Achieve an A* in A-level mathematics.

Over 500 exam style questions categorised by topic and difficulty. Aimed at students targeting A* grade.
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E
Forces & Motion - Sample Question
Two particles, PP and QQ, of masses 8 kg and 11 kg respectively, hang freely over a fixed smooth pulley, connected by a light inextensible string. They are held at rest with the string taut and QQ 1.5 metres above the ground. The string is assumed to be long enough so that PP will not hit the pulley when the system is released.
The system is released.
(a)
Write the equations of motion for PP and QQ.
(4)
(b)
Find the tension in the string and the acceleration of QQ
(2)
(c)
Find the time taken for QQ to hit the ground.
(3)
M
Differentiation - Sample Question
Shown is the graph of the lemniscate of Bernoulli described parametrically by the equations: x=cos⁡t1+sin⁡2tx=\cfrac{\cos t}{1 + \sin^2t} and y=sin⁡tcos⁡t1+sin⁡2ty=\cfrac{\sin t \cos t}{1+\sin^2t}
(a)
Prove that the turning points are found when cos⁡t=23\cos t=\sqrt{\cfrac{2}{3}}
(8)
(b)
Hence find the exact coordinates of the turning point(s) and label them on a copy of the above graph.
(5)
A general lemniscate of Bernoulli has the parametic form of: x=acos⁡t1+sin⁡2tx=\cfrac{a\cos t}{1 + \sin^2t} and y=asin⁡tcos⁡t1+sin⁡2ty=\cfrac{a\sin t \cos t}{1+\sin^2t} where aa is a constant.
(c)
Determine the cartesian equation of the general lemniscate of Bernoulli and therefore that of. Note that your equation should only contain the squares of a,xa,x and yy.
(5)
(d)
Comment on the significance of the constant aa and how varying it will change the graph.
(2)
H
Constant Acceleration - Sample Question
A bouncy ball is dropped vertically from a platform 56 metres above the ground. On impact with the ground, it bounces but its rebound velocity after each impact follows a geometric sequence with a common ratio of 55\cfrac{\sqrt{5}}{5}.
(a)
Find the exact rebound velocity after the 3rd 3^{\text {rd }} bounce.
(4)
(b)
Find the maximum height the ball will reach after its 2nd 2^{\text {nd }} bounce.
(4)
(c)
Show that the maximum heights the ball will reach also form a geometric sequence and find its common ratio.
(4)
(d)
Assuming the ball continues to bounce in this way, find the total distance travelled by the ball after it has impacted the ground on 11 occasions.
(4)

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Albert

Hey! I'm having trouble with this question, can anyone help out?

Sienna

I was doing that last week, I found this video useful! 🙂

Novica

The key for this one is to use a substitution, try setting u = ln(x) and see if that simplifies your working

Albert

Thanks! That's much clearer now 👍

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