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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	3	5	6	6	6	6	8	9	7	8	13	10	87
Score

Question 1 Code: 9709/12/M/J/11/1, Topic: Integration

Find $\displaystyle\int\left(x^{3}+\frac{1}{x^{3}}\right) \mathrm{d} x$. $[3]$

Question 2 Code: 9709/11/M/J/14/4, Topic: Differentiation

A curve has equation $\displaystyle y=\frac{4}{(3 x+1)^{2}}$. Find the equation of the tangent to curve at the point where the line $x=-1$ intersects the curve. $[5]$

Question 3 Code: 9709/13/M/J/14/4, Topic: Trigonometry

$\text{(i)}$ Prove the identity $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x} \equiv \sin x+\cos x$. $[3]$

$\text{(ii)}$ Hence solve the equation $\displaystyle\frac{\tan x+1}{\sin x \tan x+\cos x}=3 \sin x-2 \cos x$ for $0 \leqslant x \leqslant 2 \pi$. $[3]$

Question 4 Code: 9709/11/M/J/19/4, Topic: Coordinate geometry

 

The diagram shows a trapezium $A B C D$ in which the coordinates of $A, B$ and $C$ are $(4,0),(0,2)$ and $(h, 3 h)$ respectively. The lines $B C$ and $A D$ are parallel, angle $A B C=90^{\circ}$ and $C D$ is parallel to the $x$-axis.

$\text{(i)}$ Find, by calculation, the value of $\mathrm{h}$. $[3]$

$\text{(ii)}$ Hence find the coordinates of $D$. $[3]$

Question 5 Code: 9709/13/M/J/13/5, Topic: Trigonometry

$\text{(i)}$ Sketch, on the same diagram, the curves $y=\sin 2 x$ and $y=\cos x-1$ for $0 \leqslant x \leqslant 2 \pi$. $[4]$

$\text{(ii)}$ Hence state the number of solutions, in the interval $0 \leqslant x \leqslant 2 \pi$, of the equations

$\text{(a)}$ $2 \sin 2 x+1=0$, $[1]$

$\text{(b)}$ $\sin 2 x-\cos x+1=0$. $[1]$

Question 6 Code: 9709/11/M/J/20/7, Topic: Trigonometry

$\text{(a)}$ Prove the identity $\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta} \equiv \frac{2}{\cos \theta}$. $[3]$

$\text{(b)}$ Hence solve the equation $\displaystyle \frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=\frac{3}{\sin \theta}$, for $0 \leqslant \theta \leqslant 2 \pi$. $[3]$

Question 7 Code: 9709/13/M/J/14/8, Topic: Quadratics

$\text{(i)}$ Express $2 x^{2}-10 x+8$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants, and use your answer to state the minimum value of $2 x^{2}-10 x+8$. $[4]$

$\text{(ii)}$ Find the set of values of $k$ for which the equation $2 x^{2}-10 x+8=k x$ has no real roots. $[4]$

Question 8 Code: 9709/12/M/J/15/8, Topic: Series

$\text{(a)}$ The first, second and last terms in an arithmetic progression are 56,53 and $-22$ respectively. Find the sum of all the terms in the progression. $[4]$

$\text{(b)}$ The first, second and third terms of a geometric progression are $2 k+6,2 k$ and $k+2$ respectively, where $k$ is a positive constant.

$\text{(i)}$ Find the value of $k$. $[3]$

$\text{(ii)}$ Find the sum to infinity of the progression. $[2]$

Question 9 Code: 9709/13/M/J/21/8, Topic: Functions

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

$$ \begin{aligned}&\mathrm{f}: x \mapsto x^{2}-1 \text { for } x<0, \\&\mathrm{~g}: x \mapsto \frac{1}{2 x+1} \text { for } x<-\frac{1}{2}.\end{aligned} $$

$\text{(a)}$ Solve the equation $\mathrm{fg}(x)=3$. $[4]$

$\text{(b)}$ Find an expression for $(\mathrm{fg})^{-1}(x)$. $[3]$

Question 10 Code: 9709/13/M/J/13/9, Topic: Series

$\text{(a)}$ In an arithmetic progression, the sum, $S_{n}$, of the first $n$ terms is given by $S_{n}=2 n^{2}+8 n$. Find the first term and the common difference of the progression. $[3]$

$\text{(b)}$ The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the $1 \mathrm{st}$ term, the 9 th term and the $n$th term respectively of an arithmetic progression. Find the value of $n$. $[5]$

Question 11 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 12 Code: 9709/13/M/J/12/11, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=8-(x-2)^{2}$, for $x \in \mathbb{R}$.

$\text{(i)}$ Find the coordinates and the nature of the stationary point on the curve $y=\mathrm{f}(x)$. $[3]$

The function $\mathrm{g}$ is such that $\mathrm{g}(x)=8-(x-2)^{2}$, for $k \leqslant x \leqslant 4$, where $k$ is a constant.

$\text{(ii)}$ State the smallest value of $k$ for which $\mathrm{g}$ has an inverse. $[1]$

For this value of $k$,

$\text{(iii)}$ find an expression for $\mathrm{g}^{-1}(x)$, $[3]$

$\text{(iv)}$ sketch, on the same diagram, the graphs of $y=\mathrm{g}(x)$ and $y=\mathrm{g}^{-1}(x)$. $[3]$