Feb/March 2020

The function $f$ is specified by $f(x) = \frac{1}{3x + 2} + x^2$ for $x < -1$.

Differentiation

For the curve, the gradient at $(x, y)$ is $\frac{dy}{dx} = 2\sqrt{x + 3} - x$. It has a stationary point at $(a, 14)$, with $a$ a positive constant.

Differentiation

Solve the equation $3\tan^2 x - 5\tan x - 2 = 0$ in the interval $0^\circ \leq x \leq 180^\circ$.

Trigonometry

The endpoints of a diameter of circle $C_1$ are $(-3, -5)$ and $(7, 3)$. Circle $C_1$ is shifted by $\begin{pmatrix}8\\4\end{pmatrix}$ to form circle $C_2$, as the diagram shows.

Coordinate geometry

The graph of $y = f(x)$ is mapped onto the graph of $y = 1 + f\left(\tfrac{1}{2}x\right)$.

Functions

The diagram depicts a portion of the curve with equation $y = x^2 + 1$. The shaded area enclosed by the curve, the $y$-axis and the line $y = 5$ is rotated through $360^\circ$ about the $y$-axis.

Integration

The curve is given by $y = x^2 - 2x - 3$. A point travels along the curve so that, at $P$, the $y$-coordinate is increasing at 4 units per second and the $x$-coordinate is increasing at 6 units per second.

Coordinate geometry

Find the values of $\theta$ that satisfy $\frac{\tan\theta + 3\sin\theta + 2}{\tan\theta - 3\sin\theta + 1} = 2$ for $0^\circ \leq \theta \leq 90^\circ$.

Trigonometry

In the expansion of $(2x + \frac{a}{x^2})^5$, the coefficient of $\frac{1}{x}$ is $720$.

Series

The diagram depicts sector $AOB$, which is a part of a circle centred at $O$ with radius $6\,\text{cm}$ and with angle $AOB = 0.8$ radians. Point $C$ on $OB$ is chosen so that $AC$ is perpendicular to $OB$. Arc $CD$ is part of a circle with centre $O$, and $D$ lies on $OA$.

Integration

A woman’s basic salary in her first year at a certain company is $\$30\,000$, and when that year ends she is also given a bonus of $\$600$.

Series

Write $2x^2 + 12x + 11$ in the form $2(x + a)^2 + b$, where $a$ and $b$ are constants.

Functions

Find the solution to the equation $2\sin(\theta + 30^\circ) + 5\cos\theta = 2\sin\theta$ for $0^\circ < \theta < 90^\circ$.

Trigonometry

Determine the quotient when $4x^3 + 17x^2 + 9x$ is divided by $x^2 + 5x + 6$, and show that the remainder is $18$.

Algebra

It is stated that $\displaystyle \int_a^{3a} \frac{2}{2x - 5} \, dx = \ln \frac{7}{2}$.

Integration

The curve is given by $3x^2 - y^2 - 4\ln(2y + 3) = 26$.

Differentiation

Sketch, on the same diagram, the graphs of $y = |x + 2k|$ and $y = |2x - 3k|$, where $k$ is a positive constant. Give, in terms of $k$, the coordinates of the points where each graph crosses the axes.

Algebra

The curve is given by $y = x^3 e^{0.2x}$ for $x \ge 0$. At the point $P$ on this curve, the gradient is $15$.

Numerical solution of equations

The diagram displays a segment of the curve with equation $y = 4\sin^2 x + 8\sin x + 3$, where $x$ is measured in radians. The curve meets the $x$-axis at point $A$, and the shaded region is enclosed by the curve together with the lines $x = 0$ and $y = 0$.

Integration

Sketch the graph of the function $y = |x - 2|$.

Algebra

The complex numbers $v$ and $w$ are defined by the equations $v + iw = 5$ and $(1 + 2i)v - w = 3i$.

Complex numbers

Solve the equation $\ln 3 + \ln(2x + 5) = 2\ln(x + 2)$. State your answer in simplified exact form.

Logarithmic and exponential functions

By sketching an appropriate pair of graphs, show that the equation $\sec x = 2 - \frac{1}{2}x$ has exactly one root in the interval $0 \le x < \frac{1}{2}\pi$.

Numerical solution of equations

Calculate $\displaystyle \int_{\pi/6}^{3\pi/4} x\sec^2 x\,\mathrm{d}x$. Present your result in a fully simplified exact form.

Integration

Show that the expression $\dfrac{\cos 3x}{\sin x} + \dfrac{\sin 3x}{\cos x}$ is equal to $2\cot 2x$.

Trigonometry

The variables $x$ and $y$ are linked by the differential equation $\frac{dy}{dx} = \frac{1 + 4y^2}{e^x}$. It is known that $y = 0$ when $x = 1$.

Differential equations

The curve is defined by $x^3 + 3xy^2 - y^3 = 5$.

Differentiation

In the diagram, $OABCDEFG$ is a cuboid with $OA = 2$ units, $OC = 3$ units and $OD = 2$ units. The unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OC$ and $OD$, respectively. Point $M$ lies on $AB$ so that $MB = 2AM$. $N$ is the midpoint of $FG$.

Vectors

Define $f(x)$ by $f(x) = \dfrac{2 + 11x - 10x^2}{(1 + 2x)(1 - 2x)(2 + x)}$.

Algebra

A lorry with mass $16000\,\text{kg}$ moves along a straight horizontal road. Its engine is operating at constant power. In $10\,\text{s}$, the work done by the driving force is $750000\,\text{J}$.

Energy, work and power

A particle $P$ with mass $0.4\,\text{kg}$ rests on a rough horizontal floor. The coefficient of friction between $P$ and the floor is $\mu$. A force of magnitude $3\,\text{N}$ acts on $P$ upwards at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac{3}{4}$. Initially the particle is at rest, and its acceleration is $2\,\text{m}\,\text{s}^{-2}$.

Kinematics of motion in a straight line

The figure gives the vertical cross-section of a surface. $A$, $B$ and $C$ are three points on the cross-section. $B$ is at a level $h\,\text{m}$ above $A$. $C$ is at a level $0.5\,\text{m}$ below $A$. A particle with mass $0.2\,\text{kg}$ is projected up the slope from $A$ with initial speed $5\,\text{m}\,\text{s}^{-1}$. As it moves from $A$ to $C$, the particle remains in contact with the surface.

Energy, work and power

A cyclist moves on a straight road with constant acceleration. He goes through points $A$, $B$ and $C$. He takes $2$ seconds to cover each of the segments $AB$ and $BC$, and his speed at $B$ is $4.5\,\text{m s}^{-1}$. The length of $AB$ is $\frac{4}{5}$ of the length of $BC$.

Kinematics of motion in a straight line

Coplanar forces with magnitudes $F\ \text{N}$, $3\ \text{N}$, $6\ \text{N}$ and $4\ \text{N}$ are applied at point $P$, as indicated in the diagram.

Forces and equilibrium

On a straight horizontal test track, driverless vehicles, with no passengers, are undergoing testing. A car of mass $1600\ \text{kg}$ is pulling a trailer of mass $700\ \text{kg}$ along the track. The brakes are applied, producing a deceleration of $12\ \text{m\ s}^{-2}$. The braking force acts only on the car. In addition to the braking force, there are constant resistance forces of $600\ \text{N}$ on the car and $200\ \text{N}$ on the trailer.

Momentum

A particle travels along a straight line through point $O$. Its displacement from $O$ at time $t$ is $s\,\text{m}$, where $s = t^2 - 3t + 2$ for $0 \le t \le 6$, and $s = \frac{24}{t} - \frac{t^2}{4} + 25$ for $t > 6$.

Kinematics of motion in a straight line

Ranuf and Saed are two of the 40 club members. Every one of the 40 members will go to a concert. 35 members will travel by coach and the other 5 will travel by car. Ranuf will be on the coach and Saed will be in the car.

Permutations and combinations

A fair ordinary die is rolled over and over until either a 1 or a 6 appears.

Discrete random variables

For one variety of apples, the weights are normally distributed with mean 82 grams. 22% of the apples in this variety weigh more than 87 grams.

The normal distribution

Richard owns $3$ blue candles, $2$ red candles and $6$ green candles. Except for colour, the candles are identical. He places the $11$ candles in a straight line.

Permutations and combinations

In Greenton, $70\%$ of adults have a car, and a random sample of $8$ adults is selected from Greenton.

The normal distribution

Box A has $7$ red balls and $1$ blue ball. Box B has $9$ red balls and $5$ blue balls. One ball is taken at random from box A and placed into box B. Then a ball is taken at random from box B. The tree diagram below shows the possible colours of the balls selected.

Probability

Helen records the lengths of $150$ fish from one species in a large pond. Their lengths, measured to the nearest centimetre, are displayed in the table below.

Representation of data

On average, the booklets from a certain publisher have 1 incorrect letter in every 30 letters, and the mistakes arise at random. A booklet chosen at random from this publisher has 12500 letters.

The Poisson distribution

The lengths of a particular lizard species are normally distributed with standard deviation $3.2\text{ cm}$. A naturalist records the lengths of a random sample of 100 lizards of this species and obtains an $\alpha\%$ confidence interval for the population mean. The full width of the interval is $1.25\text{ cm}$.

Hypothesis testing

Before the one-way system was brought in, Freda’s mean time for a certain daily trip was $39.2$ minutes. Now she wants to check whether the mean time for that journey has gone down. She records the times, $t$ minutes, for 40 journeys chosen at random and summarises them as follows: $n = 40$, $\sum t = 1504$, $\sum t^2 = 57760$.

Hypothesis testing

The count of accidents on one particular road follows a Poisson distribution with mean $0.4$ per $50$-day period.

The Poisson distribution

Bottles of Lanta each hold about $300$ ml of juice. If the juice volume, in millilitres, in a bottle is $300 + X$, then $X$ is a random variable with probability density function defined by $f(x) = \begin{cases} \frac{3}{4000}(100 - x^2), & -10 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$

Continuous random variables

For large and small cups of tea, the volumes in millilitres are represented by the distributions $N(200, 30)$ and $N(110, 20)$, respectively.

Linear combinations of random variables

A nationwide survey indicates that $95\%$ of year 12 students use social media. Arvin thinks that the proportion of year 12 students at his college who use social media is below the national percentage. He takes a random sample of $20$ students from his college and records how many use social media. He then carries out a test at the $2\%$ significance level.

Hypothesis testing