Oct/Nov 2021

Expand the expression $\left(1 - \frac{1}{2x}\right)^2$.

Series

Evaluate $\displaystyle \int_{1}^{\infty} \frac{1}{(3x - 2)^{\frac{3}{2}}}\,dx$.

Integration

The curve is defined by $y = kx^2 + 2x - k$, while the line is defined by $y = kx - 2$, where $k$ is a constant.

Quadratics

Solve, by factorising, $6\cos\theta\tan\theta - 3\cos\theta + 4\tan\theta - 2 = 0$, for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

The arithmetic progression starts with first term $a$ and has common difference $-4$. The geometric progression begins with first term $5a$ and has common ratio $-\frac{1}{4}$. Its sum to infinity is the same as the sum of the first eight terms of the arithmetic progression.

Series

The diagram displays a section of the graph of $y = a\cos(bx) + c$.

Trigonometry

The diagram depicts a metal plate $ABC$ whose boundary is formed by the straight side $AB$ together with the arcs $AC$ and $BC$. The length of $AB$ is $6\text{ cm}$. Arc $AC$ is an arc of a circle centred at $B$ with radius $6\text{ cm}$, and arc $BC$ is an arc of a circle centred at $A$ with radius $6\text{ cm}$.

Circular measure

A circle centred at $(5, 2)$ passes through $(7, 5)$.

Coordinate geometry

The one-one function $f$ is given by $f : x \mapsto -3x^2 + 12x + 2$ for $x \leq k$.

Functions

A curve is defined by the equation $y = f(x)$, and it is given that $f'(x) = 2x^2 - 7 - \frac{4}{x^2}$.

Differentiation

Find the values of $\theta$ that satisfy $2\cos\theta = 7 - \frac{3}{\cos\theta}$ for $-90^\circ < \theta < 90^\circ$.

Trigonometry

The function $f$ is given by $f(x) = x^2 + \frac{k}{x} + 2$ for $x > 0$.

Differentiation

The diagram illustrates the line $x = \frac{5}{2}$, a segment of the curve $y = \frac{1}{2}x + \frac{7}{10} - \frac{1}{(x-2)^3}$ and the normal to the curve at the point $A\left(3, \frac{6}{5}\right)$.

Integration

The figure illustrates the circle defined by equation $x^2 + y^2 - 6x + 4y - 27 = 0$ together with the tangent to the circle at point $P\,(5, 4)$.

Coordinate geometry

The graph of $y = f(x)$ is changed into the graph of $y = f(2x) - 3$.

Functions

The function $f$ is given by $f(x) = \frac{x + 3}{x - 1}$ whenever $x > 1$.

Functions

A curve is defined by \[ \frac{dy}{dx} = \frac{8}{(3x+2)^2}. \] It passes through the point $(2, 5\tfrac{2}{3})$.

Integration

In an arithmetic progression, the first, third and fifth terms are $2\cos x$, $-6\sqrt{3}\sin x$ and $10\cos x$ respectively, with $\frac{1}{2}\pi < x < \pi$.

Series

A geometric progression has second term $54$ and sum to infinity $243$, while its common ratio is greater than $\tfrac{1}{2}$.

Series

In the diagram, $AB$ and $AC$ each have length $15\text{ cm}$. Point $P$ is the point where the perpendicular from $C$ meets $AB$. Also, $CP = 9\text{ cm}$. A circle arc centred at $B$ passes through $C$ and cuts $AB$ at $Q$.

Circular measure

For the expansion of $(4 + 2x)(2 - ax)^5$, the coefficient of $x^2$ is $-15$. Find the possible values of $a$.

Series

The volume $V\ \text{m}^3$ of a large circular mound of iron ore with radius $r\ \text{m}$ is represented by $V = \frac{3}{2}\left(r - \frac{1}{2}\right)^3 - 1$ for $r \geq 2$. Iron ore is being added to the mound at a steady rate of $1.5\ \text{m}^3$ per second.

Differentiation

The graph of $y = f(x)$ is changed into the graph of $y = 3 - f(x)$. Give a complete description, in the correct order, of the two transformations that have been combined.

Functions

The curve is defined by $y = f(x)$, and it is stated that $f'(x) = \left(\frac{1}{2}x + k\right)^{-2} - (1 + k)^{-2}$, where $k$ is a constant. The curve has a minimum point when $x = 2$.

Differentiation

State the first three terms, in ascending powers of $x$, of the expansion of $(1 + ax)^6$.

Series

Express $5y^2 - 30y + 50$ in the form $5(y + a)^2 + b$, with $a$ and $b$ as constants.

Differentiation

The initial term of an arithmetic progression is $84$, and its common difference is $-3$.

Series

In the diagram, $X$ and $Y$ lie on the line $AB$ so that $BX = 9\text{ cm}$ and $AY = 11\text{ cm}$. Arc $BC$ is an arc of a circle with centre $X$ and radius $9\text{ cm}$, and $CX$ is perpendicular to $AB$. Arc $AC$ is an arc of a circle with centre $Y$ and radius $11\text{ cm}$.

Functions

The diagram displays the graph of $y = f(x)$.

Functions

Show that the equation $\frac{\tan x + \cos x}{\tan x - \cos x} = k$, with $k$ a constant, may be rewritten as $(k + 1)\sin^2 x + (k - 1)\sin x - (k + 1) = 0$.

Trigonometry

The diagram presents the curves with equations $y = x^{-\frac{1}{2}}$ and $y = \dfrac{5}{2} - x^{\frac{1}{2}}$. The two curves cross at the points $A\left(\dfrac{1}{4}, 2\right)$ and $B\left(4, \dfrac{1}{2}\right)$.

Differentiation

The circle $x^2 + y^2 = 20$ cuts the line $y = 2x + 5$ at the points $A$ and $B$.

Coordinate geometry

Find the exact value of the definite integral $\int_{-1}^{2} \left(4e^{2x} - 2e^{-x}\right)\,dx$.

Integration

On one diagram, sketch the graphs of $y = 3x$ and $y = |x - 3|$.

Algebra

The variables $x$ and $y$ are linked by the equation $a y = kx$, where $a$ and $k$ are constants. The plot of $y$ against $\ln x$ is a straight line that goes through the points $(1.03, 6.36)$ and $(2.58, 9.00)$, as shown in the diagram.

Numerical solution of equations

The graph of $y = x e^{2x} + 5 e^{-x}$ contains a minimum point $M$.

Numerical solution of equations

The sketch shows the curve with parametric equations $x = \ln(2t + 3)$ and $y = \dfrac{2t - 3}{2t + 3}$. The curve meets the $y$-axis at the point $A$ and meets the $x$-axis at the point $B$.

Differentiation

The polynomials $f(x)$ and $g(x)$ are given by $f(x) = 4x^3 + ax^2 + 8x + 15$ and $g(x) = x^2 + bx + 18$, with $a$ and $b$ as constants.

Algebra

Starting from the expansion of $\cos(2\theta + \theta)$, prove that $\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$.

Trigonometry

The polynomial $p(x)$ is given by $p(x) = ax^3 + bx - 10$, where $a$ and $b$ are constants. It is stated that $(x + 2)$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $(x + 3)$ is $-55$.

Algebra

On one set of axes, sketch the graphs of $y = x + 3$ and $y = |2x - 1|$.

Logarithmic and exponential functions

The graph given by $y = 5x - 2\tan(2x)$ has one and only one stationary point for $0 \leq x \leq \frac{\pi}{4}$.

Differentiation

If $\int_a^{a+14} \frac{1}{3x}\,dx = \ln 2$, determine the positive constant $a$.

Integration

The curve is given by the equation $x^2 + 4x\cos(3y) = 6$.

Differentiation

By drawing an appropriate pair of graphs on the same diagram, show that the equation $\ln x = 2e^{-x}$ has exactly one root.

Numerical solution of equations

Prove that $4\sin x\sin\left(x + \tfrac{1}{6}\pi\right) \equiv \sqrt{3} - \sqrt{3}\cos 2x + \sin 2x$.

Trigonometry

Work out the exact value of $\int_{-1}^{2} \left(4e^{2x} - 2e^{-x}\right)\,dx$.

Integration

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

Algebra

Let $x$ and $y$ be related by the equation $a^y = kx$, with $a$ and $k$ treated as constants. The graph of $y$ against $\ln x$ is linear and passes through the points $(1.03, 6.36)$ and $(2.58, 9.00)$, as displayed in the diagram.

Numerical solution of equations

The curve whose equation is $y = x e^{2x} + 5e^{-x}$ has a minimum point $M$.

Numerical solution of equations

The diagram depicts the curve given by the parametric equations $x = \ln(2t + 3)$ and $y = \frac{2t - 3}{2t + 3}$. It meets the $y$-axis at $A$ and crosses the $x$-axis at $B$.

Differentiation

The polynomials $f(x)$ and $g(x)$ are given by $f(x) = 4x^3 + ax^2 + 8x + 15$ and $g(x) = x^2 + bx + 18$, with $a$ and $b$ as constants.

Trigonometry

By expanding $\cos(2\theta + \theta)$ first, prove that $\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$.

Trigonometry

Solve the equation $4\lvert 5^x - 1 \rvert = 5^x$, and give your answers correct to $3$ decimal places.

Logarithmic and exponential functions

Let the complex number $1 + 2i$ be called $u$. The polynomial $p(x)$ is $2x^3 + ax^2 + 4x + b$, where $a$ and $b$ are real constants. It is known that $u$ is a root of the equation $p(x) = 0$.

Complex numbers

Rewrite $5\sin x - 3\cos x$ in the form $R\sin(x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. State the exact value of $R$ and give $\alpha$ correct to $2$ decimal places.

Trigonometry

The curve defined by $y = x e^{1-2x}$ has a single stationary point.

Differentiation

By using the substitution $u = \sqrt{x}$, determine the exact value of the integral.

Integration

Demonstrate that the equation $\cot 2\theta + \cot \theta = 2$ may be rewritten as a quadratic equation in $\tan \theta$.

Trigonometry

If $(a + bx)\sqrt{1 + 4x}$, where $a$ and $b$ are constants, is written out in ascending powers of $x$, the coefficients of $x$ and $x^2$ are 3 and $-6$ respectively.

Algebra

For $y = \ln(\ln x)$, show that $\frac{dy}{dx} = \frac{1}{x\ln x}$.

Differential equations

The constant $a$ is defined so that $\int_{1}^{a} \frac{\ln x}{\sqrt{x}}\,dx = 6$.

Numerical solution of equations

The two lines $l$ and $m$ are represented by $\mathbf{r} = 3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + s(4\mathbf{i} - \mathbf{j} + 3\mathbf{k})$ and $\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + t(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})$ respectively.

Vectors

Find the value of $x$ that satisfies $3(2^{1-x}) = 7^x$. Give the answer in the form $\frac{\ln a}{\ln b}$, where $a$ and $b$ are integers.

Logarithmic and exponential functions

With regard to the origin $O$, the position vectors for the points $A$ and $B$ are $vec{OA} = \begin{pmatrix}1\\2\\-1\end{pmatrix}$ and $\vec{OB} = \begin{pmatrix}0\\3\\1\end{pmatrix}$.

Vectors

The curve is defined by $y = \sqrt{\tan x}$, for $0 \leq x < \tfrac{1}{2}\pi$.

Numerical solution of equations

Solve the inequality $|3x - a| > 2|x + 2a|$, given that $a$ is a positive constant.

Algebra

For the complex numbers $u = a + ib$ and $w = c + id$, with $a$, $b$, $c$ and $d$ all real.

Complex numbers

Express the fraction $\frac{4x^2 - 13x + 13}{(2x - 1)(x - 3)}$ in partial fractions.

Algebra

On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy $|z - 3 - 2i| \leq 1$ together with $\mathrm{Im}\, z \geq 2$.

Complex numbers

Using the expansions of $\sin(3x + 2x)$ and $\sin(3x - 2x)$, show that the identity $\tfrac{1}{2}(\sin 5x + \sin x) = \sin 3x \cos 2x$ holds.

Integration

The variables $x$ and $y$ are related by the differential equation $e^{2x} \frac{dy}{dx} = 4xy^2$, and the condition $y = 1$ is true when $x = 0$.

Differential equations

If you first expand $(\cos^2 \theta + \sin^2 \theta)^2$, then deduce that $\cos^4 \theta + \sin^4 \theta = 1 - \frac{1}{2} \sin^2 2\theta$.

Trigonometry

A curve has equation $y e^{2x} - y^2 e^x = 2$.

Differentiation

Determine the quotient and remainder for $2x^4 + 1$ divided by $x^2 - x + 2$.

Algebra

A plantation with total area $20\text{ km}^2$ is being affected by a plant disease. At time $t$ years, the diseased area is $x\text{ km}^2$, and the rate at which $x$ increases is proportional to the ratio of the infected area to the area that is still unaffected. When $t = 0$, $x = 1$ and $\frac{dx}{dt} = 1$.

Differential equations

The complex number $-\sqrt{3} + i$ is called $u$.

Complex numbers

Sketch the graph for $y = |2x - 3|$.

Algebra

Solve the equation $4^{x-2} = 4^x - 4^2$, and give your answer rounded to $3$ decimal places.

Logarithmic and exponential functions

Find the exact value for $\int_{\frac{\pi}{3}}^{\pi} x \sin \frac{1}{2}x\, dx$.

Integration

Find the values of $\sin \theta = 3 \cos 2\theta + 2$, for $0^{\circ} \leq \theta \leq 360^{\circ}$.

Trigonometry

After expanding $\cos(x - 60^{\circ})$, demonstrate that $2\cos(x - 60^{\circ}) + \cos x$ can be expressed in the form $R\cos(x - \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.

Trigonometry

The curve is defined by $\ln(x + y) = x - 2y$.

Differentiation

The diagram shows $OABCD$ as a pyramid with vertex $D$. Its horizontal base $OABC$ is a square with side $4$ units. Edge $OD$ is vertical, and $OD = 4$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$ respectively. $M$ is the midpoint of $AB$, and $N$ is a point on $CD$ such that $DN = 3NC$.

Vectors

Define $f(x) = \frac{1}{(9 - x)\sqrt{x}}$.

Integration

A bus starts from rest and accelerates uniformly for $12\ \text{s}$. It then continues at constant speed for $30\ \text{s}$ before slowing down uniformly to rest over a further $6\ \text{s}$. The total distance covered is $585\ \text{m}$.

Kinematics of motion in a straight line

Two small smooth spheres $A$ and $B$, with equal radii and masses $km\ \text{kg}$ and $m\ \text{kg}$ respectively, where $k > 1$, are able to move freely on a smooth horizontal plane. $A$ moves towards $B$ at speed $6\ \text{m s}^{-1}$, and $B$ moves towards $A$ at speed $2\ \text{m s}^{-1}$. Following the collision, $A$ and $B$ coalesce and then travel with speed $4\ \text{m s}^{-1}$.

Momentum

Four coplanar forces of magnitudes $24\ \text{N}$, $P\ \text{N}$, $20\ \text{N}$ and $36\ \text{N}$ act at a point in the directions indicated in the diagram. The system is in equilibrium. Knowing that $\sin \alpha = \frac{3}{5}$, determine the values of $P$ and $\theta$.

Forces and equilibrium

A particle of mass $12\text{ kg}$ is at rest on a rough plane inclined at an angle of $25^\circ$ to the horizontal. A force of magnitude $P\text{ N}$, acting parallel to a line of greatest slope on the plane, is applied to stop the particle from sliding down the plane. The coefficient of friction between the particle and the plane is $0.35$.

Forces and equilibrium

A car with mass $1600\text{ kg}$ moves at a constant speed of $20\text{ m s}^{-1}$ up a straight road that makes an angle of $\sin^{-1} 0.12$ to the horizontal.

Energy, work and power

A particle $P$ travels along a straight line from a point $O$ and is brought to rest $14\,\text{s}$ afterwards. If $t\,\text{s}$ have elapsed since leaving $O$, the velocity $v\,\text{m s}^{-1}$ of $P$ is defined by $v = pt^2 - qt \quad 0 \leq t \leq 6,$ $v = 63 - 4.5t \quad 6 \leq t \leq 14,$ where $p$ and $q$ are positive constants. The acceleration of $P$ is zero when $t = 2$.

Kinematics of motion in a straight line

Particles $A$ and $B$, having masses $2\,\text{kg}$ and $3\,\text{kg}$ respectively, are joined by a light inextensible string. Particle $B$ rests on a smooth fixed plane inclined at $18^\circ$ to the horizontal ground. The string goes over a fixed smooth pulley at the top of the plane. Particle $A$ hangs vertically beneath the pulley and is $0.45\,\text{m}$ above the ground (see diagram). The system is released from rest while the string is taut. When $A$ reaches the ground, the string breaks.

Energy, work and power

The diagram presents a velocity-time graph that describes a car’s motion. It is made up of six straight-line sections. Starting from rest, the car speeds up to $20\,\text{m s}^{-1}$ in $5\,\text{s}$, and then continues at this speed for another $20\,\text{s}$. It next slows to $6\,\text{m s}^{-1}$ in $5\,\text{s}$. That speed is then kept for a further $(T - 30)\,\text{s}$. After that, the car speeds up again to $20\,\text{m s}^{-1}$ over $(50 - T)\,\text{s}$, before slowing to rest in $10\,\text{s}$.

Kinematics of motion in a straight line

A van with mass $3600\,\text{kg}$ is pulling a trailer of mass $1200\,\text{kg}$ along a straight horizontal road by means of a light horizontal rope. The van experiences a resistance force of $700\,\text{N}$ and the trailer experiences a resistance force of $300\,\text{N}$.

Forces and equilibrium

The diagram depicts a semi-circular track $ABC$ of radius $1.8\,\text{m}$, fixed in a vertical plane. Points $A$ and $C$ are at the same horizontal level, while point $B$ is at the lowest part of the track. Section $AB$ is smooth, whereas section $BC$ is rough. A small block is released from rest at $A$.

Energy, work and power

A cyclist sets off from rest at point $A$ and moves along the straight road $AB$, then comes to rest again at $B$. The cyclist’s displacement from $A$ after $t$ seconds is $s\,\text{m}$, where $s = 0.004(75t^2 - t^3)$.

Kinematics of motion in a straight line

A railway engine with mass $75\,000\,\text{kg}$ is ascending a straight hill that makes an angle $\alpha$ with the horizontal, where $\sin\alpha = 0.01$. The engine moves at a steady speed of $30\,\text{m s}^{-1}$. Its power output is $960\,\text{kW}$. A constant force opposes the motion of the engine.

Energy, work and power

A block with mass $5\,\text{kg}$ is kept in equilibrium close to a vertical wall by two light strings and a horizontal force of magnitude $X\,\text{N}$, as the diagram shows. Each string is inclined at $60^{\circ}$ to the vertical.

Forces and equilibrium

Particles $P$ and $Q$ have masses $m\,\text{kg}$ and $2m\,\text{kg}$ respectively. Initially, they are both at rest and are $6.4\,\text{m}$ apart on the same line of greatest slope of a rough plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = 0.8$ (see diagram). Particle $P$ is released from rest and moves down the line of greatest slope. At the same time, particle $Q$ is projected up that same line of greatest slope with speed $10\,\text{m s}^{-1}$. The coefficient of friction between each particle and the plane is $0.6$.

Momentum

A metal post is forced straight down into the ground by allowing a heavy object to fall onto it from above. The object has mass $120\text{ kg}$ and the post has mass $40\text{ kg}$ (see diagram). The object strikes the post at speed $8\text{ m s}^{-1}$ and stays in contact with it after the collision.

Momentum

A particle with mass $8\text{ kg}$ is held at rest in equilibrium by two light inextensible strings, which are inclined at $60^\circ$ and $45^\circ$ above the horizontal.

Forces and equilibrium

A ball with mass $1.6\,\text{kg}$ is let go from rest from a position $5\,\text{m}$ above horizontal ground. As soon as it strikes the ground, it instantly loses $8\,\text{J}$ of kinetic energy and then moves upwards.

Energy, work and power

A car with mass $1400\,\text{kg}$ is travelling on a straight road while a constant force of $1250\,\text{N}$ opposes its motion.

Energy, work and power

A particle $P$ travels along a straight line, beginning from rest at a point $O$ on the line. After $t\text{ s}$ from leaving $O$, the acceleration of $P$ is $k(16 - t^2)\text{ m s}^{-2}$, where $k$ is a positive constant, and its displacement from $O$ is $s\text{ m}$. The velocity of $P$ is $8\text{ m s}^{-1}$ when $t = 4$.

Kinematics of motion in a straight line

The diagram depicts a particle of mass $5\text{ kg}$ resting on a rough horizontal table, with two light inextensible strings attached to it and passing over smooth pulleys fixed at the table’s edges. Particles of masses $4\text{ kg}$ and $6\text{ kg}$ hang freely from the two string ends. The particle of mass $6\text{ kg}$ is $0.5\text{ m}$ above the ground. The system is in limiting equilibrium.

Newton's laws of motion

Two fair coins are tossed together. The random variable $X$ represents the number of paired tosses needed until two tails are seen at the same time for the first time.

Discrete random variables

For 40 values of $x$, the following totals are given: $\sum (x - k) = 520$, $\sum (x - k)^2 = 9640$, with $k$ a constant.

Representation of data

At bedtime, Suki drinks either chocolate, tea or milk, with probabilities $0.45$, $0.35$ and $0.2$ respectively. If she chooses chocolate, the chance that she also takes a biscuit is $0.3$. If she chooses tea, the chance that she also takes a biscuit is $0.6$. If she chooses milk, she never takes a biscuit.

Probability

One fair spinner has edges labelled $0, 1, 2, 2$. A second fair spinner has edges labelled $-1, 0, 1$. Both spinners are spun once. For each spinner, the number on the edge where it stops is recorded. The random variable $X$ represents the total of the two numbers.

Discrete random variables

Raman and Sanjay are two of the $9$ people in a quiz team altogether. Two photographs of the quiz team are to be taken. In the first photograph, the $9$ members are arranged in a straight line.

Permutations and combinations

The masses, in kg, of 15 rugby players from the Rebels club and 15 soccer players from the Sharks club are given below.

Representation of data

Karli's daily time on social media, measured in minutes, follows a normal distribution with mean 125 and standard deviation 24.

The normal distribution

There are exactly 180 students at a college, and each student plays one, and only one, of the piano, the guitar, or the drums. The numbers of male and female students who play the piano, the guitar, and the drums are shown in the table below: Male: Piano 25, Guitar 44, Drums 11; Female: Piano 42, Guitar 38, Drums 20. One student from the college is selected at random.

Probability

A set of $6$ people is to be selected from $4$ men and $11$ women.

Permutations and combinations

The bag has $5$ yellow marbles and $4$ green marbles. Three marbles are chosen at random from the bag, without replacement.

Discrete random variables

How many distinct arrangements are possible for the 9 letters in TELESCOPE?

Permutations and combinations

For a particular region, each day in October has a wet-day probability of $0.16$, independently of the other days.

Probability

The completion times, measured in minutes, for this task among workers at a large company are normally distributed with mean $32.2$ and standard deviation $9.6$.

The normal distribution

The distances, $x$ m, that $140$ children travelled to school were recorded. The findings are summarised in the table below: distances $x \leq 200, x \leq 300, x \leq 500, x \leq 900, x \leq 1200, x \leq 1600$ with matching cumulative frequencies $16, 46, 88, 122, 134, 140$.

Representation of data

The local sports club has 26 members, including Mr and Mrs Khan and their son Abad. A party is being organised for Abad’s birthday, but the venue can accommodate only 20 guests.

Permutations and combinations