Questions - SPS - A-Level Maths

SPS SPS SM 2025 October Q7

7. In this question you must show detailed reasoning. Solve the following equations.

$\quad y ^ { 6 } + 7 y ^ { 3 } - 8 = 0$
$\quad 9 ^ { x + 1 } + 3 ^ { x } = 8$
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SPS SPS SM 2025 October Q8

8. In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.

$2 \log _ { 3 } ( x + 1 ) = 1 + \log _ { 3 } ( x + 7 )$
$\log _ { y } \left( \frac { 1 } { 8 } \right) = - \frac { 3 } { 2 }$
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SPS SPS SM 2025 October Q9

9.

Show that the equation $x ^ { 2 } + k x - k ^ { 2 } = 0$ has real roots for all real values of $k$.
Show that the roots of the equation $x ^ { 2 } + k x - k ^ { 2 } = 0$ are $\left( \frac { - 1 \pm \sqrt { 5 } } { 2 } \right) k$.
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SPS SPS SM 2025 October Q10

10. $f ( x ) = x ^ { 4 } + b x + c$
$( x - 2 )$ is a factor of $f ( x )$.
$f ( - 3 ) = 35$.

Find b and c.
Hence express $\mathrm { f } ( \mathrm { x } )$ as the product of linear and cubic factors.
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SPS SPS SM 2025 October Q11

11. A student dissolves 0.5 kg of salt in a bucket of water. Water leaks out of a hole in the bucket so the student lets fresh water flow in so that the bucket stays full. They assume that the salty water remaining in the bucket mixes with the fresh water that flows in, so the concentration of salt is uniform throughout the bucket. They model the mass $M \mathrm {~kg}$ of salt remaining after $t$ minutes by $M = a k ^ { t }$ where $a$ and $k$ are constants.

Show that the model for $M$ can be rewritten in the form $\log _ { 10 } M = t \log _ { 10 } k + \log _ { 10 } a$. The student measures the concentration of salt in the bucket at certain times to estimate the mass of the salt remaining. The results are shown in the table below.
$t$ minutes 8 13 21 35 50
$M \mathrm {~kg}$ 0.4 0.3 0.2 0.1 0.05
The student uses this data and plots $y = \log _ { 10 } M$ against $x = t$ using graph drawing software. The software gives $y = - 0.0214 x - 0.2403$ for the equation of the line of best fit.
1. Find the values of $a$ and $k$ that follow from the equation of the line.
2. Interpret the value of $k$ in context.
It is known that when $t = 0$ the mass of salt in the bucket is 0.5 kg . Comment on the accuracy when the model is used to estimate the initial mass of the salt.
Use the model to predict the value of $t$ at which $M = 0.01 \mathrm {~kg}$.
Rewrite the model for $M$ in the form $M = a \mathrm { e } ^ { - h t }$ where $h$ is a constant to be determined.
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SPS SPS SM 2025 October Q12

12. An arithmetic progression has first term $a$ and common difference $d$, where $a$ and $d$ are non-zero. The first, third and fourth terms of the arithmetic progression are consecutive terms of a geometric progression with common ratio $r$.

1. Show that $r = \frac { a + 2 d } { a }$.
2. Find $d$ in terms of $a$.
Find the common ratio of the geometric progression.
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SPS SPS SM 2025 October Q13

13. The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 4 y + 1 = 0$$

Find
1. the coordinates of the centre of $C$
2. the exact radius of $C$ The line with equation $y = k$, where $k$ is a constant, cuts $C$ at two distinct points.
Find the range of values for $k$, giving your answer in set notation.
The line with equation $y = m x + 4$ is a tangent to C . Find possible exact values of m .
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SPS SPS FM 2026 November Q4

4. (a) The curves $\mathrm { e } ^ { x } - 2 \mathrm { e } ^ { y } = 1$ and $2 \mathrm { e } ^ { x } + 3 \mathrm { e } ^ { 2 y } = 41$ intersect at the point $P$. Show that the $y$-coordinate of $P$ satisfies the equation $3 \mathrm { e } ^ { 2 y } + 4 \mathrm { e } ^ { y } - 39 = 0$.
(b) In this question you must show detailed reasoning. Hence find the exact coordinates of $P$.
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SPS SPS FM 2026 November Q8

5 marks

8. Prove by induction that $7 \times 9 ^ { n } - 15$ is divisible by 4 , for all integers $n \geq 0$.
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SPS SPS FM 2026 November Q10

10. (a) Find the first 4 terms, in ascending powers of $x$, in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where $k$ is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of $( 1 + k x ) ^ { 10 }$ the coefficient $x ^ { 3 }$ is 3 times the coefficient of $x$, (b) find the possible values of $k$.
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SPS SPS SM 2025 November Q3

3. The equation $k x ^ { 2 } + 4 x + ( 5 - k ) = 0$, where k is a constant, has 2 different real solutions for $x$. Find the set of possible values of $k$.
Write your answer using set notation.

SPS SPS SM 2025 November Q6

6. A sequence $t _ { 1 } , t _ { 2 } , t _ { 3 } , t _ { 4 } , t _ { 5 } , \ldots$ is given by $$t _ { n + 1 } = a t _ { n } + 3 n + 2 , \quad t \in \mathbb { N } , \quad t _ { 1 } = - 2 ,$$ where $a$ is a non zero constant.
a) Given that $\sum _ { r = 1 } ^ { 3 } \left( r ^ { 3 } + t _ { r } \right) = 12$, determine the possible values of $a$.
b) Evaluate $\sum _ { r = 8 } ^ { 31 } \left( t _ { r + 1 } - a t _ { r } \right)$.

SPS SPS SM 2025 November Q8

8. The circles $C _ { 1 }$ and $C _ { 2 }$ have respective equations $$\begin{aligned} & x ^ { 2 } + y ^ { 2 } - 6 x - 2 y = 15
& x ^ { 2 } + y ^ { 2 } - 18 x + 14 y = 95 \end{aligned}$$ a) By considering the coordinates of the centres and the lengths of the radii of $C _ { 1 }$ and $C _ { 2 }$, show that $C _ { 1 }$ and $C _ { 2 }$ touch internally at some point $P$.
b) Determine the coordinates of $P$.
c) Find the equation of the common tangent to the circles at P .

SPS SPS FM Pure 2026 January Q1

1
6 \end{array} \right) + \lambda \left( \begin{array} { r }

SPS SPS FM Pure 2026 January Q2

2
- 3
1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point $P$ lies on $l _ { 1 }$. Given that $\overrightarrow { O P }$ is perpendicular to $l _ { 1 }$, calculate the coordinates of $P$.
(ii) Relative to a fixed origin $O$, the line $l _ { 2 }$ is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }

SPS SPS FM Pure 2026 January Q4

4
- 3
12 \end{array} \right) + \mu \left( \begin{array} { r }

SPS SPS FM Pure 2026 January Q6

6 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 3
1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point $P$ lies on $l _ { 1 }$. Given that $\overrightarrow { O P }$ is perpendicular to $l _ { 1 }$, calculate the coordinates of $P$.
(ii) Relative to a fixed origin $O$, the line $l _ { 2 }$ is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4
- 3

SPS SPS FM Pure 2026 January Q12

3 marks

12 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 3
4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point $A$ does not lie on $l _ { 2 }$. Given that the vector $\overrightarrow { O A }$ is parallel to the line $l _ { 2 }$ and $| \overrightarrow { O A } | = \sqrt { 2 }$ units, calculate the possible position vectors of the point $A$.
[0pt] [BLANK PAGE] Q7.
The simultaneous equations $$\begin{aligned} & 2 x - y = 1
& 3 x + k y = b \end{aligned}$$ are represented by the matrix equation $\mathbf { M } \binom { x } { y } = \binom { 1 } { b }$.

State the value of $k$ for which $\mathbf { M } ^ { - 1 }$ does not exist and find $\mathbf { M } ^ { - 1 }$ in terms of $k$ when $\mathbf { M } ^ { - 1 }$ exists.
Use $\mathbf { M } ^ { - 1 }$ to solve the simultaneous equations when $k = 5$ and $b = 21$.
The two equations can be interpreted as representing two lines in the $x - y$ plane. Describe the relationship between these two lines
(A) when $k = 5$ and $b = 21$,
(B) when $k = - \frac { 3 } { 2 }$ and $b = 1$,
(C) when $k = - \frac { 3 } { 2 }$ and $b = \frac { 3 } { 2 }$.
[0pt] [BLANK PAGE] Q8.
You are given the matrix $\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6
0.6 & - 0.8 \end{array} \right)$.
Calculate $\mathbf { M } ^ { 2 }$. You are now given that the matrix $\mathbf { M }$ represents a reflection in a line through the origin.
Explain how your answer to part (i) relates to this information.
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By investigating the invariant points of the reflection, find the equation of the mirror line. [3]
Describe fully the transformation represented by the matrix $\mathbf { P } = \left( \begin{array} { r r } 0.8 & - 0.6
0.6 & 0.8 \end{array} \right)$.
A composite transformation is formed by the transformation represented by $\mathbf { P }$ followed by the transformation represented by $\mathbf { M }$. Find the single matrix that represents this composite transformation.
The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?
[0pt] [BLANK PAGE] Q9.
(a) Two points, $A$ and $B$, on an Argand diagram are represented by the complex numbers $2 + 3 \mathrm { i }$ and $- 4 - 5 \mathrm { i }$ respectively. Given that the points $A$ and $B$ are at the ends of a diameter of a circle $C _ { 1 }$, express the equation of $C _ { 1 }$ in the form $\left| z - z _ { 0 } \right| = k$.
(4 marks)
(b) A second circle, $C _ { 2 }$, is represented on the Argand diagram by the equation $| z - 5 + 4 i | = 4$. Sketch on one Argand diagram both $C _ { 1 }$ and $C _ { 2 }$.
(3 marks)
(c) The points representing the complex numbers $z _ { 1 }$ and $z _ { 2 }$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively and are such that $\left| z _ { 1 } - z _ { 2 } \right|$ has its maximum value. Find this maximum value, giving your answer in the form $a + b \sqrt { 5 }$.
(5 marks)
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SPS SPS FM Pure 2025 September Q2

2. Prove by induction that $11 \times 7 ^ { n } - 13 ^ { n } - 1$ is divisible by 3 , for all integers $n \geq 0$.
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SPS SPS FM Pure 2025 September Q4

6 marks

4. The curve $C$ has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$. The point $P$ lies on $C$ where $t = \frac { 2 \pi } { 3 }$
The line $l$ is the normal to $C$ at $P$.
Show that an equation for $l$ is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line $l$ intersects the curve $C$ again at the point $Q$.
Find the exact coordinates of $Q$. You must show clearly how you obtained your answers.
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On the Argand diagram below, sketch the locus, $L$, of points satisfying the equation $$\arg ( z + \mathrm { i } ) = \frac { \pi } { 6 }$$ [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{75e73922-4324-441c-b942-c709a71d9025-12_1310_1354_513_463}
$\quad z _ { 1 }$ is a point on $L$ such that $| z |$ is a minimum. Find the exact value of $z _ { 1 }$ in the form $a + b \mathrm { i }$
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SPS SPS FM Pure 2025 September Q7

8 marks

7. (a) Prove the identity $\frac { \cos x } { \sec x + 1 } + \frac { \cos x } { \sec x - 1 } \equiv 2 \cot ^ { 2 } x$
[0pt] [3 marks]
(b) Hence, solve the equation $$\frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) + 1 } = \cot \left( 2 \theta + \frac { \pi } { 3 } \right) - \frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) - 1 }$$ in the interval $0 \leq \theta \leq 2 \pi$, giving your values of $\theta$ to three significant figures where appropriate.
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SPS SPS FM Pure 2025 September Q8

8. A population of meerkats is being studied. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where $P$, in thousands, is the population of meerkats and $t$ is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double.
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SPS SPS FM Pure 2025 September Q9

6 marks

9. A curve $C$ has equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$

1. Show that the equation of the normal to $C$ at the point where $C$ crosses the $y$-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
The equation $\mathrm { f } ( x ) = 0$ has one positive root, $\alpha$.
1. Show that $\alpha$ lies between 2 and 3 Fully justify your answer.
2. Show that the roots of $\mathrm { f } ( x ) = 0$ satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with $x _ { 1 } = 2$ to find the values of $x _ { 2 }$ and $x _ { 3 }$ giving your answers to three decimal places.
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4. Figure 1 below shows a sketch of the graphs of $y = e - e ^ { - \frac { x } { 2 } }$ 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] \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{75e73922-4324-441c-b942-c709a71d9025-22_1236_1566_1519_360}
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SPS SPS SM 2025 November Q2

2 Express $y = 2 \sin 2 x - 3 \cos 2 x$ in the form $y = R \sin ( 2 x - \alpha )$,
where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$ In this question you must show all of your algebraic steps clearly. $$f ( x ) = \frac { 1 } { \sqrt { 1 + 2 x } }$$

Expand $f ( x )$ in accending powers of $x$ up to and including the term in $x ^ { 3 }$.
Hence, show that $\frac { 2 - 5 x } { \sqrt { 1 + 2 x } } \approx 2 - 7 x + A x ^ { 2 } + B x ^ { 3 }$, where $A$ and $B$ are constants to be found.
State the set of values of $x$ for which the expansion in part (ii) is valid.

SPS SPS FM Mechanics 2026 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b3e54459-0d05-4858-8978-60fe3d4d1719-04_501_693_242_635} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform L-shaped lamina $O A B C D E$, shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and $a$ metres wide. Giving each answer in terms of $a$, find the distance of the centre of mass of the lamina from

$O E$,
$O A$. The lamina is freely suspended from $O$ and hangs in equilibrium with $O E$ at an angle $\theta$ to the downward vertical through $O$, where $\tan \theta = \frac { 4 } { 3 }$.
Find the value of $a$.
[0pt] [Question 2 Continued]

Questions — SPS (1106 questions)

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SPS SPS SM 2025 October Q7

SPS SPS SM 2025 October Q8

SPS SPS SM 2025 October Q9

SPS SPS SM 2025 October Q10

SPS SPS SM 2025 October Q11

SPS SPS SM 2025 October Q12

SPS SPS SM 2025 October Q13

SPS SPS FM 2026 November Q4

SPS SPS FM 2026 November Q8

SPS SPS FM 2026 November Q10

SPS SPS SM 2025 November Q3

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SPS SPS SM 2025 November Q8

SPS SPS FM Pure 2026 January Q1

SPS SPS FM Pure 2026 January Q2

SPS SPS FM Pure 2026 January Q4

SPS SPS FM Pure 2026 January Q6

SPS SPS FM Pure 2026 January Q12

SPS SPS FM Pure 2025 September Q2

SPS SPS FM Pure 2025 September Q4

SPS SPS FM Pure 2025 September Q7

SPS SPS FM Pure 2025 September Q8

SPS SPS FM Pure 2025 September Q9

SPS SPS SM 2025 November Q2

SPS SPS FM Mechanics 2026 January Q2

\(t\) minutes	8	13	21	35	50
\(M \mathrm {~kg}\)	0.4	0.3	0.2	0.1	0.05