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

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	2	6	6	6	7	8	8	7	8	10	9	11	88
Score

Question 1

 

The diagram shows part of the graph of $y=a+b \sin x.$ State the values of the constants $a$ and $b$. $[2]$

Question 2

The point $A$ has coordinates $(-2,6)$. The equation of the perpendicular bisector of the line $A B$ is $2 y=3 x+5$.

$\text{(i)}$ Find the equation of $A B$. $[3]$

$\text{(ii)}$ Find the coordinates of B. $[3]$

Question 3

 

The diagram shows a prism $A B C D P Q R S$ with a horizontal square base $A P S D$ with sides of length $6 \mathrm{~cm}$. The cross-section $A B C D$ is a trapezium and is such that the vertical edges $A B$ and $D C$ are of lengths $5 \mathrm{~cm}$ and $2 \mathrm{~cm}$ respectively. Unit vectors $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are parallel to $A D, A P$ and $A B$ respectively.

$\text{(i)}$ Express each of the vectors $\overrightarrow{C P}$ and $\overrightarrow{C Q}$ in terms of $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$. $[2]$

$\text{(ii)}$ Use a scalar product to calculate angle $P C Q$. $[4]$

Question 4

$\text{(i)}$ Express the equation $3 \sin \theta=\cos \theta$ in the form $\tan \theta=k$ and solve the equation for $0^{\circ}< \theta <180^{\circ}$. $[2]$

$\text{(ii)}$ Solve the equation $3 \sin ^{2} 2 x=\cos ^{2} 2 x$ for $0^{\circ} < x < 180^{\circ}$. $[4]$

Question 5

The line with gradient $-2$ passing through the point $P(3 t, 2 t)$ intersects the $x$-axis at $A$ and the $y$-axis at $B$.

$\text{(i)}$ Find the area of triangle $A O B$ in terms of $t$. $[3]$

The line through $P$ perpendicular to $A B$ intersects the $x$-axis at $C$.

$\text{(ii)}$ Show that the mid-point of $P C$ lies on the line $y=x$. $[4]$

Question 6

A curve for which $\displaystyle\frac{\mathrm{d} y}{\mathrm{~d} x}=7-x^{2}-6 x$ passes through the point $(3,-10)$.

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

$\text{(ii)}$ Express $7-x^{2}-6 x$ in the form $a-(x+b)^{2}$, where $a$ and $b$ are constants. $[2]$

$\text{(iii)}$ Find the set of values of $x$ for which the gradient of the curve is positive. $[3]$

Question 7

The function $\mathrm{f}: x \mapsto x^{2}-4 x+k$ is defined for the domain $x \geqslant p$, where $k$ and $p$ are constants.

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

$\text{(ii)}$ State the range of $\mathrm{f}$ in terms of $k$. $[1]$

$\text{(iii)}$ State the smallest value of $p$ for which $\mathrm{f}$ is one-one. $[1]$

$\text{(iv)}$ For the value of $p$ found in part $\text{(iii)}$, find an expression for $\mathrm{f}^{-1}(x)$ and state the domain of $\mathrm{f}^{-1}$, giving your answers in terms of $k$. $[4]$

Question 8

Points $A$ and $B$ have coordinates $(h, h)$ and $(4 h+6,5 h)$ respectively. The equation of the perpendicular bisector of $A B$ is $3 x+2 y=k$. Find the values of the constants $\mathrm{h}$ and $k$. $[7]$

Question 9

$\text{(a)}$ The first term of an arithmetic progression is $-2222$ and the common difference is 17. Find the value of the first positive term. $[3]$

$\text{(b)}$ The first term of a geometric progression is $\sqrt{3}$ and the second term is $2 \cos \theta$, where $0 < \theta < \pi$. Find the set of values of $\theta$ for which the progression is convergent. $[5]$

Question 10

 

The function $\mathrm{f}: x \mapsto p \sin ^{2} 2 x+q$ is defined for $0 \leqslant x \leqslant \pi$, where $p$ and $q$ are positive constants. The diagram shows the graph of $y=\mathrm{f}(x)$.

$\text{(i)}$ In terms of $p$ and $q$, state the range of $\mathrm{f}$. $[2]$

$\text{(ii)}$ State the number of solutions of the following equations.

$\quad\text{(a)}$ $\mathrm{f}(x)=p+q$ $[1]$

$\quad\text{(b)}$ $\mathrm{f}(x)=q$ $[1]$

$\quad\text{(c)}$ $\displaystyle \mathrm{f}(x)=\frac{1}{2} p+q$ $[1]$

$\text{(iii)}$ For the case where $p=3$ and $q=2$, solve the equation $\mathrm{f}(x)=4$, showing all necessary working. $[5]$

Question 11

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 12

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto 2 x+1 \\ &\mathrm{~g}: x \mapsto x^{2}-2 \end{aligned} $$

$\text{(i)}$ Find and simplify expressions for $\mathrm{fg}(x)$ and $\operatorname{gf}(x)$. $[2]$

$\text{(ii)}$ Hence find the value of $a$ for which $\mathrm{fg}(a)=\operatorname{gf}(a)$. $[3]$

$\text{(iii)}$ Find the value of $b(b \neq a)$ for which $\mathrm{g}(b)=b$. $[2]$

$\text{(iv)}$ Find and simplify an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$. $[2]$

The function $\mathrm{h}$ is defined by

$$ \mathrm{h}: x \mapsto x^{2}-2, \quad \text { for } x \leqslant 0 $$

$\text{(v)}$ Find an expression for $\mathrm{h}^{-1}(x)$. $[2]$