## Achieve an A* in A-level mathematics.

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### Never be surprised by a test again

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: $x=\frac{\mathrm{cos}t}{1+{\mathrm{sin}}^{2}t}x=\cfrac\left\{\cos t\right\}\left\{1 + \sin^2t\right\}$ and $y=\frac{\mathrm{sin}t\mathrm{cos}t}{1+{\mathrm{sin}}^{2}t}y=\cfrac\left\{\sin t \cos t\right\}\left\{1+\sin^2t\right\}$
(a)
Prove that the turning points are found when $\mathrm{cos}t=\sqrt{\frac{2}{3}}\cos t=\sqrt\left\{\cfrac\left\{2\right\}\left\{3\right\}\right\}$
(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: $x=\frac{a\mathrm{cos}t}{1+{\mathrm{sin}}^{2}t}x=\cfrac\left\{a\cos t\right\}\left\{1 + \sin^2t\right\}$ and $y=\frac{a\mathrm{sin}t\mathrm{cos}t}{1+{\mathrm{sin}}^{2}t}y=\cfrac\left\{a\sin t \cos t\right\}\left\{1+\sin^2t\right\}$ 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 $\frac{\sqrt{5}}{5}\cfrac\left\{\sqrt\left\{5\right\}\right\}\left\{5\right\}$.
(a)
Find the exact rebound velocity after the bounce.
(4)
(b)
Find the maximum height the ball will reach after its 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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# maths-buddies

###### 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 👍