Questions - SPS SPS SM - A-Level Maths

SPS SPS SM 2025 February Q1

Given that $( x - 2 )$ is a factor of $2 x ^ { 3 } + k x - 4$, find the value of the constant $k$.
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(a)
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-06_412_919_187_694}

The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism $A B C D E F$. The base $A C F D$ of the roof is a horizontal rectangle, and the cross-section $A B C$ of the roof is an isosceles triangle with $A B = B C$. The lengths of $A C$ and $C F$ are $2 x \mathrm {~cm}$ and $y \mathrm {~cm}$ respectively, and the height of $B E$ above the base of the roof is $x \mathrm {~cm}$. The total surface area of the five faces of the roof is $600 \mathrm {~cm} ^ { 2 }$ and the volume of the roof is $V \mathrm {~cm} ^ { 3 }$. Show that $V = k x \left( 300 - x ^ { 2 } \right)$, where $k = \sqrt { a } + b$ and $\alpha$ and $b$ are integers to be determined.
(b) Use differentiation to determine the value of $x$ for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in $\mathrm { cm } ^ { 3 }$, correct to the nearest integer.
(d) Explain why, for this roof, $x$ must be less than a certain value, which you should state.
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SPS SPS SM 2025 February Q3

3.
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-08_469_471_130_877} The diagram shows a sector $A O B$ of a circle with centre $O$. The length of the $\operatorname { arc } A B$ is 6 cm and the area of the sector $A O B$ is $24 \mathrm {~cm} ^ { 2 }$. Find the area of the shaded segment enclosed by the $\operatorname { arc } A B$ and the chord $A B$, giving your answer correct to 3 significant figures.
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SPS SPS SM 2025 February Q4

4. (a) The number $K$ is defined by $K = n ^ { 3 } + 1$, where $n$ is an integer greater than 2 . Given that $n ^ { 3 } + 1 \equiv ( n + 1 ) \left( n ^ { 2 } + b n + c \right)$, find the constants $b$ and $c$.
(b) Prove that $K$ has at least two distinct factors other than 1 and $K$.
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SPS SPS SM 2025 February Q5

5. A curve has the following properties:

The gradient of the curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 x$.
The curve passes through the point $( 4 , - 13 )$.

Determine the coordinates of the points where the curve meets the line $y = 2 x$.
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SPS SPS SM 2025 February Q6

6. For all real values of $x$, the functions f and g are defined by $\mathrm { f } ( x ) = x ^ { 2 } + 8 a x + 4 a ^ { 2 }$ and $\mathrm { g } ( x ) = 6 x - 2 a$, where $a$ is a positive constant.

Find $\mathrm { fg } ( x )$. Determine the range of $\mathrm { fg } ( x )$ in terms of $a$.
If $f g ( 2 ) = 144$, find the value of $a$.
Determine whether the function fg has an inverse.
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SPS SPS SM 2025 February Q7

7. (a) Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$ (b) Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of $\theta$ between $0 ^ { \circ }$ and $180 ^ { \circ }$, correct to 1 decimal place.
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SPS SPS SM 2025 February Q8

8.
\includegraphics[max width=\textwidth, alt={}, center]{9eff9a1d-7d5c-4cee-87c9-8811dad16ffb-18_680_942_118_651} The diagram shows the curve with equation $y = 5 x ^ { 4 } + a x ^ { 3 } + b x$, where $a$ and $b$ are integers. The curve has a minimum at the point $P$ where $x = 2$. The shaded region is enclosed by the curve, the $x$-axis and the line $x = 2$. Given that the area of the shaded region is 48 units $^ { 2 }$, determine the $y$-coordinate of $P$.
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SPS SPS SM 2024 November Q3

3 Integrate with respect to $x$

$\quad \int \frac { 3 } { 2 } ( 2 x - 7 ) ^ { 5 } d x$
$\quad \int \frac { 3 x } { 2 } \left( 2 x ^ { 2 } - 7 \right) ^ { 5 } d x$
Express $\frac { 5 - x } { 1 - x - 2 x ^ { 2 } }$ in partial fractions.
Hence find the exact value of $\int _ { 1 } ^ { 2 } \frac { 5 - x } { 1 - x - 2 x ^ { 2 } } d x$
Using your answer to part a, find a quadratic approximation for the expression $\frac { 5 - x } { 1 - x - 2 x ^ { 2 } }$,
giving your answer in the form $p + q x + r x ^ { 2 }$, where $p , q$ and $r$ are constants to be found. The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } + 2 \cos x \text { for } - \pi \leq x \leq \pi$$ Determine whether the curve with equation $y = \mathrm { f } ( x )$ has a point of inflection at the point where $x = 0$ Fully justify your answer.
a) 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 $\cos \theta \neq 0$
b) 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 $5 \quad \theta = 0 ^ { \circ } , 104.5 ^ { \circ } , 255.5 ^ { \circ } , 360 ^ { \circ }$
Explain why the student should reject one of their values for $\cos \theta$ in Step 4 .
State the correct solutions to the equation $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3 \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$$ A particle moves in the $x - y$ plane so that at time $t$ seconds, where $t \geqslant 0$, its coordinates are given by $x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }$.
(a) Explain why the path of the particle never crosses the $x$-axis.
(b) Determine the exact values of $t$ when the path of the particle intersects the $y$-axis.
(c) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }$.
(d) Hence find the coordinates of the particle when its path is parallel to the $y$-axis. \section*{In this question you must show detailed reasoning.} (a) Express $\cos x + \sqrt { 3 } \sin x$ in the form $R \sin ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. Give the values of $R$ and $\alpha$ in exact form.
(b) Hence solve the equation $\cos x = \sqrt { 3 } ( 1 - \sin x )$ for values of $x$ in the interval $- \pi \leqslant x \leqslant \pi$. Give the roots of this equation in exact form. A curve has equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$ a) Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$ where $A$ and $B$ are constants to be found.
b) Hence show that the $x$ coordinates of the turning points of the curve are solutions of the equation $$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$ In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{2af0aec0-4e21-4050-b26f-52f2c9c68b51-17_732_472_233_778} The diagram shows the curve $y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }$, for $x \geqslant 0$, and the normal to the curve at the point $\left( \frac { 1 } { 4 } \pi , 0 \right)$. Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the $y$-axis is $\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }$.

SPS SPS SM 2025 October Q1

Express each of the following in the form $p x ^ { q }$, where $p$ and $q$ are constants.
1. $\frac { 2 } { \sqrt [ 4 ] { x } }$
2. $\quad ( 5 x \sqrt { x } ) ^ { 3 }$
3. $\sqrt { 2 x ^ { 3 } } \times \sqrt { 8 x ^ { 5 } }$
4. $\quad x ^ { 5 } \left( 27 x ^ { 6 } \right) ^ { \frac { 2 } { 3 } }$
5. In this question you must show detailed reasoning.
Simplify $10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }$, giving your answer in the form $a + b \sqrt { 5 }$.

SPS SPS SM 2025 October Q3

3. The line $l$ passes through the points $A ( - 3,0 )$ and $B \left( \frac { 5 } { 2 } , 22 \right)$

Find the equation of $l$ giving your answer in the form $y = m x + c$ where $m$ and $c$ are constants. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2fa9e78c-8210-456c-9b70-5378609ac47d-04_728_959_447_625} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the line $l$ and the curve $C$, which intersect at $A$ and $B$.
Given that
- $C$ has equation $y = 2 x ^ { 2 } + 5 x - 3$
- the region $R$, shown shaded in Figure 2, is bounded by $l$ and $C$
- use inequalities to define $R$.
  (2)
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SPS SPS SM 2025 October Q4

(a) A sequence has terms $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ defined by $u _ { 1 } = 3$ and $u _ { n + 1 } = u _ { n } ^ { 2 } - 5$ for $n \geqslant 1$.
1. Find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
2. Describe the behaviour of the sequence.
  (b) The second, third and fourth terms of a geometric progression are 12,8 and $\frac { 16 } { 3 }$. Determine the sum to infinity of this geometric progression.
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3. In this question you must show detailed reasoning.
  \includegraphics[max width=\textwidth, alt={}, center]{2fa9e78c-8210-456c-9b70-5378609ac47d-08_464_645_210_283}
The diagram shows the cuboid $A B C D E F G H$ where $A D = 3 \mathrm {~cm} , A F = ( 2 x + 1 ) \mathrm { cm }$ and $D C = ( x - 2 ) \mathrm { cm }$. The volume of the cuboid is at most $9 \mathrm {~cm} ^ { 3 }$.
Find the range of possible values of $x$. Give your answer in interval notation.
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SPS SPS SM 2025 October Q6

6. Sketch the graph of $$y = ( x - k ) ^ { 2 } ( x + 2 k )$$ where $k$ is a positive constant.
Label the coordinates of the points where the graph meets the axes.
\includegraphics[max width=\textwidth, alt={}, center]{2fa9e78c-8210-456c-9b70-5378609ac47d-10_1253_1207_596_395}
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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 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 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 SM 2021 January Q1

4 marks

1. The graph below shows the velocity of an object moving in a straight line over a 20 second journey.
\includegraphics[max width=\textwidth, alt={}, center]{82e828ab-0812-4de1-b066-720f3632046f-04_581_1678_395_191}

Find the maximum magnitude of the acceleration of the object.
The object is at its starting position at times $0 , t _ { 1 }$ and $t _ { 2 }$ seconds. Find $t _ { 1 }$ and $t _ { 2 }$
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SPS SPS SM 2021 January Q2

8 marks

2. In this question use $g = 9.8 \mathrm {~ms} ^ { - 2 }$
A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85

The boy applies a horizontal force of 150 N . Show that the crate remains stationary.
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Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N , at an angle of $15 ^ { \circ }$ above the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{82e828ab-0812-4de1-b066-720f3632046f-04_245_915_2161_575} Determine whether the crate remains stationary.
Fully justify your answer.
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Questions — SPS SPS SM (145 questions)

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