Questions - SPS - A-Level Maths

SPS SPS SM Pure 2024 September Q4

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. Solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ (Total for Question 4 is 4 marks)

SPS SPS SM Pure 2024 September Q5

5. Find the coefficient of the term in $x ^ { 7 }$ of the binomial expansion of $$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.
(3) \section*{(Total for Question 5 is 3 marks)}

SPS SPS SM Pure 2024 September Q6

The curve $C _ { 1 }$ has equation $y = \mathrm { f } ( x )$.

Given that

$\mathrm { f } ( x )$ is a quadratic expression
$C _ { 1 }$ has a maximum turning point at $( 2,20 )$
$C _ { 1 }$ passes through the origin
1. sketch a graph of $C _ { 1 }$ showing the coordinates of any points where $C _ { 1 }$ cuts the coordinate axes,
2. find an expression for $\mathrm { f } ( x )$.

The curve $C _ { 2 }$ has equation $y = x \left( x ^ { 2 } - 4 \right)$
Curve $C _ { 1 }$ and $C _ { 2 }$ meet at the origin, and at the points $P$ and $Q$
Given that the $x$ coordinate of the point $P$ is negative,

using algebra and showing all stages of your working, find the coordinates of $P$
(3)

SPS SPS SM Pure 2024 September Q7

7. The circle $C$

has centre $A ( 3,5 )$
passes through the point $B ( 8 , - 7 )$

The points $M$ and $N$ lie on $C$ such that $M N$ is a chord of $C$.
Given that $M N$

lies above the $x$-axis
is parallel to the $x$-axis
has length $4 \sqrt { 22 }$

Find an equation for the line passing through points $M$ and $N$.
(5)
(Total for Question 7 is 5 marks)

SPS SPS SM Pure 2024 September Q8

8. (a) Sketch the curve with equation $$y = a ^ { - x } + 4$$ where $a$ is a constant and $a > 1$
On your sketch show

the coordinates of the point of intersection of the curve with the $y$-axis
the equation of any asymptotes to the curve.
(3)
(b) Use the trapezium rule with 5 trapeziums to find an approximate value for

$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(3)
(c) Using the answer to part (b), find an approximate value for

$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x$
$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x$

SPS SPS SM Pure 2024 September Q9

9. The sum to infinity of the geometric series $$a + a r + a r ^ { 2 } + \ldots$$ is 10 .
The sum to infinity of the series formed by the squares of the terms is 100/9.
a) Show that $r = 4 / 5$ and find $a$.
b) Find the sum to infinity of the series formed by the cubes of the terms. \section*{(Total for Question 9 is 5 marks)}

SPS SPS SM Pure 2024 September Q10

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Given that $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ \section*{(Total for Question 10 is 6 marks)}

SPS SPS SM Pure 2024 September Q11

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

Solve, for $0 \leq x < 360 ^ { \circ }$, the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-24_641_732_778_680} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where $A$ is a constant and $\theta$ is measured in radians.
The points $P , Q$ and $R$ lie on the curve and are shown in Figure 1.
Given that the $y$ coordinate of $P$ is 7
(a) state the value of $A$,
(b) find the exact coordinates of $Q$,
(c) find the value of $\theta$ at $R$, giving your answer to 3 significant figures.

SPS SPS SM Pure 2024 September Q12

12. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-26_390_630_351_721} \captionsetup{labelformat=empty} \caption{Diagram NOT accurately drawn}

\end{figure} The diagram shows a quadrilateral $O A C B$ in which $$\overrightarrow { O A } = 4 \mathbf { a } \quad \overrightarrow { O B } = 3 \mathbf { b } \quad \overrightarrow { B C } = 2 \mathbf { a } + \mathbf { b }$$ The point $P$ lies on $A C$ such that $A P : P C = 3 : 2$
The point $Q$ is such that $O P Q$ and $B C Q$ are straight lines.
Using a vector method, find $\overrightarrow { O Q }$ in terms of $\mathbf { a }$ and $\mathbf { b }$
Give your answer in its simplest form.
Show your working clearly.

SPS SPS SM Pure 2024 September Q13

13. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-28_633_725_475_676} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point $P$ is the only stationary point on the curve.
The line $l$ passes through the point $P$ and is parallel to the $x$-axis.
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line $l$ and the line with equation $x = 4$ Use algebraic integration to find the exact area of $R$.
(8) \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET }

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 ASFM Statistics 2025 January Q1

$\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c$,
if $X$ and $Y$ are independent then $\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )$.

\section*{Non-parametric tests} Goodness-of-fit test and contingency tables: $\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } } \sim \chi _ { v } ^ { 2 }$
Approximate distributions for large samples
Wilcoxon Signed Rank test: $T \sim \mathrm {~N} \left( \frac { 1 } { 4 } n ( n + 1 ) , \frac { 1 } { 24 } n ( n + 1 ) ( 2 n + 1 ) \right)$
Wilcoxon Rank Sum test (samples of sizes $m$ and $n$, with $m \leq n$ ): $$W \sim \mathrm {~N} \left( \frac { 1 } { 2 } m ( m + n + 1 ) , \frac { 1 } { 12 } m n ( m + n + 1 ) \right)$$ \section*{Discrete distributions} $X$ is a random variable taking values $x _ { i }$ in a discrete distribution with $\mathrm { P } \left( X = x _ { i } \right) = p _ { i }$
Expectation: $\mu = \mathrm { E } ( X ) = \sum x _ { i } p _ { i }$
Variance: $\sigma ^ { 2 } = \operatorname { Var } ( X ) = \sum \left( x _ { i } - \mu \right) ^ { 2 } p _ { i } = \sum x _ { i } ^ { 2 } p _ { i } - \mu ^ { 2 }$ $n = 8 \quad \sum p = 28.5 \quad \sum q = 26.7 \quad \sum p ^ { 2 } = 136.35 \quad \sum q ^ { 2 } = 116.35 \quad \sum p q = 116.70$
\includegraphics[max width=\textwidth, alt={}, center]{76f751ed-394d-41cb-b98f-bc8efcf3365e-08_705_1164_1139_267}

State which, if either, of the variables $p$ and $q$ is independent.
Calculate the equation of the regression line of $q$ on $p$.
1. Use the regression line to estimate the value of $q$ for an investment account for which $p = 2.5$.
2. Give two reasons why this estimate could be considered reliable.
Comment on the reliability of using the regression line to predict the value of $q$ when $p = 7.0$. Total: $\_\_\_\_$ / 9 marks \section*{Question 4} After a holiday organised for a group, the company organising the holiday obtained scores out of 10 for six different aspects of the holiday. The company obtained responses from 100 couples and 100 single travellers. The total scores for each of the aspects are given in the following table. After further investigation, the statistician decides to use a different model for the distribution of $F$. In this model it is now assumed that $\mathrm { P } ( F = 0 )$ is still 0.200 , but that if one failure occurs, there is an increased probability that further failures occur.
Explain the effect of this assumption on the value of $\mathrm { P } ( F = 1 )$. Total: $\_\_\_\_$ / 10 marks \section*{Question 6} In a fashion competition, two judges gave marks to a large number of contestants.
The value of Spearman's rank correlation coefficient, $r _ { s }$, between the marks given to 7 randomly chosen contestants is $\frac { 27 } { 28 }$.
An excerpt from the table of critical values of $r _ { s }$ is shown below. \section*{Critical values of Spearman's rank correlation coefficient}
1-tail test 5\% 2.5\% 1\% 0.5\%
2-tail test 10\% 5\% 2\% 1\%
\multirow{3}{*}{$n$} 6 0.8286 0.8857 0.9429 1.0000
7 0.7143 0.7857 0.8929 0.9286
8 0.6429 0.7381 0.8333 0.8810
Test whether there is evidence, at the $1 \%$ significance level, that the judges agree with each another. The marks given by the two judges to the 7 randomly chosen contestants were as follows, where $x$ is an integer.
Contestant $A$ $B$ $C$ $D$ $E$ $F$ $G$
Judge 1 64 65 67 78 79 80 86
Judge 2 61 63 78 80 81 90 $x$
Use the value $r _ { s } = \frac { 27 } { 28 }$ to determine the range of possible values of $x$.
Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context. Total: $\_\_\_\_$ / 9 marks \section*{Question 7} A bag contains $2 m$ yellow and $m$ green counters. Three counters are chosen at random, without replacement. The probability that exactly two of the three counters are yellow is $\frac { 28 } { 55 }$. Determine the value of $m$. Total: $\_\_\_\_$ End of Paper

SPS SPS FM Statistics 2025 January Q1

6 marks

$\mathrm { E } ( a X + b Y + c ) = a \mathrm { E } ( X ) + b \mathrm { E } ( Y ) + c$,
if $X$ and $Y$ are independent then $\operatorname { Var } ( a X + b Y + c ) = a ^ { 2 } \operatorname { Var } ( X ) + b ^ { 2 } \operatorname { Var } ( Y )$.

\section*{Non-parametric tests} Goodness-of-fit test and contingency tables: $\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } } \sim \chi _ { v } ^ { 2 }$
Approximate distributions for large samples
Wilcoxon Signed Rank test: $T \sim \mathrm {~N} \left( \frac { 1 } { 4 } n ( n + 1 ) , \frac { 1 } { 24 } n ( n + 1 ) ( 2 n + 1 ) \right)$
Wilcoxon Rank Sum test (samples of sizes $m$ and $n$, with $m \leq n$ ): $$W \sim \mathrm {~N} \left( \frac { 1 } { 2 } m ( m + n + 1 ) , \frac { 1 } { 12 } m n ( m + n + 1 ) \right)$$ \section*{Percentage points of the normal distribution} If $Z$ has a normal distribution with mean 0 and variance 1 then, for each value of $p$, the table gives the value of $z$ such that $P ( Z \leq z ) = p$. Test whether there is evidence, at the $1 \%$ significance level, that the judges agree with each another. The marks given by the two judges to the 7 randomly chosen contestants were as follows, where $x$ is an integer.

Contestant	$A$	$B$	$C$	$D$	$E$	$F$	$G$
Judge 1	64	65	67	78	79	80	86
Judge 2	61	63	78	80	81	90	$x$

(b) Use the value $r _ { s } = \frac { 27 } { 28 }$ to determine the range of possible values of $x$.
(c) Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context. Total: $\_\_\_\_$ / 9 marks \section*{Question 7} A circle, centre $O$, has radius $x \mathrm {~cm}$, where $x$ is an observation from the random variable $X$ which has a uniform distribution on $[ 0 , \pi ]$
(a) Find the probability that the area of the circle is greater than $10 \mathrm {~cm} ^ { 2 }$
(b) State, giving a reason, whether the median area of the circle is greater or less than $10 \mathrm {~cm} ^ { 2 }$ The triangle $O A B$ is drawn inside the circle with $O A$ and $O B$ as radii of length $x \mathrm {~cm}$ and angle $A O B$ is $x$ radians.
(c) Use algebraic integration to find the expected value of the area of triangle $O A B$. Give your answer as an exact value. Detailed working is required and calculator integration is not allowed.
[0pt] [6] Total: $\_\_\_\_$ / 10 marks End of Paper

SPS SPS FM Mechanics 2025 January Q1

1. A smooth uniform sphere $A$, of mass $5 m$ and radius $r$, is at rest on a smooth horizontal plane. A second smooth uniform sphere $B$, of mass $3 m$ and radius $r$, is moving in a straight line on the plane with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes $A$. Immediately before the impact the direction of motion of $B$ makes an angle of $60 ^ { \circ }$ with the line of centres of the spheres. The direction of motion of $B$ is turned through an angle of $30 ^ { \circ }$ by the impact. Find

the speed of $B$ immediately after the impact,
the coefficient of restitution between the spheres.

SPS SPS FM Mechanics 2025 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7ecb33af-c165-4b72-98f2-7574fad3fdd5-04_506_613_246_671} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $2 a$ and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point $A$. The particle moves with constant speed $v$ in a horizontal circle with centre $O$, where $O$ is vertically below $A$ and $O A = 2 a$, as shown in Figure 2.

Show that the extension in the string is $\frac { 2 } { 5 } a$.
Find $v ^ { 2 }$ in terms of $a$ and $g$.
[0pt] [Question 2 Continued]

SPS SPS FM Mechanics 2025 January Q3

3. A particle $P$ of mass $2 m$ is moving in a straight line with speed $3 u$ on a smooth horizontal table. A second particle $Q$ of mass $3 m$ is moving in the opposite direction to $P$ along the same straight line with speed $u$. The particle $P$ collides directly with $Q$. The direction of motion of $P$ is reversed by the collision. The coefficient of restitution between $P$ and $Q$ is $e$.

Show that the speed of $Q$ immediately after the collision is $\frac { u } { 5 } ( 8 e + 3 )$
Find the range of possible values of $e$. The total kinetic energy of the particles before the collision is $T$. The total kinetic energy of the particles after the collision is $k T$. Given that $e = \frac { 1 } { 2 }$
find the value of $k$.
[0pt] [Question 3 Continued]

SPS SPS FM Mechanics 2025 January Q4

4. One end $A$ of a light elastic string $A B$, of modulus of elasticity $m g$ and natural length $a$, is fixed to a point on a rough plane inclined at an angle $\theta$ to the horizontal. The other end $B$ of the string is attached to a particle of mass $m$ which is held at rest on the plane. The string $A B$ lies along a line of greatest slope of the plane, with $B$ lower than $A$ and $A B = a$. The coefficient of friction between the particle and the plane is $\mu$, where $\mu < \tan \theta$. The particle is released from rest.

Show that when the particle comes to rest it has moved a distance $2 a ( \sin \theta - \mu \cos \theta )$ down the plane.
Given that there is no further motion, show that $\mu \geqslant \frac { 1 } { 3 } \tan \theta$.
[0pt] [Question 4 Continued]

SPS SPS FM Mechanics 2025 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7ecb33af-c165-4b72-98f2-7574fad3fdd5-10_881_1301_173_397} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform solid right circular cylinder has height $h$ and radius $r$. The centre of one plane face is $O$ and the centre of the other plane face is $Y$. A cylindrical hole is made by removing a solid cylinder of radius $\frac { 1 } { 4 } r$ and height $\frac { 1 } { 4 } h$ from the end with centre $O$. The axis of the cylinder removed is parallel to $O Y$ and meets the end with centre $O$ at $X$, where $O X = \frac { 1 } { 4 } r$. One plane face of the cylinder removed coincides with the plane face through $O$ of the original cylinder. The resulting solid $S$ is shown in Figure 3.

Show that the centre of mass of $S$ is at a distance $\frac { 85 h } { 168 }$ from the plane face
containing $O$. containing $O$. The solid $S$ is freely suspended from $O$. In equilibrium the line $O Y$ is inclined at an angle $\arctan ( 17 )$ to the horizontal.
Find $r$ in terms of $h$.
[0pt] [Question 5 Continued]
[0pt] [Question 5 Continued]

SPS SPS SM Statistics 2025 January Q1

Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.

$A$ is the event that Isobel's parents watch a match.
$B$ is the event that Isobel scores in a match.

You are given that $\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }$ and $\mathrm { P } ( A ) = \frac { 7 } { 10 }$.

Calculate $\mathrm { P } ( A \cap B )$. The probability that Isobel does not score and her parents do not attend is 0.1 .
Draw a Venn diagram showing the events $A$ and $B$, and mark in the probability corresponding to each of the regions of your diagram.
Are events $A$ and $B$ independent? Give a reason for your answer.
By comparing $\mathrm { P } ( B \backslash A )$ with $\mathrm { P } ( B )$, explain why Isobel should ask her parents not to attend.
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q2

2. On average, $25 \%$ of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by $X$.

State two conditions needed for $X$ to be modelled by the distribution $\mathrm { B } ( 12,0.25 )$. In the rest of this question you should assume that these conditions are satisfied.
Find $\mathrm { P } ( X \leqslant 6 )$. In order to claim a free gift, 7 vouchers are needed.
Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q3

3. The continuous random variable $T$ has mean $\mu$ and standard deviation $\sigma$. It is known that $\mathrm { P } ( T < 140 ) = 0.01$ and $\mathrm { P } ( T < 300 ) = 0.8$.

Assuming that $T$ is normally distributed, calculate the values of $\mu$ and $\sigma$. In fact, $T$ represents the time, in minutes, taken by a randomly chosen runner in a public marathon, in which about $10 \%$ of runners took longer than 400 minutes.
State with a reason whether the mean of $T$ would be higher than, equal to, or lower than the value calculated in part (i).
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q4

4. The table shows information about the time, $t$ minutes correct to the nearest minute, taken by 50 people to complete a race.

Time (minutes)	$t \leqslant 27$	$28 \leqslant t \leqslant 30$	$31 \leqslant t \leqslant 35$	$36 \leqslant t \leqslant 45$	$46 \leqslant t \leqslant 60$	$t \geqslant 61$
Number of people	0	4	28	14	4	0

In a histogram illustrating the data, the height of the block for the $31 \leqslant t \leqslant 35$ class is 5.6 cm . Find the height of the block for the $28 \leqslant t \leqslant 30$ class. (There is no need to draw the histogram.)
The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
Calculate estimates of the mean and standard deviation of the data.
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q5

5. The masses, $m$ grams, of 52 apples of a certain variety were found and summarised as follows. $$n = 52 \quad \Sigma ( m - 150 ) = - 182 \quad \Sigma ( m - 150 ) ^ { 2 } = 1768$$ Calculate the variance and thus the exact value of $\sum m ^ { 2 }$
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q6

6. A television company believes that the proportion of households that can receive Channel C is 0.35 .

In a random sample of 14 households it is found that 2 can receive Channel C. Test, at the $2.5 \%$ significance level, whether there is evidence that the proportion of households that can receive Channel C is less than 0.35 .
On another occasion the test is carried out again, with the same hypotheses and significance level as in part (i), but using a new sample, of size $n$. It is found that no members of the sample can receive Channel C. Find the largest value of $n$ for which the null hypothesis is not rejected. Show all relevant working.
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 January Q7

7. The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities.

Local Authority	Underground, metro, light rail or tram	Train	Bus	Drive	Walk or cycle
A	0.3\%	4.5\%	17\%	52.8\%	11\%
B	0.2\%	1.7\%	1.7\%	63.4\%	11\%
C	35.2\%	3.0\%	12\%	11.7\%	16\%
D	8.9\%	1.4\%	9\%	54.7\%	10\%

One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.

Which one of the Local Authorities is a London borough? Give a reason for your answer.
Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
[0pt] [BLANK PAGE]

Time (minutes)	\(t \leqslant 27\)	\(28 \leqslant t \leqslant 30\)	\(31 \leqslant t \leqslant 35\)	\(36 \leqslant t \leqslant 45\)	\(46 \leqslant t \leqslant 60\)	\(t \geqslant 61\)
Number of people	0	4	28	14	4	0

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