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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	5	6	6	6	5	7	12	12	9	9	10	10	97
Score

Question 1 Code: 9709/11/M/J/19/2, Topic: Quadratics

The line $4 y=x+c$, where $c$ is a constant, is a tangent to the curve $y^{2}=x+3$ at the point $P$ on the curve.

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

$\text{(ii)}$ Find the coordinates of $P$. $[2]$

Question 2 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 3 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 4 Code: 9709/11/M/J/19/5, Topic: Quadratics

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=-2 x^{2}+12 x-3$ for $x \in \mathbb{R}$.

$\text{(i)}$ Express $-2 x^{2}+12 x-3$ in the form $-2(x+a)^{2}+b$, where $a$ and $b$ are constants. $[2]$

$\text{(ii)}$ State the greatest value of $\mathrm{f}(x)$. The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=2 x+5$ for $x \in \mathbb{R}$. $[1]$

$\text{(iii)}$ Find the values of $x$ for which $\operatorname{gf}(x)+1=0$. $[3]$

Question 5 Code: 9709/11/M/J/21/6, Topic: Quadratics

The equation of a curve is $y=(2 k-3) x^{2}-k x-(k-2)$, where $k$ is a constant. The line $y=3 x-4$ is a tangent to the curve.

Find the value of $k$ $[5]$

Question 6 Code: 9709/12/M/J/19/7, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} \mathrm{f}: x & \mapsto 3 x-2, \quad x \in \mathbb{R} \\ \mathrm{g}: x & \mapsto \frac{2 x+3}{x-1}, \quad x \in \mathbb{R}, x \neq 1 \end{aligned} $$

$\text{(i)}$ Obtain expressions for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Solve the equation $\mathrm{fg}(x)=\frac{7}{3}$. $[3]$

Question 7 Code: 9709/13/M/J/11/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 3 x-4, \quad x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto 2(x-1)^{3}+8, \quad x>1 \end{aligned} $$

$\text{(i)}$ Evaluate $\mathrm{fg(2)}$. $[2]$

$\text{(ii)}$ Sketch in a single diagram the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$, making clear the relationship between the graphs. $[3]$

$\text{(iii)}$ Obtain an expression for $\mathrm{g}^{\prime}(x)$ and use your answer to explain why $\mathrm{g}$ has an inverse. $[3]$

$\text{(iv)}$ Express each of $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$ in terms of $x$. $[4]$

Question 8 Code: 9709/12/M/J/12/10, Topic: Functions

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+5 \quad \text { for } x \in \mathbb{R} \\ &\mathrm{g}: x \mapsto \frac{8}{x-3} \quad \text { for } x \in \mathbb{R}, x \neq 3 \end{aligned} $$

$\text{(i)}$ Obtain expressions, in terms of $x$, for $\mathrm{f}^{-1}(x)$ and $\mathrm{g}^{-1}(x)$, stating the value of $x$ for which $\mathrm{g}^{-1}(x)$ is not defined. $[4]$

$\text{(ii)}$ Sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the same diagram, making clear the relationship between the two graphs. $[3]$

$\text{(iii)}$ Given that the equation $f g(x)=5-k x$, where $k$ is a constant, has no solutions, find the set of possible values of $k$. $[5]$

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

The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto 2 x+k, x \in \mathbb{R}$, where $k$ is a constant.

$\text{(i)}$ In the case where $k=3$, solve the equation $\mathrm{ff}(x)=25$. $[2]$

The function $\mathrm{g}$ is defined by $\mathrm{g}: x \mapsto x^{2}-6 x+8, x \in \mathbb{R}$.

$\text{(ii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=\mathrm{g}(x)$ has no real solutions. $[3]$

The function $\mathrm{h}$ is defined by $\mathrm{h}: x \mapsto x^{2}-6 x+8, x>3$.

$\text{(iii)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[4]$

Question 10 Code: 9709/13/M/J/16/10, Topic: Functions

The function $\mathrm{f}$ is such that $\mathrm{f}(x)=2 x+3$ for $x \geqslant 0$. The function $\mathrm{g}$ is such that $\mathrm{g}(x)=a x^{2}+b$ for $x \leqslant q$, where $a, b$ and $q$ are constants. The function fg is such that fg $(x)=6 x^{2}-21$ for $x \leqslant q$

$\text{(i)}$ Find the values of $a$ and $b$. $[3]$

$\text{(ii)}$ Find the greatest possible value of $q$. $[2]$

It is now given that $q=-3$.

$\text{(iii)}$ Find the range of $\mathrm{fg}$. $[1]$

$\text{(iv)}$ Find an expression for $\mathrm{(f g)^{-1}}(x)$ and state the domain of $(f g)^{-1}$. $[3]$

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

The function $\mathrm{f}: x \mapsto 4-3 \sin x$ is defined for the domain $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)=2$. $[3]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. $[2]$

$\text{(iii)}$ Find the set of values of $k$ for which the equation $\mathrm{f}(x)=k$ has no solution. $[2]$

The function $\mathrm{g}: x \mapsto 4-3 \sin x$ is defined for the domain $\frac{1}{2} \pi \leqslant x \leqslant A$.

$\text{(iv)}$ State the largest value of $A$ for which $\mathrm{g}$ has an inverse. $[1]$

$\text{(v)}$ For this value of $A$, find the value of $g^{-1}(3)$. $[3]$

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 $y=\mathrm{g}^{-1}(x)$. $[3]$