Questions - SPS - A-Level Maths

State the median reaction time.
Calculate the interquartile range of these reaction times.
Find the mean and standard deviation of these reaction times.
State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
i) the median
ii) the mean
ii) the standard deviation
[0pt] [4]
Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
[0pt] [2]

SPS SPS FM Statistics 2025 April Q3

2 marks

3. Miguel has six numbered tiles, labelled $2,2,3,3,4,4$. He selects two tiles at random, without replacement. The variable $M$ denotes the sum of the numbers on the two tiles.

Show that $P ( M = 6 ) = \frac { 1 } { 3 }$
[0pt] [2] The table shows the probability distribution of $M$
$m$ 4 5 6 7 8
$P ( M = m )$ $\frac { 1 } { 15 }$ $\frac { 4 } { 15 }$ $\frac { 1 } { 3 }$ $\frac { 4 } { 15 }$ $\frac { 1 } { 15 }$
Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable $S$ denotes the sum of the numbers on the two tiles that Sofia selects.
Find $P ( M = S )$
Find $P ( S = 7 \mid M = S )$

SPS SPS FM Statistics 2025 April Q4

4. The discrete random variable $X$ has a geometric distribution. It is given that $\operatorname { Var } ( X ) = 20$. Determine $\mathrm { P } ( X \geqslant 7 )$.

SPS SPS FM Statistics 2025 April Q5

7 marks

5. An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.

If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions.
[0pt] [2]
Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7 -mark questions are the first four questions on the paper, but are arranged in random order. The 9 -mark questions are the last four questions, but are arranged in random order. Find the probability that
the questions on geometric distributions and on binomial distributions are next to one another,
[0pt] [2]
the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
[0pt] [3] \section*{6.} The random variable $X$ represents the weight in kg of a randomly selected male dog of a particular breed. $X$ is Normally distributed with mean 30.7 and standard deviation 3.5.
i) Find the $90 ^ { \text {th } }$ percentile for the weights of these dogs.
ii) Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg . The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg .
iii) Given that $5 \%$ of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
iv) Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.

SPS SPS FM Statistics 2025 April Q7

7. The random variable $y$ has probability density function $\mathrm { f } ( y )$ given by $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leq y \leq 3
0 & \text { otherwise } \end{array} \right.$$ where $k$ and $a$ are positive constants.

1. Explain why $a \geq 3$
2. Show that $k = \frac { 2 } { 9 ( a - 2 ) }$ Given that $\mathrm { E } ( Y ) = 1.75$
Find the values of a and k .
Write down the mode of Y

SPS SPS SM Statistics 2025 April Q1

It is known that, under standard conditions, $12 \%$ of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample.
2. Test, at the $5 \%$ significance level, whether the proportion of defective light bulbs has decreased under the new process.
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The histogram shows information about the lengths, $l$ centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{a18b06b1-053e-45b2-9c28-2f125cf6cbba-06_805_1151_269_280} The number of worms in the sample with lengths in the class $3 \leqslant l < 4$ is 30 .
Find the number of worms in the sample with lengths in the class $0 \leqslant l < 2$.
Find an estimate of the number of worms in the sample with lengths in the range $4.5 \leqslant l < 5.5$.
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SPS SPS SM Statistics 2025 April Q3

1 marks

3. A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form $a X + b$, where $X$ represents the original height in centimetres.
Given the following information about the encoded data:
The mean of the encoded heights is 5.4 cm
The standard deviation of the encoded heights is 2.0 cm
The researcher knows that the transformation used was $0.2 X - 30$

Find the mean of the original heights in the sample.
Find the standard deviation of the original heights in the sample.
If an encoded height value is 6.8 , what was the original height in centimetres?
[0pt] [1]
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SPS SPS SM Statistics 2025 April Q4

4. A manufacturing plant produces electronic circuit boards that need to pass two quality checks - a mechanical inspection and an electrical test. Historical data shows that $15 \%$ of boards fail the mechanical inspection. Of those that pass the mechanical inspection, $8 \%$ fail the electrical test. Of those that fail the mechanical inspection, $60 \%$ fail the electrical test.

If a board is randomly selected from production, what is the probability that it passes both inspections?
If a board is selected at random and is found to have passed the electrical test, what is the probability that it also passed the mechanical inspection?
The company continues to test boards from a large batch until finding one that passes both inspections. Each board is tested independently of all others. What is the probability that they need to test exactly 3 boards to find one that passes both inspections?
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 April Q5

4 marks

5. In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: $20 \mid 3579$
$21 \mid 02568$
$22 \mid 134579$
$23 \mid 0258$
$24 \mid 1467$
$25 \mid 25$ Key: 21 | 0 represents a reaction time of 210 milliseconds

State the median reaction time.
Calculate the interquartile range of these reaction times.
Find the mean and standard deviation of these reaction times.
State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table.
One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
a. the median
b. the mean
c. the standard deviation
[0pt] [4]
Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context
[0pt] [BLANK PAGE]

SPS SPS SM Statistics 2025 April Q6

6. A retail bakery makes cherry muffins where, due to the production process, $15 \%$ of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.

State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid.
Using your model from part (a), find the probability that a randomly selected box contains:
1. exactly 3 muffins with a lower than expected quantity of cherries,
2. at least 5 muffins with a lower than expected quantity of cherries.
The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries.
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SPS SPS SM Statistics 2025 April Q7

7. Miguel has six numbered tiles, labelled $2,2,3,3,4,4$. He selects two tiles at random, without replacement. The variable $M$ denotes the sum of the numbers on the two tiles.

Show that $P ( M = 6 ) = \frac { 1 } { 3 }$ The table shows the probability distribution of $M$
$m$ 4 5 6 7 8
$P ( M = m )$ $\frac { 1 } { 15 }$ $\frac { 4 } { 15 }$ $\frac { 1 } { 3 }$ $\frac { 4 } { 15 }$ $\frac { 1 } { 15 }$
Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable $S$ denotes the sum of the numbers on the two tiles that Sofia selects.
Find $P ( M = S )$
Find $P ( S = 7 \mid M = S )$
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SPS SPS FM Pure 2025 June Q1

The complex number $z$ satisfies the equation $z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }$. Find $z$, giving your answer in the form $x + \mathrm { i } y$.
[0pt] [BLANK PAGE]
(a) Use binomial expansions to show that $\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$

A student substitutes $x = \frac { 1 } { 2 }$ into both sides of the approximation shown in part (a) in an attempt to find an approximation to $\sqrt { 6 }$
(b) Give a reason why the student should not use $x = \frac { 1 } { 2 }$
(c) Substitute $x = \frac { 1 } { 11 }$ into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to $\sqrt { 6 }$. Give your answer as a fraction in its simplest form.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2025 June Q3

3. Describe a sequence of transformations which maps the graph of $$y = | 2 x - 5 |$$ onto the graph of $$y = | x |$$ [BLANK PAGE]

SPS SPS FM Pure 2025 June Q4

4. Given that $$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where $A$ is a rational constant to be found.
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SPS SPS FM Pure 2025 June Q5

5. Two matrices $\mathbf { A }$ and $\mathbf { B }$ satisfy the equation $$\mathbf { A B } = \boldsymbol { I } + 2 \mathbf { A }$$ where $\boldsymbol { I }$ is the identity matrix and $\mathbf { B } = \left[ \begin{array} { c c } 3 & - 2
- 4 & 8 \end{array} \right]$ \section*{Find $\mathbf { A }$.} [BLANK PAGE]

SPS SPS FM Pure 2025 June Q6

6. (a) Prove that $$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$, the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.
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SPS SPS FM Pure 2025 June Q7

7. Fig. 10 shows the graph of $x ^ { 3 } + y ^ { 3 } = x y$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-16_538_527_251_785} \captionsetup{labelformat=empty} \caption{Fig. 10}

\end{figure}

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
P is the maximum point on the curve. The parabola $y = k x ^ { 2 }$ intersects the curve at P . Find the value of the constant $k$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2025 June Q8

8. (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{14f14bf3-88ee-413c-a62d-0914f41a485d-18_1339_1383_370_402}
(b) There is a unique complex number $w$ that satisfies both $$| w - 3 | = 2 \quad \text { and } \quad \arg ( w + 1 ) = \alpha$$ where $\alpha$ is a constant such that $0 < \alpha < \pi$
(b) (i) Find the value of $\alpha$.
(b) (ii) Express $w$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$.
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2025 June Q9

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-20_707_823_130_701} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve $C$ with equation $y = x \ln x , \quad x > 0$ The line $l$ is the normal to $C$ at the point $P ( \mathrm { e } , \mathrm { e } )$ The region $R$, shown shaded in Figure 2, is bounded by the curve $C$, the line $l$ and the $x$-axis. Show that the exact area of $R$ is $A \mathrm { e } ^ { 2 } + B$ where $A$ and $B$ are rational numbers to be found.
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SPS SPS FM Pure 2025 June Q10

10. Prove by induction that $f ( n ) = 2 ^ { 4 n } + 5 ^ { 2 n } + 7 ^ { n }$ is divisible by 3 for all positive integers $n$.
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SPS SPS FM Pure 2025 June Q11

11. Fig. 15 shows the graph of $\mathrm { f } ( x ) = 2 x + \frac { 1 } { x } + \ln x - 4$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{14f14bf3-88ee-413c-a62d-0914f41a485d-24_1008_771_212_669} \captionsetup{labelformat=empty} \caption{Fig. 15}

\end{figure}

Show that the equation $$2 x + \frac { 1 } { x } + \ln x - 4 = 0$$ has a root, $\alpha$, such that $0.1 < \alpha < 0.9$.
Obtain the following Newton-Raphson iteration for the equation in part (i). $$x _ { r + 1 } = x _ { r } - \frac { 2 x _ { r } ^ { 3 } + x _ { r } + x _ { r } ^ { 2 } \left( \ln x _ { r } - 4 \right) } { 2 x _ { r } ^ { 2 } - 1 + x _ { r } }$$
Explain why this iteration fails to find $\alpha$ using each of the following starting values.
(A) $x _ { 0 } = 0.4$
(B) $x _ { 0 } = 0.5$
(C) $x _ { 0 } = 0.6$
[0pt] [BLANK PAGE] \section*{12. In this question you must show detailed reasoning.}
\includegraphics[max width=\textwidth, alt={}]{14f14bf3-88ee-413c-a62d-0914f41a485d-26_819_589_173_826}
The curve $C$ has parametric equations $$x = \frac { 1 } { \sqrt { 2 + t } } , \quad y = \ln ( 1 + t ) , \quad 2 \leq t < \infty$$ The point $P$ on curve $C$ has $x$-coordinate $\frac { 1 } { 2 }$.
(a) Find the exact $y$-coordinate of $P$. The tangent to $C$ at $P$ meets the $y$-axis at point $Y$.
(b) Determine the exact coordinates of $Y$. The curve $C$ and the line segment $P Y$ are rotated $2 \pi$ radians about the $y$-axis.
(c) Determine the exact volume of the solid generated. Give your answer in the form $\pi ( \ln p + q )$, where $p$ and $q$ are rational numbers.
[0pt] [You are given that the volume of a cone with radius $r$ and height $h$ is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ ]
[0pt] [BLANK PAGE]

SPS SPS FM Pure 2025 June Q13

13.

Using a suitable substitution, find $$\int \sqrt { 1 - x ^ { 2 } } d x$$
Show that the differential equation $$\frac { d y } { d x } = 2 \sqrt { 1 - x ^ { 2 } - y ^ { 2 } + x ^ { 2 } y ^ { 2 } }$$ given that $y = 0$ when $x = 0 , | x | < 1$ and $| y | < 1$, has the solution $$y = x \cos \left( x \sqrt { 1 - x ^ { 2 } } \right) + \sqrt { 1 - x ^ { 2 } } \sin \left( x \sqrt { 1 - x ^ { 2 } } \right) .$$ [BLANK PAGE]

SPS SPS FM Pure 2025 June Q14

14. The three dimensional non-zero vector $\boldsymbol { u }$ has the following properties:

The angle $\theta$ between $\boldsymbol { u }$ and the vector $\left( \begin{array} { l } 1
5
9 \end{array} \right)$ is acute.
The (non-reflex) angle between $\boldsymbol { u }$ and the vector $\left( \begin{array} { l } 9
5
1 \end{array} \right)$ is $2 \theta$.
$\boldsymbol { u }$ is perpendicular to the vector $\left( \begin{array} { l } 1
1
1 \end{array} \right)$.

Find the angle $\theta$.
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SPS SPS FM Pure 2024 September Q1

(a) Sketch the graph with equation

$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of $x$ which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of $x$ which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.
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\(m\)	4	5	6	7	8
\(P ( M = m )\)	\(\frac { 1 } { 15 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 15 }\)

\(m\)	4	5	6	7	8
\(P ( M = m )\)	\(\frac { 1 } { 15 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 4 } { 15 }\)	\(\frac { 1 } { 15 }\)

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