Questions - AQA Paper 1 - A-Level Maths

It is given that $$P ( x ) = 4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4$$ Use the factor theorem to show that $( 2 x + 1 )$ is a factor of $\mathrm { P } ( x )$
13
Express $\mathrm { P } ( x )$ in the form $$\mathrm { P } ( x ) = ( 2 x + 1 ) \left( a x ^ { 2 } + b x + c \right)$$ where $a$, $b$ and $c$ are constants to be found.
13
Given that $n$ is a positive integer, use your answer to part (b) to explain why $4 n ^ { 3 } + 8 n ^ { 2 } + 11 n + 4$ is never prime.
[0pt] [2 marks]

AQA Paper 1 2024 June Q14

2 marks

The equation $$x ^ { 3 } = \mathrm { e } ^ { 6 - 2 x }$$ has a single solution, $x = \alpha$
By considering a suitable change of sign, show that $\alpha$ lies between 0 and 4
14
Show that the equation $x ^ { 3 } = \mathrm { e } ^ { 6 - 2 x }$ can be rearranged to give $$x = 3 - \frac { 3 } { 2 } \ln x$$ 14
1. Use the iterative formula $$x _ { n + 1 } = 3 - \frac { 3 } { 2 } \ln x _ { n }$$ with $x _ { 1 } = 4$, to find $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$
  Give your answers to three decimal places.
  14
(ii) Figure 1 below shows a sketch of parts of the graphs of $$y = 3 - \frac { 3 } { 2 } \ln x \text { and } y = x$$ On Figure 1, draw a staircase or cobweb diagram to show how convergence takes place.
Label, on the $x$-axis, the positions of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$
[0pt] [2 marks] \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-22_1328_1390_744_395}
\end{figure} 14
(iii) Explain why the iterative formula $$x _ { n + 1 } = 3 - \frac { 3 } { 2 } \ln x _ { n }$$ fails to converge to $\alpha$ when the starting value is $x _ { 1 } = 0$

AQA Paper 1 2024 June Q15

4 marks

Show that the expression $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta$$ can be written as $$4 \cos \theta - \sec \theta$$ where $\sin \theta \neq 0$ and $\cos \theta \neq 0$
[0pt] [4 marks]
15
A student is attempting to solve the equation $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3 \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$$ They use the result from part (a), and write the following incorrect solution: $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3$$ Step $1 \quad 4 \cos \theta - \sec \theta = 3$
Step $24 \cos \theta - \frac { 1 } { \cos \theta } - 3 = 0$
Step $34 \cos ^ { 2 } \theta - 3 \cos \theta - 1 = 0$ Step $4 \cos \theta = 1$ or $\cos \theta = - 0.25$ Step $5 \theta = 0 ^ { \circ } , 104.5 ^ { \circ } , 255.5 ^ { \circ } , 360 ^ { \circ }$ 15
1. Explain why the student should reject one of their values for $\cos \theta$ in Step 4. 15
(ii) State the correct solutions to the equation $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3 \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$$ Figure 2 below shows a 1.5 metre length of pipe. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-26_335_693_502_740}
\end{figure} The symmetrical cross-section of the pipe is shown below, in Figure 3, where $x$ and $y$ are measured in centimetres. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-26_652_734_1247_717}
\end{figure} Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe.
$\boldsymbol { x }$ 0 0.4 0.8 1.2 1.6 2
$\boldsymbol { y }$ - 3 - 2.943 - 2.752 - 2.353 - 1.572 0

AQA Paper 1 2024 June Q17

2 marks

17 The function f is defined by $$\mathrm { f } ( x ) = | x | + 1 \text { for } x \in \mathbb { R }$$ The function g is defined by $$g ( x ) = \ln x$$ where g has its greatest possible domain. 17

Using set notation, state the range of f 17
State the domain of g
17
The composite function h is given by $$\mathrm { h } ( x ) = \operatorname { gf } ( x ) \text { for } x \in \mathbb { R }$$ 17
1. Write down an expression for $\mathrm { h } ( x )$ in terms of $x$
  17
(ii) Determine if h has an inverse. Fully justify your answer.
[0pt] [2 marks]

AQA Paper 1 2024 June Q18

Use a suitable substitution to show that $$\int _ { 0 } ^ { 4 } ( 4 x + 1 ) ( 2 x + 1 ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$ can be written as $$\frac { 1 } { 2 } \int _ { a } ^ { 9 } \left( 2 u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$ where $a$ is a constant to be found.
18
Hence, or otherwise, show that $$\int _ { 0 } ^ { 4 } ( 4 x + 1 ) ( 2 x + 1 ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1322 } { 15 }$$ 18
A graph has the equation $$y = ( 4 x + 1 ) \sqrt { 2 x + 1 }$$ A student uses four rectangles to approximate the area under the graph between the lines $x = 0$ and $x = 4$ The rectangles are all the same width.
All the rectangles are drawn under the curve as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-32_1031_698_744_735} The total area of the four rectangles is $A$ The student decides to improve their approximation by increasing the number of rectangles used. Explain why the value of the student's improved approximation will be greater than $A$, but less than $\frac { 1322 } { 15 }$

AQA Paper 1 2024 June Q19

19 A curve has equation $$y ^ { 3 } \mathrm { e } ^ { 2 x } + 2 y - 16 x = k$$ where $k$ is a constant. The curve has a stationary point on the $y$-axis.
Determine the value of $k$
2 A gardener stores rainwater in a cylindrical container. The container has a height of 130 centimetres.
The gardener empties the water from the container through a hose.
The hose is attached 5 centimetres from the bottom of the container.
At time $t$ minutes after the hose is switched on, the depth of water, $h$ centimetres, in the container decreases at a rate which is proportional to $h - 5$ Initially the container of water is full, and the depth of water is decreasing at a rate of 1.5 centimetres per minute.

AQA Paper 1 2024 June Q20

5 marks

Show that $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.012 ( h - 5 )$$ 20
Solve the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.012 ( h - 5 )$$ to find an expression for $h$ in terms of $t$
[0pt] [5 marks]
20
Find the time taken for the container to be half empty. Give your answer to the nearest minute.
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-38_2495_1915_169_123} Question number Additional page, if required. Write the question numbers in the left-hand margin.

AQA Paper 1 Specimen Q1

1 marks

1 Find the gradient of the line with equation $2 x + 5 y = 7$
Circle your answer.
[0pt] [1 mark] $$\begin{array} { l l l l } \frac { 2 } { 5 } & \frac { 5 } { 2 } & - \frac { 2 } { 5 } & - \frac { 5 } { 2 } \end{array}$$

AQA Paper 1 Specimen Q2

1 marks

2 A curve has equation $y = \frac { 2 } { \sqrt { x } }$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
Circle your answer.
[0pt] [1 mark] $$\frac { \sqrt { x } } { 3 } \quad \frac { 1 } { x \sqrt { x } } \quad - \frac { 1 } { x \sqrt { x } } \quad - \frac { 1 } { 2 x \sqrt { x } }$$

AQA Paper 1 Specimen Q3

3 marks

3 When $\theta$ is small, find an approximation for $\cos 3 \theta + \theta \sin 2 \theta$, giving your answer in the form $a + b \theta ^ { 2 }$
[0pt] [3 marks]

AQA Paper 1 Specimen Q4

6 marks

Use the factor theorem to prove that $x + 3$ is a factor of $\mathrm { p } ( x )$
[0pt] [2 marks] 4
Simplify the expression $\frac { 2 x ^ { 3 } + 7 x ^ { 2 } + 2 x - 3 } { 4 x ^ { 2 } - 1 } , x \neq \pm \frac { 1 } { 2 }$
[0pt] [4 marks]

AQA Paper 1 Specimen Q5

8 marks

5 The diagram shows a sector $A O B$ of a circle with centre $O$ and radius $r \mathrm {~cm}$.
\includegraphics[max width=\textwidth, alt={}, center]{66b1c52c-0475-493a-8c0a-abf4c2c7b07d-06_414_584_516_863} The angle $A O B$ is $\theta$ radians
The sector has area $9 \mathrm {~cm} ^ { 2 }$ and perimeter 15 cm . 5

Show that $r$ satisfies the equation $2 r ^ { 2 } - 15 r + 18 = 0$
[0pt] [4 marks]
5
Find the value of $\theta$. Explain why it is the only possible value.
[0pt] [4 marks]

AQA Paper 1 Specimen Q6

6 Sam goes on a diet. He assumes that his mass, $m \mathrm {~kg}$ after $t$ days, decreases at a rate that is inversely proportional to the cube root of his mass. 6

Construct a differential equation involving $m$, $t$ and a positive constant $k$ to model this situation.
6
Explain why Sam's assumption may not be appropriate.

AQA Paper 1 Specimen Q7

4 marks

7 Find the values of $k$ for which the equation $( 2 k - 3 ) x ^ { 2 } - k x + ( k - 1 ) = 0$ has equal roots.
[0pt] [4 marks]

AQA Paper 1 Specimen Q8

6 marks

Given that $u = 2 ^ { x }$, write down an expression for $\frac { \mathrm { d } u } { \mathrm {~d} x }$
8
Find the exact value of $\int _ { 0 } ^ { 1 } 2 ^ { x } \sqrt { 3 + 2 ^ { x } } \mathrm {~d} x$ Fully justify your answer.
[0pt] [6 marks]
Turn over for the next question

AQA Paper 1 Specimen Q9

8 marks

9 A curve has equation $y = \frac { 2 x + 3 } { 4 x ^ { 2 } + 7 }$ 9

1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
  [0pt] [2 marks]
  L
  L
  LL
  LU
  L
  LL 9
(ii) Hence show that $y$ is increasing when $4 x ^ { 2 } + 12 x - 7 < 0$
[0pt] [4 marks]
9
Find the values of $x$ for which $y$ is increasing.
[0pt] [2 marks] Turn over for the next question

AQA Paper 1 Specimen Q10

10 marks

10 The function f is defined by $$\mathrm { f } ( x ) = 4 + 3 ^ { - x } , x \in \mathbb { R }$$ 10

Using set notation, state the range of f
[0pt] [2 marks]
10
The inverse of f is $\mathrm { f } ^ { - 1 }$ 10
1. Using set notation, state the domain of $f ^ { - 1 }$
  [0pt] [1 mark]
  10
(ii) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$
[0pt] [3 marks]
10
The function $g$ is defined by $$g ( x ) = 5 - \sqrt { x } , ( x \in \mathbb { R } : x > 0 )$$ 10
1. Find an expression for gf (x)
  [0pt] [1 mark]
  10
(ii) Solve the equation $\mathrm { gf } ( x ) = 2$, giving your answer in an exact form.
[0pt] [3 marks]

AQA Paper 1 Specimen Q11

8 marks

11 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } + 8 x - 12 y = 12$ 11

Find the coordinates of $C$ and the radius of the circle.
[0pt] [3 marks] 11
The points $P$ and $Q$ lie on the circle.
The origin is the midpoint of the chord $P Q$.
Show that $P Q$ has length $n \sqrt { 3 }$, where $n$ is an integer.
[0pt] [5 marks]

AQA Paper 1 Specimen Q12

12 A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation $x ^ { 2 } + 2 x y + 2 y ^ { 2 } = 10$, where $x$ and $y$ are measured in metres. The $x$ and $y$ axes are horizontal and vertical respectively.
\includegraphics[max width=\textwidth, alt={}, center]{66b1c52c-0475-493a-8c0a-abf4c2c7b07d-18_214_451_776_858} Find the maximum vertical height above the platform of the sculpture.

AQA Paper 1 Specimen Q14

10 marks

14 An open-topped fish tank is to be made for an aquarium.
It will have a square horizontal base, rectangular vertical sides and a volume of $60 \mathrm {~m} ^ { 3 }$
The materials cost:

$\pounds 15$ per $\mathrm { m } ^ { 2 }$ for the base
$\pounds 8$ per $\mathrm { m } ^ { 2 }$ for the sides.

Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer.
[0pt] [8 marks] 14
1. In reality, the thickness of the base and sides of the tank is 2.5 cm
  Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank of the tank.
  [0pt] [1 mark]
  LIH
  L
  LL 14
(ii) How would your refinement affect your answer to part (a)?
[0pt] [1 mark]

AQA Paper 1 Specimen Q15

8 marks

15 The height $x$ metres, of a column of water in a fountain display satisfies the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 8 \sin 2 t } { 3 \sqrt { x } }$, where $t$ is the time in seconds after the display begins. 15

Solve the differential equation, given that initially the column of water has zero height.
Express your answer in the form $x = \mathrm { f } ( t )$
[0pt] [7 marks]
15
Find the maximum height of the column of water, giving your answer to the nearest cm .
[0pt] [1 mark]

AQA Paper 1 Specimen Q16

10 marks

16 A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number. 16

Identify the rational number for which the student's argument is not true. 16
Prove that the student is right for all rational numbers other than the one you have identified in part (a).
[0pt] [4 marks]
$17 \quad \mathrm { f } ( x ) = \sin x$
Using differentiation from first principles find the exact value of $f ^ { \prime } \left( \frac { \pi } { 6 } \right)$
Fully justify your answer.
[0pt] [6 marks] \section*{DO NOT WRITE ON THIS PAGE} ANSWER IN THE SPACES PROVIDED

\(\boldsymbol { x }\)	0	0.4	0.8	1.2	1.6	2
\(\boldsymbol { y }\)	- 3	- 2.943	- 2.752	- 2.353	- 1.572	0

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