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AQA Further Paper 1 2021 June Q11

Find the acute angle between the lines $L _ { 1 }$ and $L _ { 2 }$, giving your answer to the nearest $0.1 ^ { \circ }$
11
The lines $L _ { 1 }$ and $L _ { 2 }$ lie in the plane $\Pi _ { 1 }$
11
1. Find the equation of $\Pi _ { 1 }$, giving your answer in the form r.n $= d$
  11
(ii) Hence find the shortest distance of the plane $\Pi _ { 1 }$ from the origin. 11
The points $A ( 4 , - 1 , - 1 ) , B ( 1,5 , - 7 )$ and $C ( 3,4 , - 8 )$ lie in the plane $\Pi _ { 2 }$
Find the angle between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer to the nearest $0.1 ^ { \circ }$

AQA Further Paper 1 2021 June Q12

4 marks

12 The matrix $\mathbf { A } = \left[ \begin{array} { c c c } 1 & 5 & 3
4 & - 2 & p
8 & 5 & - 11 \end{array} \right]$, where $p$ is a constant.
12

Given that $\mathbf { A }$ is a non-singular matrix, find $\mathbf { A } ^ { - 1 }$ in terms of $p$.
State any restrictions on the value of $p$.
12
The equations below represent three planes. $$\begin{aligned} x + 5 y + 3 z & = 5
4 x - 2 y + p z & = 24
8 x + 5 y - 11 z & = - 30 \end{aligned}$$ 12
1. Find, in terms of $p$, the coordinates of the point of intersection of the three planes.
  [0pt] [4 marks]
  12
(ii) In the case where $p = 2$, show that the planes are mutually perpendicular.

When the extension of string $A P$ is $e _ { A }$ metres, the tension in $A P$ is $24 m e _ { A }$
When the extension of string $B P$ is $e _ { B }$ metres, the tension in $B P$ is $10 m e _ { B }$
The natural length of string $A P$ is 1 metre
The natural length of string $B P$ is 1.3 metres
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-24_92_1082_1030_479}

Show that when $A P = 1.5$ metres, the tension in $A P$ is equal to the tension in $B P$.
15
$\quad P$ is held at the point between $A$ and $B$ where $A P = 1.9$ metres, and then released from rest. At time $t$ seconds after $P$ is released, $A P = ( 1.5 + x )$ metres.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-25_140_1068_493_484} Show that when $P$ is moving towards $A$, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 34 x = 6.664$$ 15
The container is then filled with oil, and $P$ is again released from rest at the point between $A$ and $B$ where $A P = 1.9$ metres. At time $t$ seconds after $P$ is released, the oil causes a resistive force of magnitude $10 m v$ newtons to act on the particle, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the particle. Find $x$ in terms of $t$ when $P$ is moving towards $A$.
\includegraphics[max width=\textwidth, alt={}, center]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-27_2492_1721_217_150}
\includegraphics[max width=\textwidth, alt={}]{8f7a5fc0-6936-4aed-a173-e221bf86e4fd-32_2486_1719_221_150}

AQA Further Paper 1 2022 June Q1

1 marks

1 The displacement of a particle from its equilibrium position is $x$ metres at time $t$ seconds. The motion of the particle obeys the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 9 x$$ Calculate the period of its motion in seconds.
Circle your answer.
[0pt] [1 mark]
$\frac { \pi } { 9 }$
$\frac { 2 \pi } { 9 }$
$\frac { \pi } { 3 }$
$\frac { 2 \pi } { 3 }$

AQA Further Paper 1 2022 June Q2

2 Simplify $$\frac { \cos \left( \frac { 6 \pi } { 13 } \right) + i \sin \left( \frac { 6 \pi } { 13 } \right) } { \cos \left( \frac { 2 \pi } { 13 } \right) - i \sin \left( \frac { 2 \pi } { 13 } \right) }$$ Tick ( $\checkmark$ ) one box. $$\begin{array} { l l } \cos \left( \frac { 8 \pi } { 13 } \right) + i \sin \left( \frac { 8 \pi } { 13 } \right) & \square
\cos \left( \frac { 8 \pi } { 13 } \right) - i \sin \left( \frac { 8 \pi } { 13 } \right) & \square
\cos \left( \frac { 4 \pi } { 13 } \right) + i \sin \left( \frac { 4 \pi } { 13 } \right) & \square
\cos \left( \frac { 4 \pi } { 13 } \right) - i \sin \left( \frac { 4 \pi } { 13 } \right) & \square \end{array}$$

AQA Further Paper 1 2022 June Q4

1 marks

4 The vector $\mathbf { v }$ is an eigenvector of the matrix $\mathbf { N }$ with corresponding eigenvalue 4
The vector $\mathbf { v }$ is also an eigenvector of the matrix $\mathbf { M }$ with corresponding eigenvalue 3
Given that $$\mathbf { N M } ^ { 2 } \mathbf { v } = \lambda \mathbf { v }$$ find the value of $\lambda$
Circle your answer.
[0pt] [1 mark]
102436144

AQA Further Paper 1 2022 June Q5

5 It is given that $z = - \frac { 3 } { 2 } + \mathrm { i } \frac { \sqrt { 11 } } { 2 }$ is a root of the equation $$z ^ { 4 } - 3 z ^ { 3 } - 5 z ^ { 2 } + k z + 40 = 0$$ where $k$ is a real number.
5

Find the other three roots.
5
Given that $x \in \mathbb { R }$, solve $$x ^ { 4 } - 3 x ^ { 3 } - 5 x ^ { 2 } + k x + 40 < 0$$

AQA Further Paper 1 2022 June Q6

Given that $| x | < 1$, prove that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$ 6
Solve the equation $$20 \operatorname { sech } ^ { 2 } x - 11 \tanh x = 16$$ Give your answer in logarithmic form.
$7 \quad$ The matrix $\mathbf { M }$ is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 7 & - 3
3 & 6 & k + 1
1 & 3 & 2 \end{array} \right]$$ where $k$ is a constant.

AQA Further Paper 1 2022 June Q7

1. Given that $\mathbf { M }$ is a non-singular matrix, find $\mathbf { M } ^ { - 1 }$ in terms of $k$
  7
(ii) State any restrictions on the value of $k$ 7
Using your answer to part (a)(i), solve $$\begin{array} { r } x + 7 y - 3 z = 6
3 x + 6 y + 6 z = 3
x + 3 y + 2 z = 1 \end{array}$$

AQA Further Paper 1 2022 June Q8

The complex number $w$ is such that $$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$ It is given that $w = x + \mathrm { i } y$, where $x$ and $y$ are real and $x > 0$
Find an equation for $y$ in terms of $x$
8
The complex number $z$ satisfies both $$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$ The region $R$ is the locus of $z$
Sketch the region $R$ on the Argand diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511} 8
$\quad z _ { 1 }$ is the point in $R$ at which $| z |$ is minimum. 8
1. Calculate the exact value of $\left| z _ { 1 } \right|$
  8
(ii) Express $z _ { 1 }$ in the form $a + \mathrm { i } b$, where $a$ and $b$ are real.

AQA Further Paper 1 2022 June Q9

4 marks

9 Roberto is solving this mathematics problem: The curve $C _ { 1 }$ has polar equation $$r ^ { 2 } = 9 \sin 2 \theta$$ for all possible values of $\theta$
Find the area enclosed by $C _ { 1 }$ Roberto's solution is as follows: $$\begin{aligned} A & = \frac { 1 } { 2 } \int _ { - \pi } ^ { \pi } 9 \sin 2 \theta \mathrm {~d} \theta
& = \left[ - \frac { 9 } { 4 } \cos 2 \theta \right] _ { - \pi } ^ { \pi }
& = 0 \end{aligned}$$ 9

$\quad$ Sketch the curve $C _ { 1 }$ 9
Explain what Roberto has done wrong.
9
$\quad$ Find the area enclosed by $C _ { 1 }$
9
$\quad P$ and $Q$ are distinct points on $C _ { 1 }$ for which $r$ is a maximum. $P$ is above the initial line. Find the polar coordinates of $P$ and $Q$
9
The matrix $\mathbf { M } = \left[ \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right]$ represents the transformation T T maps $C _ { 1 }$ onto a curve $C _ { 2 }$
9
1. T maps $P$ onto the point $P ^ { \prime }$
  Find the polar coordinates of $P ^ { \prime }$
  [0pt] [4 marks]
  9
(ii) Find the area enclosed by $C _ { 2 }$ Fully justify your answer.

AQA Further Paper 1 2022 June Q10

10 In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at $A ( 7,2 , - 3 )$ and the other end at $B ( 9 , - 3 , - 2 )$ A laser beam travels from $A$ to $B$ and continues in a straight line towards a large thin sheet of glass. The sheet of glass lies within a plane $\Pi _ { 1 }$ which is modelled by the equation $$4 x + p y + 5 z = 9$$ where $p$ is an integer.
10

The laser beam hits $\Pi _ { 1 }$ at an acute angle $\alpha$, where $\sin \alpha = \frac { \sqrt { 15 } } { 75 }$
Find the value of $p$
10
A second large sheet of glass lies on the other side of $\Pi _ { 1 }$ This second sheet lies within a plane $\Pi _ { 2 }$ which is modelled by the equation $$4 x + p y + 5 z = - 5$$ Calculate the distance between the sheets of glass.
10
The point $A ( 7,2 , - 3 )$ is reflected in $\Pi _ { 1 }$
Find the coordinates of the image of $A$ after reflection in $\Pi _ { 1 }$

AQA Further Paper 1 2022 June Q11

11 In this question use $g$ as $10 \mathrm {~m \mathrm {~s} ^ { - 2 }$} A smooth plane is inclined at $30 ^ { \circ }$ to the horizontal.
The fixed points $A$ and $B$ are 3.6 metres apart on the line of greatest slope of the plane, with $A$ higher than $B$ A particle $P$ of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points $A$ and $B$ respectively. The particle $P$ moves on a straight line that passes through $A$ and $B$
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-18_417_709_774_669} The natural length of the string $A P$ is 1.4 metres.
When the extension of the string $A P$ is $e _ { A }$ metres, the tension in the string $A P$ is $7 e _ { A }$ newtons.
The natural length of the string $B P$ is 1 metre.
When the extension of the string $B P$ is $e _ { B }$ metres, the tension in the string $B P$ is $9 e _ { B }$ newtons. The particle $P$ is held at the point between $A$ and $B$ which is 0.2 metres from its equilibrium position and lower than its equilibrium position.
The particle $P$ is then released from rest.
At time $t$ seconds after $P$ is released, its displacement towards $B$ from its equilibrium position is $x$ metres. 11

Show that during the subsequent motion the object satisfies the equation $$\ddot { x } + 50 x = 0$$ Fully justify your answer. 11
The experiment is repeated in a large tank of oil.
During the motion the oil causes a resistive force of $k v$ newtons to act on the particle, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the particle. The oil causes critical damping to occur.
11
1. Show that $k = \frac { 16 \sqrt { 2 } } { 5 }$
  11
(ii) Find $x$ in terms of $t$, giving your answer in exact form.
11
(iii) Calculate the maximum speed of the particle.

AQA Further Paper 1 2022 June Q12

4 marks

12 The Argand diagram shows the solutions to the equation $z ^ { 5 } = 1$
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-22_1079_995_354_520} 12

Solve the equation $$z ^ { 5 } = 1$$ giving your answers in the form $z = \cos \theta + \mathrm { i } \sin \theta$, where $0 \leq \theta < 2 \pi$
[0pt] [2 marks] 12
Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{a889963c-266c-497e-b7fc-99a249ba9e58-23_2484_1726_219_141}
12
The Argand diagram on page 22 is repeated below.
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-24_1079_1000_354_520} Explain, with reference to the Argand diagram, why the expression $$16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1$$ has a repeated quadratic factor.
12
$O$ is the centre of a regular pentagon $A B C D E$ such that $O A = O B = O C = O D =$ $O E = 1$ unit.
The distance from $O$ to $A B$ is $h$
By solving the equation $16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1 = 0$, show that $$h = \frac { \sqrt { 5 } + 1 } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-26_2492_1721_217_150}

AQA Further Paper 1 2023 June Q1

1 Find the number of solutions of the equation $\tanh x = \cosh x$
Circle your answer.
0
1

AQA Further Paper 1 2023 June Q2

1 marks

2
3 2 The diagram below shows a locus on an Argand diagram.
\includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-02_855_962_1085_539} Which of the equations below represents the locus shown above?
Circle your answer.
[0pt] [1 mark]
$| z - 2 + 3 \mathrm { i } | = 2$
$| z + 2 - 3 \mathrm { i } | = 2$
$| z - 2 + 3 \mathrm { i } | = 4$
$| z + 2 - 3 \mathrm { i } | = 4$

AQA Further Paper 1 2023 June Q3

3 The matrix $\mathbf { A } = \left[ \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right]$ represents a transformation.
Which one of the points below is an invariant point under this transformation?
Circle your answer.
$( 1,1 )$
$( 0,2 )$
$( 3,0 )$
$( 2,1 )$

AQA Further Paper 1 2023 June Q4

4 The solution of a second order differential equation is $\mathrm { f } ( t )$
The differential equation models heavy damping.
Which one of the statements below could be true?
Tick ( $\checkmark$ ) one box. $$\begin{aligned} & \mathrm { f } ( t ) = 2 \mathrm { e } ^ { - t } \cos ( 3 t ) + 5 \mathrm { e } ^ { - t } \sin ( 3 t )
& \mathrm { f } ( t ) = 3 \mathrm { e } ^ { - t } + 4 t \mathrm { e } ^ { - t }
& \mathrm { f } ( t ) = 7 \mathrm { e } ^ { - t } + 2 \mathrm { e } ^ { - 2 t }
& \mathrm { f } ( t ) = 8 \mathrm { e } ^ { - t } \cos ( 3 t - 0.1 ) \end{aligned}$$ □
□
□
□

AQA Further Paper 1 2023 June Q5

5 The function f is defined by $$f ( r ) = 2 ^ { r } ( r - 2 ) \quad ( r \in \mathbb { Z } )$$ 5

Show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r 2 ^ { r }$$ 5
Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 ^ { n + 1 } ( n - 1 ) + 2$$

AQA Further Paper 1 2023 June Q6

6 The matrix $\mathbf { M }$ is given by $$\mathbf { M } = \frac { 1 } { 10 } \left[ \begin{array} { c c c } a & a & - 6
0 & 10 & 0
9 & 14 & - 13 \end{array} \right]$$ where $a$ is a real number. The vectors $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$, and $\mathbf { v } _ { 3 }$ are eigenvectors of $\mathbf { M }$
The corresponding eigenvalues are $\lambda _ { 1 } , \lambda _ { 2 }$, and $\lambda _ { 3 }$ respectively.
It is given that $\lambda _ { 2 } = 1$ and $\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 1
0
3 \end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { l } 1
1
1 \end{array} \right]$ and $\mathbf { v } _ { 3 } = \left[ \begin{array} { l } c
0
1 \end{array} \right]$,
where $c$ is an integer. 6

1. Find the value of $\lambda _ { 1 }$
  6
(ii) Find the value of $a$
6
Find the integer $c$ and the value of $\lambda _ { 3 }$
6
Find matrices $\mathbf { U } , \mathbf { D }$ and $\mathbf { U } ^ { - 1 }$, such that $\mathbf { D }$ is diagonal and $\mathbf { M } = \mathbf { U D U } ^ { - 1 }$

AQA Further Paper 1 2023 June Q7

7 The function f is defined by $$f ( x ) = \left| \sin x + \frac { 1 } { 2 } \right| \quad ( 0 \leq x \leq 2 \pi )$$ Find the set of values of $x$ for which $$f ( x ) \geq \frac { 1 } { 2 }$$ Give your answer in set notation.

AQA Further Paper 1 2023 June Q8

8 The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { \sin x } \quad ( 0 \leq x \leq 2 \pi )$$ The diagram below shows the graph of $y = \mathrm { g } ( x )$
\includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-09_369_593_548_721} 8

Find the $x$-coordinate of each of the stationary points of the graph of $y = \mathrm { g } ( x )$, giving your answers in exact form. 8
Use Simpson's rule with 3 ordinates to estimate $$\int _ { 0 } ^ { \pi } g ( x ) d x$$ giving your answer to two decimal places.
8
Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b).

AQA Further Paper 1 2023 June Q9

9 The position vectors of the points $A , B$ and $C$ are $$\begin{aligned} & \mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }
& \mathbf { b } = - \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }
& \mathbf { c } = - 2 \mathbf { j } \end{aligned}$$ respectively.
9

Find the area of the triangle $A B C$
9
The points $A , B$ and $C$ all lie in the plane $\Pi$ Find an equation of the plane $\Pi$, in the form $\mathbf { r } \cdot \mathbf { n } = d$
$\mathbf { 9 ( c ) } \quad$ The point $P$ has position vector $\mathbf { p } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$
Find the exact distance of $P$ from $\Pi$

Questions — AQA Further Paper 1 (97 questions)

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AQA Further Paper 1 2021 June Q11

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