May/June 2020

The first nine terms of an arithmetic progression add to 117. The next four terms add to 91.

Series

The coordinates of points $A$ and $B$ are $(-1,-2)$ and $(7,4)$, respectively.

Coordinate geometry

The diagram depicts a section of the curve $y = \frac{8}{x+2}$ together with the line $2y + x = 8$, which meet at $A$ and $B$. Point $C$ is on the curve, and the tangent at $C$ runs parallel to $AB$.

Integration

When $(kx + \frac{1}{x})^5 + (1 - \frac{2}{x})^8$ is expanded, the coefficient attached to $\frac{1}{x}$ is 74.

Series

Each year, the selling price of a diamond necklace rises by 5% compared with the price in the previous year. In the year 2000, the necklace was priced at $36000$.

Series

The diagram presents the graph of $y = f(x)$, where $f(x) = \frac{3}{2}\cos 2x + \frac{1}{2}$ for $0 \leq x \leq \pi$.

Functions

The line has equation $y = mx + c$, with $m$ and $c$ as constants, and the curve is defined by $xy = 16$.

Differentiation

The functions $f$ and $g$ are given for $x \in \mathbb{R}$ by $f: x \mapsto \frac{1}{2}x - a$ and $g: x \mapsto 3x + b$, with $a$ and $b$ constant.

Functions

Prove that $\displaystyle \frac{1 + \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 + \sin\theta} = \frac{2}{\cos\theta}$.

Trigonometry

The diagram shows $ABC$ as a semicircle with diameter $AC$, centre $O$ and radius $6\text{ cm}$. Arc $AB$ measures $15\text{ cm}$. Point $X$ is on $AC$ and $BX$ is perpendicular to $AX$.

Coordinate geometry

The curve is described by $y = (3 - 2x)^3 + 24x$.

Differentiation

Find the coefficient of $x^2$ when $(x - \frac{2}{x})^6$ is expanded.

Series

The curve is given by the equation $y = 54x - (2x - 7)^3$.

Differentiation

For the circle with centre $C$, the equation is $x^2 + y^2 - 8x + 4y - 5 = 0$.

Coordinate geometry

Express the equation $3 \cos \theta = 8 \tan \theta$ in quadratic form with respect to $\sin \theta$.

Trigonometry

A spherical weather balloon is being inflated by a pump. Its volume is rising at a constant rate of $600\,\text{cm}^3$ per second. At the start of pumping, the balloon was empty.

Differentiation

The formula for the $n$th term of an arithmetic progression is $\frac{1}{2}(3n - 15)$.

Series

For $x \in \mathbb{R}$, the function $f$ is defined by $f: x \mapsto a - 2x$, where $a$ is a constant.

Functions

The curve is described by $y = 2x^2 + kx + k - 1$, with $k$ as a constant.

Differentiation

The diagram shows $OAB$ as a sector of a circle with centre $O$ and radius $2r$, with angle $AOB = \frac{\pi}{6}$ radians. Point $C$ lies halfway along $OA$.

Circular measure

The diagram shows a section of the curve $y = \frac{6}{x}$. The points $(1, 6)$ and $(3, 2)$ are on the curve. The shaded area is enclosed by the curve and the lines $y = 2$ and $x = 1$.

Integration

Let functions $f$ and $g$ be defined by $f(x) = 2 - 3\sin 2x$ for $0 \leq x \leq \pi$, and by $g(x) = -2f(x)$ for $0 \leq x \leq \pi$.

Functions

Find the values of $m$ for which the line $y = mx + 1$ and the curve $y = 3x^2 + 2x + 4$ meet at two distinct points.

Quadratics

Points A and B have coordinates $(-7, 3)$ and $(5, 11)$, respectively.

Coordinate geometry

The sketch displays a section of the curve with equation $y = x^3 - 2bx^2 + b^2x$ together with the line $OA$, and $A$ is the maximum point on the curve. The $x$-coordinate of $A$ is $a$, and the curve also has a minimum point at $(b, 0)$, where $a$ and $b$ are positive constants.

Integration

The curve is defined by $\frac{dy}{dx} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}$, and the point $(4, 7)$ lies on it.

Integration

In each of parts (a), (b) and (c), the graph drawn with solid lines has equation $y = f(x)$. The graph drawn with broken lines is a transformed version of $y = f(x)$.

Functions

Expand $(1 + a)^5$ in ascending powers of $a$ as far as and including the term in $a^3$.

Series

The diagram depicts a cord looped around a pulley and a pin. The pulley is represented by a circle with centre $O$ and radius $5\text{ cm}$. The thickness of the cord and the dimensions of the pin $P$ may be ignored. The pin is placed $13\text{ cm}$ vertically beneath $O$. Points $A$ and $B$ lie on the circle so that $AP$ and $BP$ are tangents to the circle. The cord runs over the major arc $AB$ of the circle and beneath the pin so that the cord stays taut.

Circular measure

A point $P$ is travelling on a curve so that the $x$-coordinate of $P$ is rising at a steady rate of $2$ units per minute. The curve is given by $y = \sqrt{5x - 1}$.

Differentiation

Show that $\frac{\tan\theta}{1 + \cos\theta} + \frac{\tan\theta}{1 - \cos\theta}$ is equal to $\frac{2}{\sin\theta \, \cos\theta}$.

Trigonometry

The initial term of the progression is $\sin^2 \theta$, with $0 < \theta < \tfrac{1}{2}\pi$. Its second term is $\sin^2 \theta \cos^2 \theta$.

Series

The functions $f$ and $g$ are given by $f(x) = x^2 - 4x + 3$ for $x > c$, where $c$ is a constant, and $g(x) = \frac{1}{x+1}$ for $x > -1$.

Functions

Solve $\ln(x + 1) - \ln x = 2 \ln 2$.

Logarithmic and exponential functions

Let the polynomial $p(x)$ be $p(x) = 6x^3 + ax^2 + 9x + b$, with $a$ and $b$ as constants. You are told that $(x - 2)$ and $(2x + 1)$ are factors of $p(x)$.

Algebra

The curve is defined parametrically by $x = e^t - 2e^{-t}$ and $y = 3e^{2t} + 1$.

Differentiation

On one diagram, sketch the graphs of $y = |3x + 2a|$ and $y = |3x - 4a|$, where $a$ is a positive constant. State the coordinates of the points at which each graph meets the axes.

Algebra

The diagram illustrates a section of the curve whose equation is $y = x^3 \cos 2x$. This curve reaches a maximum at the point $M$.

Numerical solution of equations

Prove that $\sin 2\theta (\cosec \theta - \sec \theta)$ can be transformed into $\sqrt{8} \cos \left(\theta + \frac{\pi}{4}\right)$.

Trigonometry

Determine the quotient when $9x^3 - 6x^2 - 20x + 1$ is divided by $(3x + 2)$, and show that the remainder is $9$.

Integration

With $2^y = 9^{3x}$ given, use logarithms to show that $y = kx$ and determine the value of $k$ correct to $3$ significant figures.

Logarithmic and exponential functions

Determine the exact coordinates of the stationary point on the curve defined by $y = 5x e^{\frac{1}{x}}$.

Differentiation

A curve is given by $\cos 3x + 5\sin y = 3$.

Differentiation

The variables $x$ and $y$ obey $y = Ax^{-2p}$, where $A$ and $p$ are constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line passing through the points $(-0.68, 3.02)$ and $(1.07, -1.53)$, as illustrated in the diagram.

Algebra

On one set of axes, sketch the graphs of $y = |2x - 3|$ and $y = 3x + 5$.

Algebra

The polynomial $p(x)$ is given by $p(x) = 6x^3 + ax^2 - 4x - 3$, with $a$ as a constant. It is known that $(x + 3)$ is a factor of $p(x)$.

Trigonometry

It is stated that $\int_0^a \left(\frac{4}{2x + 1} + 8x\right) \, dx = 10$, with $a$ a positive constant.

Numerical solution of equations

Show that $3\sin 2\theta \cot \theta = 6\cos^2 \theta$.

Trigonometry

Since $2^y = 9^{3x}$,

Logarithmic and exponential functions

Determine the exact coordinates of the stationary point on the curve given by $y = 5x e^{\frac{1}{2}x}$.

Differentiation

The curve has equation $\cos 3x + 5\sin y = 3$.

Differentiation

The variables $x$ and $y$ are related by the equation $y = Ax^{-2p}$, where $A$ and $p$ are constants. The graph of $\ln y$ plotted against $\ln x$ is a straight line passing through the points $(-0.68, 3.02)$ and $(1.07, -1.53)$, as illustrated in the diagram.

Logarithmic and exponential functions

On one set of axes, sketch the graphs of $y = |2x - 3|$ and $y = 3x + 5$.

Algebra

The polynomial $p(x)$ is specified by $p(x)=6x^3+ax^2-4x-3$, with $a$ as a constant. You are told that $(x+3)$ is a factor of $p(x)$.

Trigonometry

We are told that $\int_0^a \left( \frac{4}{2x + 1} + 8x \right) \, dx = 10$, where $a$ is a positive constant.

Numerical solution of equations

Show that, after simplification, $3 \sin 2\theta \cot \theta \equiv 6 \cos^2 \theta$.

Trigonometry

Find the set of $x$ values for which $2\left(3^{1-2x}\right) < 5^x$. Write your answer in simplified exact form.

Logarithmic and exponential functions

The complex number $u$ is given by $u = \dfrac{3i}{a + 2i}$, with $a$ taken to be real.

Complex numbers

Expand $(2 - 3x)^{-2}$ in increasing powers of $x$, stopping at and including the term in $x^2$, and simplify the coefficients.

Algebra

Rearrange $ an(\theta + 60^\circ) = 2 + \tan(60^\circ - \theta)$ into a quadratic in $\tan \theta$, then solve the equation for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

The curve defined by $y = e^{2x}(\sin x + 3\cos x)$ has a stationary point for $0 \leq x \leq \pi$.

Differentiation

Determine the quotient and remainder when $2x^3 - x^2 + 6x + 3$ is divided by $x^2 + 3$.

Integration

A circle with centre $O$ and radius $r$ is shown. The tangents to the circle at $A$ and $B$ intersect at $T$, and angle $AOB$ is $2x$ radians. The shaded region lies between the tangents $AT$ and $BT$, and the minor arc $AB$. Its area is equal to the area of the circle.

Numerical solution of equations

Define $f(x) = \dfrac{\cos x}{1 + \sin x}$.

Integration

A particular curve has gradient at a point $(x, y)$ proportional to $\dfrac{y}{x\sqrt{x}}$. It passes through the points $(1, 1)$ and $(4, e)$.

Differential equations

Relative to the origin $O$, triangle $ABC$ has vertices with position vectors $\overrightarrow{OA} = 2i + 5k$, $\overrightarrow{OB} = 3i + 2j + 3k$ and $\overrightarrow{OC} = i + j + k$.

Vectors

Determine the quotient and remainder when $6x^4 + x^3 - x^2 + 5x - 6$ is divided by $2x^2 - x + 1$.

Algebra

Using $O$ as the origin, the position vectors of $A$ and $B$ are $ \overrightarrow{OA} = 6\mathbf{i} + 2\mathbf{j}$ and $\overrightarrow{OB} = 2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$. $M$ is the midpoint of $OA$. Point $N$, which lies on $AB$ between $A$ and $B$, satisfies $AN = 2NB$.

Vectors

The variables $x$ and $y$ obey the equation $y^2 = A e^{kx}$, where $A$ and $k$ are constants. The graph of $\ln y$ plotted against $x$ is a straight line, and it goes through the points $(1.5,\,1.2)$ and $(5.24,\,2.7)$, as shown in the diagram.

Differential equations

Determine the exact value of $\int_{1}^{4} x^2 \ln x\, dx$.

Integration

The curve is defined by $y = \cos x \sin 2x$.

Differentiation

Express $\sqrt{2} \cos x - \sqrt{5} \sin x$ as $R \cos (x + \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$. State the exact value of $R$ and the value of $\alpha$ accurate to 3 d.p.

Trigonometry

The diagram depicts the curve $y = \dfrac{x}{1 + 3x^4}$, for $x \geq 0$, together with its maximum point $M$.

Integration

The variables $x$ and $y$ are linked by the differential equation $\dfrac{dy}{dx} = \dfrac{y - 1}{(x + 1)(x + 3)}$. It is stated that $y = 2$ when $x = 0$.

Differential equations

Solve the equation $(1 + 2i)w + i w^* = 3 + 5i$. Express your answer in the form $x + iy$, with $x$ and $y$ real.

Complex numbers

The diagram displays the curves $y = \cos x$ and $y = \dfrac{k}{1 + x}$, where $k$ is a constant, for $0 \leq x \leq \frac{1}{2}\pi$. The curves meet at the point for which $x = p$.

Numerical solution of equations

Solve $|2x - 1| > 3|x + 2|$.

Algebra

A tank of water has the form of a hemisphere. Its axis is vertical, the lowest point is $A$ and the radius is $r$, as shown in the diagram. The water depth at time $t$ is $h$. When $t = 0$ the tank is completely full, so the water depth is $r$. At this moment a tap at $A$ is opened and the water starts to leave at a rate proportional to $\sqrt{h}$. The tank is empty at time $t = 14$. When the depth is $h$, the volume of water in the tank is $V$. It is given that $V = \frac{1}{3}\pi(3rh^2 - h^3)$.

Differential equations

Determine the exact value of $\int_{0}^{1} (2 - x)e^{-2x}\,dx$.

Integration

Show that $\ln(1 + e^{-x}) + 2x = 0$ may be rewritten as a quadratic equation in $e^{x}$.

Logarithmic and exponential functions

The curve is defined by the equation $y = x \tan^{-1}\left(\frac{1}{2}x\right)$.

Differentiation

Start by rewriting $\tan\theta \tan(\theta + 45^\circ) = 2 \cot 2\theta$ as a quadratic in $\tan\theta$, then solve it for $0^\circ < \theta < 90^\circ$.

Trigonometry

Show by sketching an appropriate pair of graphs that the equation $x^5 = 2 + x$ has one and only one real root.

Numerical solution of equations

Define the function $f(x)$ by $f(x) = \frac{2}{(2x - 1)(2x + 1)}$.

Integration

With respect to the origin $O$, the position vectors of $A$, $B$ and $D$ are $\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}$, $\overrightarrow{OB} = 2\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}$ and $\overrightarrow{OD} = 3\mathbf{i} + 2\mathbf{k}$. A further point $C$ is arranged so that $ABCD$ forms a parallelogram.

Vectors

The complex numbers $u$ and $w$ satisfy $u - w = 2i$ and $uw = 6$.

Complex numbers

At point $A$, three coplanar forces with magnitudes $100\,\text{N}$, $50\,\text{N}$ and $50\,\text{N}$ act, as shown in the diagram. The value of $\cos \alpha$ is $\frac{4}{5}$.

Forces and equilibrium

A car with mass $1800\,\text{kg}$ is pulling a trailer of mass $400\,\text{kg}$ on a straight horizontal road. The car and trailer are joined by a light rigid tow-bar. The car has acceleration $1.5\,\text{m}\,\text{s}^{-2}$. Constant resistance forces act, with $250\,\text{N}$ on the car and $100\,\text{N}$ on the trailer.

Energy, work and power

A particle $P$ is launched vertically upwards at a speed of $5\,\text{m s}^{-1}$ from a point $A$, which lies $2.8\,\text{m}$ above the horizontal ground.

Kinematics of motion in a straight line

The diagram depicts a ring of mass $0.1\,\text{kg}$ passed onto a fixed horizontal rod. The rod is rough, with coefficient of friction between the ring and the rod equal to $0.8$. A force of magnitude $T\,\text{N}$ acts on the ring at an angle of $30^\circ$ to the rod, directed downwards in the vertical plane that contains the rod. At the start, the ring is at rest.

Forces and equilibrium

A child with mass $35\,\text{kg}$ is on a rope swing. Model the child as a particle $P$, and model the rope as a light inextensible string of length $4\,\text{m}$. At the start, $P$ is being held at an angle of $45^\circ$ to the vertical (see diagram).

Energy, work and power

A particle travels along the straight line $AB$. Its velocity $v\,\text{m s}^{-1}$ after $t\,\text{s}$ from leaving $A$ is $v = k(t^2 - 10t + 21)$, where $k$ is a constant. When $t = 3$, the particle's displacement from $A$, measured towards $B$, is $2.85\,\text{m}$, and when $t = 6$ it is $2.4\,\text{m}$.

Kinematics of motion in a straight line

A particle $P$ with mass $0.3\ \text{kg}$ is placed on a smooth plane that is inclined at $30^{\circ}$ to the horizontal, and it is let go from rest. $P$ travels $2.5\ \text{m}$ down the slope before arriving on a horizontal plane. Its speed does not change as $P$ reaches the horizontal plane. A particle $Q$ of mass $0.2\ \text{kg}$ is at rest on the horizontal plane, $1.5\ \text{m}$ from the lower end of the inclined plane (see diagram). $P$ then collides directly with $Q$.

Momentum

A tram begins at rest and accelerates uniformly for $20\text{ s}$. It then continues at a steady speed, $V\text{ m s}^{-1}$, for $170\text{ s}$ before coming to rest under a uniform deceleration whose magnitude is twice the acceleration. The tram covers a total distance of $2.775\text{ km}$.

Kinematics of motion in a straight line

Four coplanar forces of magnitudes $20\text{ N}$, $P\text{ N}$, $3P\text{ N}$ and $4P\text{ N}$ act at a point in the directions shown in the diagram. The system remains in equilibrium.

Forces and equilibrium

A particle with mass $2.5\text{ kg}$ is kept in equilibrium on a rough plane inclined at $20^\circ$ to the horizontal by a force of magnitude $T\text{ N}$ acting at an angle of $60^\circ$ to a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is $0.3$.

Forces and equilibrium

Two small smooth spheres $A$ and $B$, with equal radii and masses of $4\,\text{kg}$ and $2\,\text{kg}$ respectively, are on a smooth horizontal plane. At the beginning, $B$ is at rest, while $A$ is travelling towards $B$ at $10\,\text{m s}^{-1}$. After they collide, $A$ keeps moving in the same direction, but at half the speed of $B$.

Momentum

A car with mass $1250\,\text{kg}$ is travelling along a straight road.

Energy, work and power

A particle $P$ moves along a straight line. The velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is given by: $v = 2t + 1$ for $0 \leq t \leq 5$, $v = 36 - t^2$ for $5 \leq t \leq 7$, $v = 2t - 27$ for $7 \leq t \leq 13.5$.

Kinematics of motion in a straight line

Particles $P$ with mass $m\,\text{kg}$ and $Q$ with mass $0.2\,\text{kg}$ are able to move freely on a smooth horizontal plane. $P$ is launched at a speed of $2\,\text{m s}^{-1}$ towards $Q$, which is at rest. After the collision, $P$ and $Q$ travel in opposite directions with speeds of $0.5\,\text{m s}^{-1}$ and $1\,\text{m s}^{-1}$ respectively.

Momentum

A minibus with mass $4000\,\text{kg}$ is moving along a straight horizontal road. The resistive force is $900\,\text{N}$.

Energy, work and power

The diagram shows four coplanar forces of magnitudes $40\,\text{N}$, $20\,\text{N}$, $50\,\text{N}$ and $F\,\text{N}$ acting at a point in the directions indicated. These four forces are in equilibrium.

Forces and equilibrium

A car begins at rest and travels in a straight line with constant acceleration $a\,\text{m s}^{-2}$ over a distance of $50\,\text{m}$. It then moves at constant velocity for $500\,\text{m}$ in $25\,\text{s}$, before slowing to rest. The deceleration magnitude is $2a\,\text{m s}^{-2}$.

Kinematics of motion in a straight line

A block $B$ with mass $4\,\text{kg}$ is driven upwards along the line of greatest slope on a smooth plane that is tilted at $30^\circ$ to the horizontal by a force applied to $B$, and this force acts in the same direction as the motion of $B$. The block goes through points $P$ and $Q$ with speeds $12\,\text{m s}^{-1}$ and $8\,\text{m s}^{-1}$, respectively. $P$ and $Q$ are $10\,\text{m}$ apart, with $P$ at a lower level than $Q$.

Energy, work and power

A particle moves along the straight line $PQ$. Its velocity $t$ s after leaving $P$ is $v$ m s$^{-1}$, where $v = 4.5 + 4t - 0.5t^2$. The particle is instantaneously at rest at $Q$.

Kinematics of motion in a straight line

Masses $3m$ kg and $2m$ kg, named $A$ and $B$, are fixed to the two ends of a light inextensible string. The string runs over a fixed smooth pulley that is attached to the edge of a plane. The plane makes an angle $\theta$ with the horizontal. $A$ is on the plane while $B$ hangs vertically, $0.8$ m above the floor. The section of string from $A$ to the pulley is parallel to a line of greatest slope on the plane (see diagram). At the beginning, both $A$ and $B$ are at rest.

Newton's laws of motion

For two fair six-sided dice, the score is the total of the numbers showing on the top faces. The dice are rolled again and again until a score of 4 is obtained. Let the random variable $X$ denote the number of throws needed.

Discrete random variables

Find the number of different arrangements that can be made from the 9 letters in the word JEWELLERY when the three Es are together and the two Ls are together.

Permutations and combinations

A firm makes small sweet boxes containing 5 jellies and 3 chocolates. Jemeel randomly picks 3 sweets from one box.

Discrete random variables

A music competition features $8$ pianists, $4$ guitarists and $6$ violinists. $7$ of these musicians are to be chosen to progress to the final.

Permutations and combinations

Each Monday evening, Rani prepares her meal. She eats a pizza, a burger, or a curry with probabilities $0.35$, $0.44$, $0.21$ respectively. If she cooks a pizza, Rani has some fruit with probability $0.3$. If she cooks a burger, she has some fruit with probability $0.8$. If she cooks a curry, she never has any fruit.

Probability

For female snakes of this species, the lengths follow a normal distribution with mean $54\text{ cm}$ and standard deviation $6.1\text{ cm}$.

The normal distribution

The daily numbers of chocolate bars sold in a cinema over 100 days are recorded in the table below: Number of chocolate bars sold: $1$-$10$, $11$-$15$, $16$-$30$, $31$-$50$, $51$-$60$; Number of days: $18$, $24$, $30$, $20$, $8$.

Representation of data

Across $n$ values of the variable $x$, it is stated that $\sum (x - 50) = 144$ and $\sum x = 944$.

Representation of data

A group of $500$ students was surveyed about which one of four colleges they attended and whether they favoured soccer or hockey. The numbers in each category are listed in the table below.

Probability

Machines $A$ and $B$ each manufacture metal rods of a particular type. The lengths, in metres, of the $19$ rods from machine $A$ and the $19$ rods from machine $B$ are shown in the back-to-back stem-and-leaf diagram below. Key: $7\;|\;22\;|\;4$ means $0.227\,\text{m}$ for machine $A$ and $0.224\,\text{m}$ for machine $B$.

Representation of data

Trees in the Redian forest are grouped into tall, medium, or short categories according to their height. Their heights may be represented by a normal distribution with mean $40\text{ m}$ and standard deviation $12\text{ m}$. Any tree with height below $25\text{ m}$ is placed in the short category.

The normal distribution

One fair three-sided spinner is labelled 1, 2, 3. Another fair five-sided spinner is labelled 1, 1, 2, 2, 3. Each spinner is spun once, and the number showing on the side where it lands is recorded. The random variable $X$ is the larger of the two numbers when they are different, and their shared value when they are the same.

Discrete random variables

Determine the number of distinct arrangements of the 10 letters in SUMMERTIME that have an E at the beginning and an E at the end.

Permutations and combinations

On any one day, the chance that Moena sends a message to her friend Pasha is $0.72$.

Discrete random variables