Feb/March 2018

Solve the inequality $|5x + 2| > |4x + 3|$.

Algebra

The curve is given by the equation $y = 4x \sin \frac{1}{2}x$.

Differentiation

Apply the trapezium rule with four intervals to estimate $\int_0^8 \ln(x + 2)\,dx$, giving the result to 3 significant figures.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x) = 4x^3 + 4x^2 - 29x - 15$.

Logarithmic and exponential functions

You are told that $\int_{-a}^{2a} 4e^{-2x}\,dx = 25$, with $a$ a positive constant.

Numerical solution of equations

Show that $\cosec 2x + \cot 2x \equiv \cot x$ by simplifying the left-hand side.

Integration

The diagram displays a section of the curve given by the parametric equations $x = t^2 + 4t$, $y = t^3 - 3t^2$. It has a minimum at $M$ and meets the $x$-axis at $P$.

Differentiation

Apply the trapezium rule using three intervals to estimate the value of $\int_{0}^{\frac{\pi}{4}} \sqrt{1-\tan x}\, dx$, giving your answer correct to $3$ decimal places.

Numerical solution of equations

Line $l$ is described by $\mathbf{r} = 4\mathbf{i} + 3\mathbf{j} - \mathbf{k} + \mu(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k})$. Plane $p$ is given by $2x - 3y - z = 4$.

Vectors

Expand $\sqrt[4]{(1-4x)}$ in ascending powers of $x$, including terms up to and including the term in $x^3$, and simplify the coefficients.

Algebra

By using the expansions of $\cos(3x+x)$ and $\cos(3x-x)$, demonstrate that $\frac{1}{2}(\cos 4x + \cos 2x) \equiv \cos 3x \cos x$.

Trigonometry

The variables $x$ and $y$ are linked by the equation $y^n = Ax^3$, with $n$ and $A$ constant. It is stated that $y = 2.58$ when $x = 1.20$, and $y = 9.49$ when $x = 2.51$.

Logarithmic and exponential functions

For the curve, the parametric equations are $x = 2t + \sin 2t$, $y = 1 - 2\cos 2t$, where $-\frac{1}{2}\pi < t < \frac{1}{2}\pi$.

Differentiation

Variables $x$ and $\theta$ are related by the differential equation $x\cos^2\theta\,\frac{dx}{d\theta} = 2\tan\theta + 1$, for $0 \le \theta < \frac{1}{2}\pi$ and $x > 0$. It is given that $x = 1$ when $\theta = \frac{1}{4}\pi$.

Differential equations

By drawing appropriate graphs, show that the equation $e^{2x} = 6 + e^{-x}$ has exactly one real root.

Numerical solution of equations

Take $f(x) = \frac{5x^2 + x + 27}{(2x + 1)(x^2 + 9)}$.

Integration

The complex number $1 + 2i$ is represented by $u$.

Complex numbers

Particles $A$ and $B$, with masses $0.8\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are linked by a light inextensible string which runs over a fixed smooth pulley. The particles are suspended vertically. The system is released from rest.

Newton's laws of motion

The three coplanar forces illustrated in the diagram are balanced.

Forces and equilibrium

A girl with mass $40\,\text{kg}$ moves down a slide in a water park. She begins at point $A$ and travels to point $B$, which lies $7.2\,\text{m}$ vertically beneath the level of $A$, as the diagram shows.

Energy, work and power

A particle with mass $12\,\text{kg}$ rests on a rough plane inclined at an angle of $25^\circ$ to the horizontal. A force of magnitude $P\,\text{N}$ is applied to the particle. This force is horizontal, and the particle is on the point of moving up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is $0.8$.

Forces and equilibrium

A small rocket is launched straight up from ground level from rest, and it travels with constant acceleration. After $10\,\text{s}$, the rocket is at a height of $200\,\text{m}$.

Kinematics of motion in a straight line

A car with mass $1200\,\text{kg}$ can move at a maximum constant speed of $60\,\text{m s}^{-1}$ on a straight horizontal road. When the car is moving at $v\,\text{m s}^{-1}$, the resistive force has magnitude $35v\,\text{N}$.

Energy, work and power

A particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = 4 + 0.2t$ for $0 \le t \le 10$, and by $v = -2 + \frac{800}{t^2}$ for $10 \le t \le 20$.

Kinematics of motion in a straight line

A uniform rectangular block has a square base $ABCD$ with $AB = BC = 0.4\,\text{m}$. The block’s height is $h\,\text{m}$. It stands on a rough plane inclined at $30^\circ$ to the horizontal with its base in contact with the plane. The block does not slide. It is given that the block is on the point of toppling when the diagonal $AC$ lies along a line of greatest slope.

Probability

An object is launched from a point on horizontal ground with speed $15\,\text{m s}^{-1}$ at an angle of $35^\circ$ above the horizontal.

Representation of data

A small ball $B$ is joined to one end of a light elastic string whose natural length is $0.4\,\text{m}$ and whose modulus of elasticity is $12\,\text{N}$. The opposite end of the string is fixed at point $A$. The ball is thrown vertically downwards at speed $1\,\text{m s}^{-1}$ from a point $0.4\,\text{m}$ vertically beneath $A$, and it attains its maximum speed at the position $0.7\,\text{m}$ below $A$.

Probability

A particle $P$ is launched from point $O$ on level ground. At time $t\,\text{s}$ after the launch, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical upward displacement is $y\,\text{m}$. The path of $P$ is given by $y = 3x - 0.05x^2$.

Representation of data

One end of a light inextensible string of length $0.4\,\text{m}$ is fixed to the lowest point of a hemisphere of radius $0.4\,\text{m}$, with the hemisphere’s axis vertical. A particle $P$ of mass $0.3\,\text{kg}$ is fastened to the other end of the string. The string is taut and is inclined at an angle of $30^\circ$ to the horizontal. $P$ travels on the smooth inner surface of the hemisphere in a horizontal circle (see diagram).

Probability

A tiny object with mass $0.2\,\text{kg}$ is initially at rest at point $O$ on a rough horizontal surface. The coefficient of friction between the object and the surface is $0.5$. A force of magnitude $P\,\text{N}$ acting at an angle $\theta$ below the horizontal is applied to the object. At time $t\,\text{s}$ after the force starts to act, the object's velocity is $v\,\text{m s}^{-1}$ away from $O$ (see diagram). It is given that $\tan\theta = \frac{3}{4}$ and that $P = 0.4t$ for $0 \leq t \leq 8$.

Representation of data

$ABCD$ denotes a uniform square lamina with side length $0.6\text{ m}$. A circular hole of radius $r\text{ m}$ is cut in the lamina. The centre of the hole is $0.3\text{ m}$ from $AB$ and $0.25\text{ m}$ from $AD$. The lamina is freely suspended from $A$ and hangs so that the axis of symmetry makes an angle of $48^\circ$ with the horizontal (see Fig. 1).

Representation of data

There are 900 students in one year group. The same puzzle is issued to every student, and the time taken, $t$ minutes, to finish it is recorded. The results are arranged in the frequency table below: Time taken, $t$ minutes: $t \leq 3$, $3 < t \leq 4$, $4 < t \leq 5$, $5 < t \leq 6$, $6 < t \leq 8$, $8 < t \leq 10$, $10 < t \leq 14$ with matching frequencies $120$, $180$, $200$, $160$, $110$, $80$, $50$.

Sampling and estimation

A set of $3$ letters is chosen from the $8$ letters in COLLIDER.

Sampling and estimation

Last Saturday, Sarah noted the colour and style of $160$ cars in a car park. Every car that was neither red nor silver was placed in the ‘other’ group. Her findings are displayed in the table below: Red - Saloon $20$, Hatchback $40$, Estate $12$; Silver - Saloon $14$, Hatchback $26$, Estate $10$; Other - Saloon $6$, Hatchback $24$, Estate $8$.

Sampling and estimation

The discrete random variable $X$ is described by the probability distribution shown below.

Linear combinations of random variables

For a set of $n$ values of $x$, the information below was obtained: $\sum(x - 20) = 136$, $\sum(x - 20)^2 = 2888$. The mean of these $n$ values of $x$ is $24.25$.

Sampling and estimation

The digits $1, 3, 5, 6, 6, 6, 8$ may be rearranged to make numerous different $7$-digit numbers.

Sampling and estimation

The packet weights for one particular type of biscuit are assumed to be normally distributed, with mean 400 grams and standard deviation $\sigma$ grams.

Continuous random variables

A survey carried out at one large college found that the fraction of students who own a car is $\frac{1}{4}$.

Sampling and estimation