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SPS SPS FM Mechanics 2021 September Q5

8 marks Standard +0.3

In this question use $g = 10 \text{ m s}^{-2}$. A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at $30°$ to the horizontal.

Find the exact value of the coefficient of friction between the particle and the plane. [2]

A horizontal force of 36 N is now applied to the particle.

Find how far down the plane the particle travels after the force has been applied for 4 s. [6]

SPS SPS FM Statistics 2021 September Q1

6 marks Moderate -0.8

5 girls and 3 boys are arranged at random in a straight line. Find the probability that none of the boys is standing next to another boy. [3 marks]
A cricket team consisting of six batsmen, four bowlers, and one wicket-keeper is to be selected from a group of 18 cricketers comprising nine batsmen, seven bowlers, and two wicket-keepers. How many different teams can be selected? [3 marks]

SPS SPS FM Statistics 2021 September Q2

9 marks Moderate -0.3

$P(E) = 0.25$, $P(F) = 0.4$ and $P(E \cap F) = 0.12$

Find $P(E'|F')$ [2 marks]
Explain, showing your working, whether or not $E$ and $F$ are statistically independent. Give reasons for your answer. [2 marks]

The event $G$ has $P(G) = 0.15$ The events $E$ and $G$ are mutually exclusive and the events $F$ and $G$ are independent.

Draw a Venn diagram to illustrate the events $E$, $F$ and $G$, giving the probabilities for each region. [3 marks]
Find $P([F \cup G]')$ [2 marks]

SPS SPS FM Statistics 2021 September Q3

11 marks Moderate -0.8

A group of students were surveyed by a principal and $\frac{2}{3}$ were found to always hand in assignments on time. When questioned about their assignments $\frac{3}{5}$ said they always start their assignments on the day they are issued and, of those who always start their assignments on the day they are issued, $\frac{11}{20}$ hand them in on time.

Draw a tree diagram to represent this information. [3 marks]
Find the probability that a randomly selected student:
1. always start their assignments on the day they are issued and hand them in on time. [2 marks]
2. does not always hand in assignments on time and does not start their assignments on the day they are issued. [4 marks]
Determine whether or not always starting assignments on the day they are issued and handing them in on time are statistically independent. Give reasons for your answer. [2 marks]

SPS SPS FM Statistics 2021 September Q4

4 marks Moderate -0.8

In a town, 54% of the residents are female and 46% are male. A random sample of 200 residents is chosen from the town. Using a suitable approximation, find the probability that more than half the sample are female. [4 marks]

SPS SPS FM Statistics 2021 September Q5

9 marks Standard +0.3

The heights of a population of men are normally distributed with mean $\mu$ cm and standard deviation $\sigma$ cm. It is known that 20% of the men are taller than 180 cm and 5% are shorter than 170 cm.

Sketch a diagram to show the distribution of heights represented by this information. [2 marks]
Find the value of $\mu$ and $\sigma$. [5 marks]
Three men are selected at random, find the probability that they are all taller than 175 cm. [2 marks]

SPS SPS SM Mechanics 2021 September Q1

8 marks Easy -1.3

A racing car starts from rest at the point $A$ and moves with constant acceleration of $11 \text{ m s}^{-2}$ for $8 \text{ s}$. The velocity it has reached after $8 \text{ s}$ is then maintained for $7 \text{ s}$. The racing car then decelerates from this velocity to $40 \text{ m s}^{-1}$ in a further $2 \text{ s}$, reaching point $B$.

Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch. [5]
Given that the distance between $A$ and $B$ is $1404 \text{ m}$, find the value of $T$. [3]

SPS SPS SM Mechanics 2021 September Q2

7 marks Easy -1.3

A particle $P$ is acted upon by three forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ given by $\mathbf{F}_1 = (6\mathbf{i} - 4\mathbf{j}) \text{ N}$, $\mathbf{F}_2 = (-3\mathbf{i} + 9\mathbf{j}) \text{ N}$ and $\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j}) \text{ N}$, where $a$ and $b$ are constants. Given that $P$ is in equilibrium,

find the value of $a$ and the value of $b$. [2]

The force $\mathbf{F}_3$ is now removed. The resultant of $\mathbf{F}_1$ and $\mathbf{F}_2$ is $\mathbf{R}$.

Find the magnitude of $\mathbf{R}$. [3]
Find the angle, to $0.1°$, that $\mathbf{R}$ makes with $\mathbf{i}$. [2]

SPS SPS SM Mechanics 2021 September Q3

12 marks Moderate -0.3

A car of mass $1200 \text{ kg}$ pulls a trailer of mass $400 \text{ kg}$ along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of $3200 \text{ N}$. The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is $0.4 \text{ m s}^{-2}$,

find the resistance to motion on the trailer, [4]
find the tension in the tow-rope. [3]

When the car and trailer are travelling at $25 \text{ m s}^{-1}$ the tow-rope breaks. Assuming that the resistances to motion remain unchanged,

find the distance the trailer travels before coming to a stop, [4]
state how you have used the modelling assumption that the tow-rope is inextensible. [1]

SPS SPS SM Mechanics 2021 September Q4

13 marks Moderate -0.3

A car starts from the point $A$. At time $t$ s after leaving $A$, the distance of the car from $A$ is $s$ m, where $s = 30t - 0.4t^2$, $0 \leq t \leq 25$. The car reaches the point $B$ when $t = 25$.

Find the distance $AB$. [2]
Show that the car travels with a constant acceleration and state the value of this acceleration. [3]

A runner passes through $B$ when $t = 0$ with an initial velocity of $2 \text{ m s}^{-1}$ running directly towards $A$. The runner has a constant acceleration of $0.1 \text{ m s}^{-2}$.

Find the distance from $A$ at which the runner and the car pass one another. [8]

SPS SPS FM Pure 2022 February Q1

7 marks Moderate -0.3

Express $\frac{1}{(2r-1)(2r+1)}$ in partial fractions. [3]
Hence find $\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}$, expressing the result as a single fraction. [4]

SPS SPS FM Pure 2022 February Q2

5 marks Challenging +1.2

$\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}$ The matrix $\mathbf{A}$ represents the linear transformation $M$. Prove that, for the linear transformation $M$, there are no invariant lines. [5]

SPS SPS FM Pure 2022 February Q3

9 marks Standard +0.3

The line $l_1$ has equation $\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}$. The plane $\Pi$ has equation $\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4$.

Find the position vector of the point of intersection of $l_1$ and $\Pi$. [3]
Find the acute angle between $l_1$ and $\Pi$. [3]

$A$ is the point on $l_1$ where $\lambda = 1$. $l_2$ is the line with the following properties. • $l_2$ passes through $A$ • $l_2$ is perpendicular to $l_1$ • $l_2$ is parallel to $\Pi$

Find, in vector form, the equation of $l_2$. [3]

SPS SPS FM Pure 2022 February Q4

9 marks Standard +0.3

$\mathbf{A}$ is a 2 by 2 matrix and $\mathbf{B}$ is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate
1. $\mathbf{AB}$
2. $\mathbf{A} + \mathbf{B}$
[2]
Given that $$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix} \begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda \mathbf{I}$$ where $a$, $b$ and $\lambda$ are constants,
1. determine • the value of $\lambda$ • the value of $a$ • the value of $b$
2. Hence deduce the inverse of the matrix $\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}$
[3]
Given that $$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leqslant \theta < \pi$$ determine the values of $\theta$ for which the matrix $\mathbf{M}$ is singular. [4]

SPS SPS FM Pure 2022 February Q5

11 marks Standard +0.3

Points $A$, $B$ and $C$ have coordinates $(4, 2, 0)$, $(1, 5, 3)$ and $(1, 4, -2)$ respectively. The line $l$ passes through $A$ and $B$.

Find a cartesian equation for $l$. [3]

$M$ is the point on $l$ that is closest to $C$.

Find the coordinates of $M$. [4]
Find the exact area of the triangle $ABC$. [4]

SPS SPS FM Pure 2022 February Q6

13 marks Challenging +1.8

The curve $C$ has equation $$r = a(p + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where $a$ and $p$ are positive constants and $p > 2$ There are exactly four points on $C$ where the tangent is perpendicular to the initial line.

Show that the range of possible values for $p$ is $$2 < p < 4$$ [5]
Sketch the curve with equation $$r = a(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi \quad \text{where } a > 0$$ [1]

John digs a hole in his garden in order to make a pond. The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20(3 + 2\cos\theta) \quad 0 \leqslant \theta < 2\pi$$ where $r$ is measured in centimetres. The depth of the pond is 90 centimetres. Water flows through a hosepipe into the pond at a rate of 50 litres per minute. Given that the pond is initially empty,

determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute. [7]

Determine the exact roots of the equation. [5]
Determine the values of $p$ and $q$. [4]

SPS SPS FM Pure 2022 February Q10

8 marks Standard +0.3

You are given that $f(x) = 4\sinh x + 3\cosh x$.

Show that the curve $y = f(x)$ has no turning points. [3]
Determine the exact solution of the equation $f(x) = 5$. [5]

SPS SPS FM Pure 2022 February Q11

12 marks Challenging +1.2

A particle $P$ of mass 2 kg can only move along the straight line segment $OA$, where $OA$ is on a rough horizontal surface. The particle is initially at rest at $O$ and the distance $OA$ is 0.9 m. When the time is $t$ seconds the displacement of $P$ from $O$ is $x$ m and the velocity of $P$ is $v$ ms$^{-1}$. $P$ is subject to a force of magnitude $4e^{-2t}$ N in the direction of $A$ for any $t \geqslant 0$. The resistance to the motion of $P$ is modelled as being proportional to $v$. At the instant when $t = \ln 2$, $v = 0.5$ and the resultant force on $P$ is 0 N.

Show that, according to the model, $\frac{dv}{dt} + v = 2e^{-2t}$. [3]
Find an expression for $v$ in terms of $t$ for $t \geqslant 0$. [5]
By considering the behaviour of $v$ as $t$ becomes large explain why, according to the model, $P$'s speed must reach a maximum value for some $t > 0$. [2]
Determine the maximum speed considered in part (c). [2]

SPS SPS FM Pure 2022 February Q12

6 marks Challenging +1.2

In this question you must show detailed reasoning.

By using an appropriate Maclaurin series prove that if $x > 0$ then $e^x > 1 + x$. [2]
Hence, by using a suitable substitution, deduce that $e^t > et$ for $t > 1$. [1]
Using the inequality in part (b), and by making a suitable choice for $t$, determine which is greater, $e^{\pi}$ or $\pi^e$. [3]