# Cambridge Additional Mathematics 2012 Past Paper May June Paper 21 | 23

## Question 1

(i) Given that $$A =\left(\begin{array}{rr}4 & -3 \\ 2 & 5\end{array}\right),$$ find the inverse matrix $$A ^{-1}$$.
(ii) Use your answer to part (i) to solve the simultaneous equations
$\begin{array}{l} 4 x-3 y=-10 \\ 2 x+5 y=21 \end{array}$

## Question 2

A cuboid has a square base of side $$(2+\sqrt{3}) cm$$ and a volume of $$(16+9 \sqrt{3}) cm ^{3}$$. Without using a calculator, find the height of the cuboid in the form $$(a+b \sqrt{3}) cm$$, where $$a$$ and $$b$$ are integers.

## Question 3

(a)The diagram shows a sketch of the curve $$y=a \sin (b x)+c$$ for $$0^{\circ} \leqslant x \leqslant 180^{\circ}$$. Find the values of $$a, b$$ and $$c .$$
(b) Given that $$f (x)=5 \cos 3 x+1,$$ for all $$x,$$ state
(i) the period of $$f$$,
(ii) the amplitude of $$f$$.

## Question 4

(i) Find $$\frac{d}{d x}\left(x^{2} \ln x\right)$$
(ii) Hence, or otherwise, find $$\int x \ln x d x$$.

## Question 5

(a) Solve the equation $$3^{2 x}=1000$$, giving your answer to 2 decimal places.
(b) Solve the equation $$\frac{36^{2 y-5}}{6^{3 y}}=\frac{6^{2 y-1}}{216^{y+6}}$$.

## Question 6

By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly.
(i) $$\quad A \cap B^{\prime}=\left(A^{\prime} \cap B\right)^{\prime}$$

(ii) $$\quad X \cap Y=X^{\prime} \cup Y^{\prime}$$

(iii) $$\quad(P \cap Q) \cup(Q \cap R)=Q \cap(P \cup R)$$

## Question 7

Given that $$f (x)=x^{2}-\frac{648}{\sqrt{x}},$$ find the value of $$x$$ for which $$f ^{\prime \prime}(x)=0$$

## Question 8

Relative to an origin $$O,$$ the position vectors of the points $$A$$ and $$B$$ are $$2 i -3 j$$ and $$11 i+42 j$$ respectively.
(i) Write down an expression for $$\overrightarrow{A B}$$.

The point $$C$$ lies on $$A B$$ such that $$\overrightarrow{A C}=\frac{1}{3} \overrightarrow{A B}$$
(ii) Find the length of $$\overrightarrow{O C}$$.
The point $$D$$ lies on $$\overrightarrow{O A}$$ such that $$\overrightarrow{D C}$$ is parallel to $$\overrightarrow{O B}$$.
(iii) Find the position vector of $$D$$.

## Question 9

A particle moves in a straight line so that, $$t$$ s after passing through a fixed point $$O$$, its velocity, $$v ms ^{-1}$$, is given by $$v=2 t-11+\frac{6}{t+1}$$. Find the acceleration of the particle when it is at instantaneous rest.

## Question 10

Solutions to this question by accurate drawing will not be accepted.

The diagram shows a trapezium $$A B C D$$ with vertices $$A(11,4), B(7,7), C(-3,2)$$ and $$D .$$ The side $$A D$$ is parallel to $$B C$$ and the side $$C D$$ is perpendicular to $$B C$$. Find the area of the trapezium $$A B C D . \quad[9]$$

## Question 11

The diagram shows a right-angled triangle $$A B C$$ and a sector $$C B D C$$ of a circle with centre $$C$$ and radius $$12 cm$$. Angle $$A C B=1$$ radian and $$A C D$$ is a straight line.
(i) Show that the length of $$A B$$ is approximately $$10.1 cm$$.
(ii) Find the perimeter of the shaded region.

## Question 12

Answer only one of the following two alternatives.

EITHER
The equation of a curve is $$\quad y=2 x^{2}-20 x+37$$
(i) Express $$y$$ in the form $$a(x+b)^{2}+c,$$ where $$a, b$$ and $$c$$ are integers.

(ii) Write down the coordinates of the stationary point on the curve.

A function $$f$$ is defined by $$f : x \mapsto 2 x^{2}-20 x+37$$ for $$x>k$$. Given that the function $$f ^{-1}(x)$$ exists.

(iii) write down the least possible value of $$k$$,
(iv) sketch the graphs of $$y= f (x)$$ and $$y= f ^{-1}(x)$$ on the axes provided,
(v) obtain an expression for $$f ^{-1}$$.

OR

A function g is defined by $$g : x \mapsto 5 x^{2}+p x+72,$$ where $$p$$ is a constant. The function can also be written as $$g : x \mapsto 5(x-4)^{2}+q$$
(i) Find the value of $$p$$ and of $$q$$.
(ii) Find the range of the function g.
(iii) Sketch the graph of the function on the axes provided.
(iv) Given that the function h is defined by $$h : x \mapsto \ln x,$$ where $$x>0,$$ solve the equation $$\operatorname{gh}(x)=12$$