$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	5	5	5	6	4	9	10	11	9	13	9	12	98
Score

Question 1 Code: 9709/13/M/J/19/2, Topic: Series

$\text{(i)}$ In the binomial expansion of $\displaystyle \left(2 x-\frac{1}{2 x}\right)^{5}$, the first three terms are $32 x^{5}-40 x^{3}+20 x$. Find the remaining three terms of the expansion. $[3]$

$\text{(ii)}$ Hence find the coefficient of $x$ in the expansion of $\displaystyle \left(1+4 x^{2}\right)\left(2 x-\frac{1}{2 x}\right)^{5}$. $[2]$

Question 2 Code: 9709/12/M/J/19/3, Topic: Differentiation, Integration

A curve is such that $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=x^{3}-\frac{4}{x^{2}}.$ The point $P(2,9)$ lies on the curve.

$\text{(i)}$ A point moves on the curve in such a way that the $x$-coordinate is decreasing at a constant rate of $0.05$ units per second. Find the rate of change of the $y$-coordinate when the point is at $P$. $[2]$

$\text{(ii)}$ Find the equation of the curve. $[3]$

Question 3 Code: 9709/12/M/J/21/4, Topic: Series

The coefficient of $x$ in the expansion of $\displaystyle \left(4 x+\frac{10}{x}\right)^{3}$ is $p$. The coefficient of $\displaystyle \frac{1}{x}$ in the expansion of $\displaystyle \left(2 x+\frac{k}{x^{2}}\right)^{5}$ is $q$.

Given that $p=6 q$, find the possible values of $k$. $[5]$

Question 4 Code: 9709/13/M/J/21/4, Topic: Trigonometry

$\text{(a)}$ Show that the equation $[2]$

$$ \frac{\tan x+\sin x}{\tan x-\sin x}=k $$

where $k$ is a constant, may be expressed as

$$ \frac{1+\cos x}{1-\cos x}=k $$

$\text{(b)}$ Hence express $\cos x$ in terms of $k$. $[2]$

$\text{(c)}$ Hence solve the equation $\displaystyle \frac{\tan x+\sin x}{\tan x-\sin x}=4$ for $-\pi < x < \pi$. $[2]$

Question 5 Code: 9709/12/M/J/21/6, Topic: Coordinate geometry

Points $A$ and $B$ have coordinates $(8,3)$ and $(p, q)$ respectively. The equation of the perpendicular bisector of $A B$ is $y=-2 x+4$.

Find the values of $p$ and $q$. $[4]$

Question 6 Code: 9709/13/M/J/20/8, Topic: Series

The first term of a progression is $\sin ^{2} \theta$, where $0 < \theta < \frac{1}{2} \pi$. The second term of the progression is $\sin ^{2} \theta \cos ^{2} \theta$.

$\text{(a)}$ Given that the progression is geometric, find the sum to infinity. It is now given instead that the progression is arithmetic. $[3]$

$\text{(b)} \quad \text{(i)}$ Find the common difference of the progression in terms of $\sin \theta$. $[3]$

$ \quad \quad \text{(ii)}$ Find the sum of the first 16 terms when $\theta=\frac{1}{3} \pi$. $[3]$

Question 7 Code: 9709/11/M/J/21/8, Topic: Circular measure

 

The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre $C$. The boundary of the plate consists of two arcs $P S$ and $Q R$ of the original circle and two semicircles with $P Q$ and $R S$ as diameters. The radius of the circle with centre $C$ is $4 \mathrm{~cm}$, and $P Q=R S=4 \mathrm{~cm}$ also.

$\text{(a)}$ Show that angle $P C S=\frac{2}{3} \pi$ radians. $[2]$

$\text{(b)}$ Find the exact perimeter of the plate. $[3]$

$\text{(c)}$ Show that the area of the plate is $\left(\frac{20}{3}\pi + 8\sqrt{3}\right)$ cm$^{2}$. $[5]$

Question 8 Code: 9709/11/M/J/21/9, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows:

$$ \begin{aligned} &\mathrm{f}(x)=(x-2)^{2}-4 \text { for } x \geqslant 2, \\ &\mathrm{~g}(x)=a x+2 \text { for } x \in \mathbb{R}, \end{aligned} $$

where $a$ is a constant.

$\text{(a)}$ State the range of $\mathrm{f}$. $[1]$

$\text{(b)}$ Find $\mathrm{f}^{-1}(x)$. $[2]$

$\text{(c)}$ Given that $a=-\frac{5}{3}$, solve the equation $\mathrm{f}(x)=\mathrm{g}(x)$. $[3]$

$\text{(d)}$ Given instead that $\operatorname{ggf}^{-1}(12)=62$, find the possible values of $a$. $[5]$

Question 9 Code: 9709/11/M/J/19/10, Topic: Integration, Differentiation

A curve for which $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 x-5$ has a stationary point at $(3,6)$.

$\text{(i)}$ Find the equation of the curve. $[6]$

$\text{(ii)}$ Find the $x$-coordinate of the other stationary point on the curve. $[1]$

$\text{(iii)}$ Determine the nature of each of the stationary points. $[2]$

Question 10 Code: 9709/13/M/J/19/10, Topic: Coordinate geometry, Integration

 

The diagram shows part of the curve with equation $y=(3 x+4)^{\frac{1}{2}}$ and the tangent to the curve at the point $A$. The $x$-coordinate of $A$ is 4.

$\text{(i)}$ Find the equation of the tangent to the curve at $A$. $[5]$

$\text{(ii)}$ Find, showing all necessary working, the area of the shaded region. $[5]$

$\text{(iii)}$ A point is moving along the curve. At the point $P$ the $y$-coordinate is increasing at half the rate at which the $x$-coordinate is increasing. Find the $x$-coordinate of $P$. $[3]$

Question 11 Code: 9709/12/M/J/20/10, Topic: Differentiation

The equation of a curve is $y=54 x-(2 x-7)^{3}$.

$\text{(a)}$ Find $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$. $[4]$

$\text{(b)}$ Find the coordinates of each of the stationary points on the curve. $[3]$

$\text{(c)}$ Determine the nature of each of the stationary points. $[2]$

Question 12 Code: 9709/12/M/J/18/11, Topic: Coordinate geometry, Integration

 

The diagram shows part of the curve $\displaystyle y=\frac{x}{2}+\frac{6}{x}$. The line $y=4$ intersects the curve at the points $P$ and $Q$.

$\text{(i)}$ Show that the tangents to the curve at $P$ and $Q$ meet at a point on the line $y=x$. $[6]$

$\text{(ii)}$ Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. Give your answer in terms of $\pi$. $[6]$