Feb/March 2019

For the expansion of $(1 - px)^5$, the coefficient of $x^3$ is $-2160$. Determine the constant $p$.

Series

The diagram depicts the curve whose equation is $y = 4x^{\frac{1}{2}}$.

Differentiation

The graph of $y = f(x)$ includes the points $(0, 2)$ and $(3, -1)$.

Differentiation

The diagram shows $CXD$ as a semicircle with centre $A$, radius $7\,\text{cm}$ and diameter $CD$. The straight line $YABX$ is perpendicular to $CD$, while arc $CYD$ lies on a circle centred at $B$ with radius $8\,\text{cm}$.

Integration

The curve is defined by $y = (2x - 1)^{-1} + 2x$.

Differentiation

The vectors $\mathbf{u}$ and $\mathbf{v}$ are given by $\mathbf{u} = \begin{pmatrix} q \\ 2 \\ 6 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 8 \\ q-1 \\ q^2-7 \end{pmatrix}$, with $q$ as a constant.

Coordinate geometry

The first two terms of a geometric progression are $p$ and $2p$ respectively, where $p$ is a positive constant. The sum of the first $n$ terms is greater than $1000p$. Show that $2^n > 1001$.

Series

Solve the equation $3\sin^2 2\theta + 8\cos 2\theta = 0$ for $0^\circ \leq \theta \leq 180^\circ$.

Trigonometry

Express the quadratic $x^2 - 4x + 7$ in the form $(x + a)^2 + b$.

Functions

The diagram illustrates a section of the curve with equation $y = \sqrt{x^3 + x^2}$. The shaded area is enclosed by the curve, the $x$-axis and the line $x = 3$.

Integration

Solve the equation $\sec^2 \theta + \tan^2 \theta = 5\tan \theta + 4$ for $0^{\circ} < \theta < 180^{\circ}$. Include all working needed.

Trigonometry

Where $x$ satisfies $|2x + 3| = |2x - 1|$, determine the value of

Algebra

The variables $x$ and $y$ are related by $y = A e^{px+p}$, with $A$ and $p$ as constants. A plot of $\ln y$ against $x$ forms a straight line through $(1, 2.835)$ and $(6, 6.585)$, as illustrated.

Logarithmic and exponential functions

Determine the quotient when $4x^3 + 8x^2 + 11x + 9$ is divided by $(2x + 1)$, and prove that the remainder equals $5$.

Algebra

The curve is defined by $y = \dfrac{e^{2x}}{4x + 1}$, and $P$ is a point on it with $y$-coordinate $10$.

Numerical solution of equations

Demonstrate that $\int_{1}^{4} \left( \frac{2}{x} + \frac{2}{2x + 1} \right) \, dx = \ln 48$.

Integration

For the curve, the parametric equations are $x = 2t - \sin 2t$, $y = 5t + \cos 2t$, where $0 \le t \le \frac{1}{2}\pi$. At point $P$ on the curve, the gradient equals $2$.

Trigonometry

Show that the equation $\log_{10}(x - 4) = 2 - \log_{10} x$ may be rearranged into a quadratic equation in $x$.

Logarithmic and exponential functions

The diagram depicts the curve $y = \sin^3 x\sqrt{\cos x}$ for $0 \leq x \leq \frac{\pi}{2}$, together with its highest point $M$.

Integration

The values in the sequence produced by the iterative formula $x_{n+1} = \frac{2x_n^6 + 12x_n}{3x_n^5 + 8}$, starting from $x_1 = 2$, converges to $\alpha$.

Numerical solution of equations

If $\sin(\theta + 45^\circ) + 2\cos(\theta + 60^\circ) = 3\cos\theta$, determine the exact value of $\tan\theta$ in surd form. There is no need to simplify your answer.

Trigonometry

Demonstrate that $\displaystyle \int_{1}^{4} x^{-\frac{3}{2}} \ln x\, dx = 2 - \ln 4$.

Integration

The variables $x$ and $y$ are linked by $\sin y = \tan x$, with $-\dfrac{\pi}{2} < y < \dfrac{\pi}{2}$.

Differentiation

The variables $x$ and $y$ are related by the differential equation $\frac{dy}{dx} = ky^3 e^{-x}$, where $k$ is a constant. It is known that $y = 1$ when $x = 0$, and that $y = \sqrt{e}$ when $x = 1$.

Differential equations

Show every step and, without a calculator, solve the equation $(1 + i)z^2 - (4 + 3i)z + 5 + i = 0$. Present your solutions in the form $x + iy$, where $x$ and $y$ are real.

Complex numbers

Consider $f(x) = \dfrac{12 + 12x - 4x^2}{(2 + x)(3 - 2x)}$.

Algebra

The two planes are given by the equations $2x + 3y - z = 1$ and $x - 2y + z = 3$.

Vectors

A ring $P$ of mass $0.03\,\text{kg}$ is fitted on a rough vertical rod. A light inextensible string is fastened to the ring and pulled upwards at an angle of $15^\circ$ to the horizontal. The tension in the string is $2.5\,\text{N}$ (see diagram). The ring is in limiting equilibrium and is just about to slide up the rod.

Forces and equilibrium

A particle is launched vertically upwards at a speed of $30\,\text{m s}^{-1}$ from a point on horizontal ground.

Kinematics of motion in a straight line

Four coplanar forces with magnitudes $F\,\text{N}$, $5\,\text{N}$, $25\,\text{N}$ and $15\,\text{N}$ act at point $P$ in the directions shown in the diagram. Since the forces are in equilibrium,

Forces and equilibrium

A $1500\text{ kg}$ car is towing a $300\text{ kg}$ trailer on a straight level road at a steady speed of $20\text{ m s}^{-1}$. The car and trailer are represented as two particles joined by a light rigid horizontal rod. The car’s engine has a power output of $6000\text{ W}$. The resistances to motion are constant, with $R\text{ N}$ acting on the car and $80\text{ N}$ acting on the trailer.

Energy, work and power

For a particle travelling along a straight line, its velocity is $v\,\text{m s}^{-1}$ at $t$ seconds after it leaves the fixed point $O$. The diagram displays a velocity-time graph that represents the particle's motion for $t=0$ to $t=16$. It is made up of five straight line sections. The acceleration from $t=0$ to $t=3$ is $3\,\text{m s}^{-2}$. At $t=5$, the particle's velocity is $7\,\text{m s}^{-1}$ and it is momentarily at rest at $t=8$. It then is at rest again at $t=16$. The smallest velocity reached is $V\,\text{m s}^{-1}$.

Kinematics of motion in a straight line

A particle travels in a straight line. It begins from rest at a fixed point $O$ on the line. At time $t$ seconds after leaving $O$, its acceleration is $a\,\text{m s}^{-2}$, where $a=0.4t^{3}-4.8t^{\frac{1}{2}}$.

Kinematics of motion in a straight line

The figure gives the vertical cross-section $PQR$ of a slide. Segment $PQ$ is a straight section of length $8\,\text{m}$, making an angle $\alpha$ with the horizontal, where $\sin\alpha = 0.8$. The straight section $PQ$ is tangent to the curved section $QR$, and $R$ is $h\,\text{m}$ above the level of $P$. The straight section $PQ$ is rough, whereas the curved section $QR$ is smooth. A particle of mass $0.25\,\text{kg}$ is projected from $P$ towards $Q$ with speed $15\,\text{m s}^{-1}$ and comes to rest at $R$. The coefficient of friction between the particle and $PQ$ is $0.5$.

Energy, work and power

A particle is projected at a speed of $24\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. Determine the speed and the direction of motion of the particle at the instant $4\,\text{s}$ after projection.

Probability

A uniform object is formed by connecting three solid cubes with edge lengths $3\,\text{m}$, $2\,\text{m}$ and $1\,\text{m}$. The object has an axis of symmetry, and the cubes are arranged one above another with the cube of edge $2\,\text{m}$ positioned between the other two cubes (see diagram).

Discrete random variables

A small ball is launched from a point $O$ on level ground. At time $t\,\text{s}$ after launch, its horizontal displacement from $O$ is $x\,\text{m}$ and its vertical displacement upwards from $O$ is $y\,\text{m}$, where $x = 4t$ and $y = 6t - 5t^2$.

Representation of data

A particle $P$ with mass $0.3\,\text{kg}$ is connected to a fixed point $A$ by a light elastic string whose natural length is $0.8\,\text{m}$ and modulus of elasticity is $16\,\text{N}$. The particle $P$ travels in a horizontal circle with centre $O$. It is given that $AO$ is vertical and that angle $OAP$ is $60^\circ$ (see diagram).

Probability

A particle $P$ with mass $0.3\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $0.6\,\text{m}$ and modulus of elasticity is $24\,\text{N}$. The other end of the string is attached to a fixed point $O$. Particle $P$ is released from rest at the point $0.4\,\text{m}$ vertically below $O$.

Probability

Fig. 1 shows the cross-section of a solid cylinder from which a cylindrical hole has been drilled to form a uniform prism. The cylinder has radius $5r$ and the hole has radius $r$. The centre of the hole is $2r$ from the centre of the cylinder.

Representation of data

A particle $P$ is projected horizontally from point $O$ along a rough horizontal surface. The coefficient of friction between $P$ and the surface is $0.2$. A horizontal force of magnitude $0.06t$ N, acting away from $O$, is exerted on $P$, where $t$ is the time after projection. $P$ is at rest when $t = 4$.

Probability

Whenever Tamar goes to work, he either wears a blue suit with probability $0.6$ or a grey suit with probability $0.4$. If his suit is blue, the probability that he wears red socks is $0.2$. If his suit is grey, the probability that he wears red socks is $0.32$.

The Poisson distribution

For $40$ readings of the variable $x$, the information given is that $\Sigma (x - c)^2 = 3099.2$, with $c$ fixed. The standard deviation of those $x$-values is $3.2$.

Sampling and estimation

Train journey times, measured in minutes, between Alphaton and Beeton follow a normal distribution with mean $140$ and standard deviation $12$.

Continuous random variables

The random variable $X$ can only have the values $-1, 1, 2, 3$. The probability that $X$ takes the value $x$ is $kx^2$, where $k$ is a constant.

The Poisson distribution

Shown below are the weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team.

Sampling and estimation

A survey carried out by a large supermarket found that $35\%$ of its customers buy online.

The Poisson distribution

Find how many distinct arrangements of all $9$ letters in the word CAMERAMAN can be made in each of the following situations.

Sampling and estimation