A Level Additional Mathematics

A Level Additional Mathematics Paper 2 Past Papers, 0 to 0

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Sittings: 1
Questions: 50
Time allowed: 180 min
Syllabus code: 6006/2

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Pass mark: 30

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Mechanics Statistical Tests and Distributions

Explained questions

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?

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