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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	5	5	5	5	5	5	5	8	7	10	10	10	80
Score

Question 1 Code: 9709/11/M/J/18/1, Topic: Series

$\text{(i)}$ Find the first three terms in the expansion, in ascending powers of $x$, of $(1-2 x)^{5}$. $[2]$

$\text{(ii)}$ Given that the coefficient of $x^{2}$ in the expansion of $\left(1+a x+2 x^{2}\right)(1-2 x)^{5}$ is 12 , find the value of the constant $a$. $[3]$

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

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=x^{2}-4 x+8$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $x^{2}-4 x+8$ in the form $(x-a)^{2}+b$. $[2]$

$\text{(ii)}$ Hence find the set of values of $x$ for which $\mathrm{f}(x)<9$, giving your answer in exact form. $[3]$

Question 3 Code: 9709/13/M/J/13/2, Topic: Circular measure

 

The diagram shows a circle $C$ with centre $O$ and radius $3 \mathrm{~cm}$. The radii $O P$ and $O Q$ are extended to $S$ and $R$ respectively so that $O R S$ is a sector of a circle with centre $O$. Given that $P S=6 \mathrm{~cm}$ and that the area of the shaded region is equal to the area of circle $C$,

$\text{(i)}$ show that angle $P O Q=\frac{1}{4} \pi$ radians, $[3]$

$\text{(ii)}$ find the perimeter of the shaded region. $[2]$

Question 4 Code: 9709/12/M/J/11/3, Topic: Quadratics

The equation $x^{2}+p x+q=0$, where $p$ and $q$ are constants, has roots $-3$ and 5.

$\text{(i)}$ Find the values of $p$ and $q$. $[2]$

$\text{(ii)}$ Using these values of $p$ and $q$, find the value of the constant $r$ for which the equation $x^{2}+p x+q+r=0$ has equal roots. $[3]$

Question 5 Code: 9709/11/M/J/13/3, Topic: Circular measure

 

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $8 \mathrm{~cm}$. Angle $B O A$ is $\alpha$ radians. $O A C$ is a semicircle with diameter $O A$. The area of the semicircle $O A C$ is twice the area of the sector $O A B$.

$\text{(i)}$ Find $\alpha$ in terms of $\pi$. $[3]$

$\text{(ii)}$ Find the perimeter of the complete figure in terms of $\pi$. $[2]$

Question 6 Code: 9709/12/M/J/19/4, Topic: Trigonometry

Angle $x$ is such that $\sin x=a+b$ and $\cos x=a-b$, where $a$ and $b$ are constants.

$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $x$. $[3]$

$\text{(ii)}$ In the case where $\tan x=2$, express $a$ in terms of $b$. $[2]$

Question 7 Code: 9709/12/M/J/11/6, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{x+3}{2 x-1}, x \in \mathbb{R}, x \neq \frac{1}{2}$.

$\text{(i)}$ Show that $\mathrm{ff}(x)=x$. $[3]$

$\text{(ii)}$ Hence, or otherwise, obtain an expression for $\mathrm{f}^{-1}(x)$. $[2]$

Question 8 Code: 9709/13/M/J/11/6, Topic: Series

$\text{(a)}$ A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term. $[4]$

$\text{(b)}$ An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms. $[4]$

Question 9 Code: 9709/11/M/J/12/6, Topic: Vectors

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are such that $\mathbf{u}=\left(\begin{array}{c}p^{2} \\ -2 \\ 6\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}2 \\ p-1 \\ 2 p+1\end{array}\right)$, where $p$ is a constant.

$\text{(i)}$ Find the values of $p$ for which $\mathbf{u}$ is perpendicular to $\mathbf{v}$. $[3]$

$\text{(ii)}$ For the case where $p=1$, find the angle between the directions of $\mathbf{u}$ and $\mathbf{v}$. $[4]$

Question 10 Code: 9709/11/M/J/17/9, Topic: Functions

The function $\mathrm{f}$ is defined by $\displaystyle\mathrm{f}: x \mapsto \frac{2}{3-2 x}$ for $x \in \mathbb{R}, x \neq \frac{3}{2}$.

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

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto 4 x+a$ for $x \in \mathbb{R}$, where $a$ is a constant.

$\text{(ii)}$ Find the value of $a$ for which $\operatorname{gf}(-1)=3$. $[3]$

$\text{(iii)}$ Find the possible values of $a$ given that the equation $\mathrm{f}^{-1}(x)=\mathrm{g}^{-1}(x)$ has two equal roots. $[4]$

Question 11 Code: 9709/12/M/J/19/10, Topic: Series

$\text{(a)}$ In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is $a$.

$\text{(i)}$ Show that the common difference of the progression is $\frac{1}{3} a$. $[4]$

$\text{(ii)}$ Given that the tenth term is 36 more than the fourth term, find the value of $a$. $[2]$

$\text{(b)}$ The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12, find the value of the fifth term. $[4]$

Question 12 Code: 9709/13/M/J/15/11, Topic: Circular measure

 

In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $r$. The point $C$ on $O B$ is such that angle $A C O$ is a right angle. Angle $A O B$ is $\alpha$ radians and is such that $A C$ divides the sector into two regions of equal area.

$\text{(i)}$ Show that $\sin \alpha \cos \alpha=\frac{1}{2} \alpha$. $[4]$

It is given that the solution of the equation in part $\text{(i)}$ is $\alpha=0.9477$, correct to 4 decimal places.

$\text{(ii)}$ Find the ratio

perimeter of region $O A C$ : perimeter of region $A C B$,

giving your answer in the form $k: 1$, where $k$ is given correct to 1 decimal place. $[5]$

$\text{(iii)}$ Find angle $A O B$ in degrees. $[1]$