Feb/March 2017

Find the set of values of $k$ for which the equation $2x^2 + 3kx + k = 0$ has two distinct real roots.

Quadratics

The diagram depicts the curve $y = f(x)$ for $x > 0$. It has a minimum at $A$ and meets the $x$-axis at $B$ and $C$. You are given that $\frac{dy}{dx} = 2x - \frac{2}{x^3}$ and that the curve goes through the point $\left(4, \frac{189}{16}\right)$.

Integration

For the expansion of $(\frac{1}{ax} + 2ax^2)^5$, the coefficient of $x$ is $5$. Determine the value of the constant $a$.

Series

The diagram depicts a water container shaped like an inverted pyramid, arranged so that when the water depth is $h$ cm, the water surface is a square with side $\frac{1}{2}h$ cm.

Differentiation

The diagram indicates that $AB = AC = 8$ cm, while angle $CAB = \frac{2}{7}\pi$ radians. Circular arc $BC$ is centred at $A$, circular arc $CD$ is centred at $B$, and $ABD$ lies on one straight line.

Circular measure

The diagram displays the graphs of $y = \tan x$ and $y = \cos x$ for $0 \leq x \leq \pi$. They meet at points $A$ and $B$.

Coordinate geometry

Taking origin $O$ as the reference point, the position vectors of $A$ and $B$ are $overrightarrow{OA} = 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$ and $\overrightarrow{OB} = 7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}$.

Coordinate geometry

For $x > 0$, the function $f$ is given by $f(x) = (4x + 1)^{\tfrac{3}{2}}$.

Differentiation

For $x \geq 0$, the functions $f$ and $g$ are given by $f : x \mapsto 2x^2 + 3$ and $g : x \mapsto 3x + 2$.

Functions

Point $A\,(2, 2)$ is located on the curve $y = x^2 - 2x + 2$.

Differentiation

Solve for $x$ in the equation $2\ln(2x) - \ln(x + 3) = \ln(3x + 5)$.

Logarithmic and exponential functions

If $\tan 2\theta \cot \theta = 8$, show that $\tan^2 \theta = \frac{3}{4}$.

Trigonometry

Solve for $x$ in the inequality $|2x - 5| < |x + 3|$.

Logarithmic and exponential functions

Determine the gradient of the curve $x^2 \sin y + \cos 3y = 4$ at the point $(2, \tfrac{1}{2}\pi)$.

Differentiation

It is stated that $a$ is a positive constant for which $\int_0^a (1 + 2x + 3e^{3x})\,dx = 250$.

Numerical solution of equations

The polynomial $p(x)$ is given by $p(x)=ax^3+bx^2-17x-a$, where $a$ and $b$ are constants. It is also known that $(x+2)$ is a factor of $p(x)$, and that the remainder on dividing $p(x)$ by $(x-2)$ is $28$.

Algebra

The diagram depicts a section of the curve $y=2\cos 2x\cos\left(2x+\frac{\pi}{6}\right)$. The shaded area is enclosed by the curve and the two coordinate axes.

Trigonometry

Solve $\ln(1 + 2^x) = 2$, and give the answer correct to 3 decimal places.

Logarithmic and exponential functions

The diagram depicts the curve $y = (\ln x)^2$. The $x$-coordinate of $P$ is $e$, and the normal to the curve at $P$ intersects the $x$-axis at $Q$.

Integration

Solve $|x - 4| < 2|3x + 1|$.

Algebra

Using suitable sketches, show that the equation $e^{-\frac{1}{2}x} = 4 - x^2$ has one positive root and one negative root.

Numerical solution of equations

Express $8\cos\theta - 15\sin\theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$, and give the exact value of $R$ together with the value of $\alpha$ correct to 2 decimal places.

Trigonometry

For the curve given by $y = e^{-ax} \tan x$, where $a$ is a positive constant, there is just one point in the interval $0 < x < \frac{1}{2}\pi$ where the tangent line is parallel to the $x$-axis.

Differentiation

The line $l$ is described by $\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \lambda(2\mathbf{i} - \mathbf{j} + \mathbf{k})$. The plane $p$ is described by $3x + y - 5z = 20$.

Vectors

The diagram shows a water tank with vertical sides and a horizontal rectangular base. The area of the base is $2\text{ m}^2$. When $t = 0$ the tank contains no water, and water starts to enter at a rate of $1\text{ m}^3$ per hour. At the same time, water also begins leaving through the base at a rate of $0.2\sqrt{h}\text{ m}^3$ per hour, where $h$ represents the depth of water in the tank after $t$ hours.

Differential equations

For this question, calculators are not allowed. Let the polynomial $z^4 + 3z^2 + 6z + 10$ be written as $p(z)$. Let the complex number $-1 + i$ be written as $u$.

Complex numbers

Take $f(x) = \frac{x(6 - x)}{(2 + x)(4 + x^2)}$.

Algebra

A particle with mass $0.4\,\text{kg}$ is launched at a speed of $12\,\text{m s}^{-1}$ up the line of greatest slope on a smooth plane inclined at $30^\circ$ to the horizontal.

Energy, work and power

Particle $P$, with mass $1.6\,\text{kg}$, hangs in equilibrium from two light inextensible strings fixed at points $A$ and $B$. These strings are inclined at $20^\circ$ and $40^\circ$ respectively to the horizontal (see diagram).

Forces and equilibrium

A particle with mass $0.6\,\text{kg}$ rests on a rough plane that makes an angle of $21^\circ$ with the horizontal. A force of magnitude $P$, acting parallel to the line of greatest slope of the plane, holds the particle in equilibrium, as shown in the diagram. The coefficient of friction between the particle and the plane is $0.3$.

Energy, work and power

A car with mass $900\text{ kg}$ travels along the straight horizontal road $ABCD$. On $AB$ and $BC$, the resistive force has constant magnitude $800\text{ N}$, while on $CD$ it has constant magnitude $R\text{ N}$. The engine supplies a constant power of $36\text{ kW}$.

Kinematics of motion in a straight line

A particle $P$ travels in a straight line from a point $O$ and is brought to rest $35\,\text{s}$ later. For $t\,\text{s}$ after leaving $O$, the velocity $v\,\text{m s}^{-1}$ of $P$ is given by $v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$ $v = 2t + 10 \quad 5 \leq t \leq 15,$ $v = a + bt^2 \quad 15 \leq t \leq 35,$ where $a$ and $b$ are constants with $a > 0$ and $b < 0$.

Kinematics of motion in a straight line

Two particles with masses $1.2\,\text{kg}$ and $0.8\,\text{kg}$ are joined by a light inextensible string that runs over a fixed smooth pulley. They hang vertically. The system starts from rest, and both particles are $0.64\,\text{m}$ above the floor (see diagram). During the motion that follows, the $0.8\,\text{kg}$ particle does not reach the pulley.

Newton's laws of motion

A small ball is launched at $15\,\text{m s}^{-1}$ at $60^\circ$ above the horizontal. Find the distance from the point of projection when the ball is moving horizontally.

Representation of data

A cylindrical container is open at the top. Its curved side and circular base are each made from the same thin material of uniform thickness. The container has radius $0.2\,\text{m}$ and height $0.9\,\text{m}$.

Representation of data

A particle $P$ is projected at a speed of $20\,\text{m s}^{-1}$, making an angle of $60^\circ$ below the horizontal, from a point $O$ that is $30\,\text{m}$ above level ground.

Probability

The diagram depicts a uniform lamina $ABCD$ with $AB = 0.75\,\text{m}$, $AD = 0.6\,\text{m}$ and $BC = 0.9\,\text{m}$. Angle $BAD = \angle ABC = 90^\circ$.

Representation of data

Particles P and Q have masses 0.4\,\text{kg} and m\,\text{kg} respectively. P is connected to the fixed point A by a light inextensible string of length 0.5\,\text{m}, which makes an angle of 60^\circ with the vertical. P and Q are linked by a light inextensible vertical string. Q is connected to the fixed point B, vertically below A, by a light inextensible string. BQ is taut and horizontal. The particles move in horizontal circles about an axis through A and B with constant angular speed \omega\,\text{rad s}^{-1} (see diagram). The tension in the string between P and Q is 1.5\,\text{N}.

Probability

$O$ and $A$ are fixed points on a rough horizontal surface, where $OA = 1\,\text{m}$. A particle $P$ of mass $0.4\,\text{kg}$ is projected from $A$ horizontally in the direction $OA$ with speed $U\,\text{m s}^{-1}$ and travels along a straight line. After projection, when $P$ is $x\,\text{m}$ from $O$, its velocity is $v\,\text{m s}^{-1}$. The coefficient of friction between the surface and $P$ is $0.4$. A force of magnitude $\frac{0.8}{x}\,\text{N}$ acts on $P$ in the direction $PO$.

Representation of data

A light elastic string has natural length $0.6\text{ m}$ and modulus of elasticity $24\text{ N}$. One end is fixed at point $O$, and the other end is attached to a particle $P$ of mass $0.4\text{ kg}$, which is in equilibrium hanging vertically beneath $O$.

Probability

The twelve values of $x$ are listed here: $1761.6,\;1758.5,\;1762.3,\;1761.4,\;1759.4,\;1759.1,\;1762.5,\;1761.9,\;1762.4,\;1761.9,\;1762.8,\;1761.0$.

Linear combinations of random variables

A bag has $10$ pink balloons, $9$ yellow balloons, $12$ green balloons and $9$ white balloons. $7$ balloons are chosen at random without replacement.

The Poisson distribution

It is given that $10\%$ of the population enjoy watching Historical Drama on television.

The Poisson distribution

The weights, measured in kilograms, of cereal packets were recorded to $4$ significant figures. The stem-and-leaf diagram below displays the data. Key: $748\;|\;5$ indicates $0.7485\text{ kg}$.

Continuous random variables

A plate of cakes contains $12$ different cakes. Determine the number of ways to divide these cakes between Alex and James if each person gets an odd number of cakes.

Sampling and estimation

Pack A contains ten cards labelled $0, 0, 1, 1, 1, 1, 1, 3, 3, 3$. Pack B contains six cards labelled $0, 0, 2, 2, 2, 2$. One card is drawn at random from each pack. The random variable $X$ is the total of the two card numbers.

Linear combinations of random variables

The middle-finger lengths, measured in centimetres, of women in Raneland follow a normal distribution with mean $\mu$ and standard deviation $\sigma$. It is given that $25\%$ of these women have fingers longer than $8.8\text{ cm}$, while $17.5\%$ have fingers shorter than $7.7\text{ cm}$.

Continuous random variables