } \right)$ intersects the $x$-axis at the point $A$ and the $y$-axis at the point $B$. Show that the area of the triangle $O A B$ is $\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }$, where $m$ and $n$ are integers.
[0pt] [6 marks]

AQA C3 2016 June Q5

5 The function f is defined by $$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$ The graph of $y = \mathrm { f } ( x )$ is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}

Find the range of f.
The composite function fg is defined by $$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$ Find an expression for $\operatorname { gg } ( x )$.

AQA C3 2016 June Q6

7 marks

Use integration by parts to find $\int \frac { \ln ( 3 x ) } { x ^ { 2 } } \mathrm {~d} x$.
The region bounded by the curve $y = \frac { \ln ( 3 x ) } { x }$, the $x$-axis from $\frac { 1 } { 3 }$ to 1 , and the line $x = 1$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid. Find the exact value of the volume of the solid generated.
[0pt] [7 marks]

AQA C3 2016 June Q7

6 marks

By writing $\sec x = ( \cos x ) ^ { - 1 }$, use the chain rule to show that, if $y = \sec x$, then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the $y$-coordinate of the stationary point of the graph of $y = \mathrm { f } ( x )$, giving your answer in the form $p \sqrt { q }$, where $p$ and $q$ are integers.
[0pt] [6 marks]

AQA C3 2016 June Q8

8 Use the substitution $u = 4 x - 1$ to find the exact value of $$\int _ { \frac { 1 } { 4 } } ^ { \frac { 1 } { 2 } } ( 5 - 2 x ) ( 4 x - 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$$

\includegraphics[max width=\textwidth, alt={}]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-18_2104_1712_603_153}

AQA C3 2016 June Q9

3 marks

It is given that $\sec x - \tan x = - 5$.
1. Show that $\sec x + \tan x = - 0.2$.
2. Hence find the exact value of $\cos x$.
Hence solve the equation $$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$ giving all values of $x$, to one decimal place, in the interval $- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }$.
[0pt] [3 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA C3 Q2

2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.

AQA C3 Q5

5 The diagram shows part of the graph of $y = \mathrm { e } ^ { 2 x } - 9$. The graph cuts the coordinate axes at ( $0 , a$ ) and ( $b , 0$ ).
\includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}

State the value of $a$, and show that $b = \ln 3$.
Show that $y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81$.
The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume of the solid formed, giving your answer in the form $\pi ( p \ln 3 + q )$, where $p$ and $q$ are integers.
Sketch the curve with equation $y = \left| \mathrm { e } ^ { 2 x } - 9 \right|$ for $x \geqslant 0$.

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