A Level Additional Mathematics Practice Questions

The position vectors of the points $A$, $B$ and $C$ relative to the origin $O$ are $a = 3i - j$, $b = 2i + j + 3k$ and $c = i - j + 6k$. Find the value of $a \cdot (b \times c)$.

For three non-zero vectors $a$, $b$ and $c$, the scalar triple product $a \cdot (b \times c)$ measures which quantity?

Let $y = \sinh^{-1} 2x$. Writing $\sinh y$ in exponential form and clearing the fraction gives which equation in $e^y$?

Given that $y = \sinh^{-1} x$, express $y$ in logarithmic form.

The standard result is $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$. Which expression gives $\sum_{r=k+1}^{3k} r^3$?

Using $\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2$, write $\sum_{r=1}^{3k} r^3$ in terms of $k$ in its simplest form.

Given that $\sum_{r=k+1}^{3k} r^3 = k^2(4k+1)(5k+2)$ for $k \geq 1$, evaluate $21^3 + 22^3 + 23^3 + \ldots + 60^3$.

The set of positive rational numbers forms a group under the operation $a * b = \frac{ab}{10}$. Find the identity element.

Under the operation $a * b = \frac{ab}{10}$ on the positive rational numbers, write the inverse of an element $a$ in terms of $a$.

The positive rational numbers form a group under $a * b = \frac{ab}{10}$. Which group axiom fails if the set is narrowed to the positive integers?

The set $G = \{0, 1, 2\}$ forms a group under addition modulo 3. State the identity element.

In the group $G = \{0, 1, 2\}$ under addition modulo 3, find the inverse of the element 2.

The set $G = \{0, 1, 2\}$ under addition modulo 3 is shown to be abelian. Which observation establishes that?

Find the value of $10^n + 12\left(4^{n+1}\right) + 5$ when $n = 1$.

For $f(n) = 10^n + 12\left(4^{n+1}\right) + 5$, the difference $f(n+1) - f(n)$ simplifies to which expression?

An induction proof shows that $10^n + 12\left(4^{n+1}\right) + 5$ is divisible by 9 for every positive integer $n$. What does the inductive step assume?

For $I_n = \int \sin^n ax\, dx$, where $n$ is a positive integer and $a$ is a constant, which reduction formula is correct?

Write $\sin^2 \theta$ in terms of $\cos 2\theta$.

Find $\int \sin^2 3x\, dx$.

Find $\int \sin^4 3x\, dx$.

Given that $y = \sinh^{-1} x$, find $\frac{dy}{dx}$.

Express $4x^2 - 4x + 5$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are integers.

A line has direction $d = 3i - k$ and passes through the point $(8, 1, -1)$. A plane has equation $r \cdot n = 5$, where $n = -i + 10j - 3k$. Which pair of checks shows that the line lies in the plane?

Find the angle between the planes $r \cdot (-i + 10j - 3k) = 5$ and $r \cdot (i + j + 3k) = 6$.

A particle is attached to one end of a light inextensible string of length $1.3$ m whose other end is fixed at A. The particle moves in a horizontal circle with the string at $35^\circ$ to the vertical. Find the radius of the circle, in metres, to 2 decimal places.

A particle of mass $2m$ kg on a string of length $L$ moves in a horizontal circle with the string at an angle $\theta$ to the vertical, at angular speed $\omega$. Resolving vertically gives $T\cos\theta = 2mg$ and horizontally gives $T\sin\theta = 2m\omega^2 L\sin\theta$. Which expression for $\omega^2$ follows from these two equations?

A particle of mass $2m$ kg hangs from a light inextensible string of length $1.3$ m attached to a fixed point A. It moves at constant speed in a horizontal circle with the string at $35^\circ$ to the vertical. Taking $g = 9.81\,\mathrm{ms^{-2}}$, calculate its angular speed in $\mathrm{rad\,s^{-1}}$, to 2 significant figures.

A uniform rectangular lamina ABCD has a mass of $10$ kg. Taking $g = 9.81\,\mathrm{ms^{-2}}$, state the weight of the lamina in newtons.

A uniform rectangular lamina ABCD has $AD = BC = 2a$ and $AB = DC = a$. Vertex A rests on a horizontal table and a force $P$ acts along the side DC. When moments are taken about A, what is the perpendicular distance from A to the line of action of $P$?

A uniform rectangular lamina ABCD of mass $10$ kg has $AD = BC = 2a$ and $AB = DC = a$. It rests in a vertical plane with A on a rough horizontal table and AD inclined at $30^\circ$ to the horizontal. A force $P$ along DC maintains equilibrium. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find $P$ in newtons, correct to 2 decimal places.

A uniform rectangular lamina of mass $10$ kg rests with one vertex A on a horizontal table. A force of $30.22$ N acts along a side that makes an angle of $30^\circ$ with the vertical, so its vertical component is upward. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find the normal reaction at A in newtons, to 2 decimal places.

A small particle is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane. It starts from rest at the highest point of the wire. Through what vertical height, in metres, has it fallen when it reaches the lowest point?

A particle of mass $1$ kg is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane with centre O. It is slightly disturbed from rest at the highest point A. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its speed in $\mathrm{ms^{-1}}$ when angle AOP is $90^\circ$, to 2 decimal places.

A particle of mass $1$ kg is threaded on a smooth circular wire of radius $0.5$ m fixed in a vertical plane. It is slightly disturbed from rest at the highest point. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its speed in $\mathrm{ms^{-1}}$ at the lowest point, to 2 decimal places.

A particle of mass $1$ kg threaded on a smooth circular wire of radius $0.5$ m reaches the lowest point of the wire with $v^2 = 2g$. Which expression gives the reaction $R$ between the wire and the particle there?

A particle of mass $2$ kg hangs at rest from a spring of natural length $1$ m and modulus of elasticity $\lambda$, fixed at O. At rest the stretch is $0.1$ m. Find $\lambda$ in terms of $g$.

A particle of mass $2$ kg hangs from a spring of natural length $1$ m and modulus $20g$, fixed at O, and rests with a stretch of $0.1$ m. It is then pulled down a further $x$ metres. Which expression gives the tension in the spring in that pulled-down position?

A particle performs simple harmonic motion satisfying $\ddot{x} = -10gx$. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find its angular frequency $\omega$ in $\mathrm{rad\,s^{-1}}$, to 2 decimal places.

A particle performs simple harmonic motion satisfying $\ddot{x} = -10gx$. Taking $g = 9.81\,\mathrm{ms^{-2}}$, find the period of the motion in seconds, to 2 decimal places.

A smooth sphere A of mass $2$ kg moving at $4\,\mathrm{ms^{-1}}$ on a smooth horizontal plane collides directly with a stationary smooth sphere B of mass $3$ kg. The coefficient of restitution is $\dfrac14$. Find the speed of B immediately after the impact, in $\mathrm{ms^{-1}}$.

A smooth sphere A of mass $2$ kg moving at $4\,\mathrm{ms^{-1}}$ collides directly with a stationary smooth sphere B of mass $3$ kg, the coefficient of restitution being $\dfrac14$. After the impact B moves off at $2\,\mathrm{ms^{-1}}$. Find the speed of A immediately after the impact, in $\mathrm{ms^{-1}}$.

A smooth sphere B of mass $3$ kg strikes a fixed vertical wall at right angles, moving at $2\,\mathrm{ms^{-1}}$. The coefficient of restitution is $\dfrac14$. Find the speed at which B rebounds, in $\mathrm{ms^{-1}}$.

At a school the pass rate in 'A' level mathematics was $40\%$. After a new teacher was hired, $4$ out of $9$ students passed. A test is to be run at the $5\%$ level for evidence of an improvement. Which pair of hypotheses is correct?

The number of passes $X$ among $9$ students is modelled by $X \sim B(9, 0.4)$. Find $P(X \ge 4)$, to 3 decimal places.

A one-tailed test at the $5\%$ level uses $H_0: p = 0.4$ against $H_1: p > 0.4$, with $4$ passes out of $9$ observed and $P(X \ge 4) = 0.517$ under $H_0$. What is the correct conclusion?

The distance travelled by a commuter driver in a day is normally distributed with mean $360$ km and standard deviation $60$ km. Find the probability that the distance travelled in a day exceeds $370$ km, to 3 decimal places.

A confidence interval for a population mean is to be given at the $98\%$ level, using a normal distribution. State the value of $z$ used, to 3 decimal places.

A random sample of $30$ days gives a mean distance of $350$ km, where the population standard deviation is known to be $60$ km. Which is the $98\%$ confidence interval for the mean distance travelled, to 1 decimal place?