Cambridge Additional Mathematics 2012 Past Paper May June Paper 21

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.

(Answer)

$\begin{aligned} y &=2 x^{2}-20 x+37 \\ &=2\left(x^{2}-10 x\right)+37 \\ &=2\left(x^{2}-5 x-5 x+5^{2}-5^{2}\right)+37 \\ &\left.=2\left[(x-5)^{2}-25\right]+37\right] \\ &\left.=2(x-5)^{2}-50+37\right] \\ &=2(x-5)^{2}-13 \end{aligned}$

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

(Answer)

Coordinate of the stationary point= (5,-13)

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 $f ^{-1}$ .

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 $\operatorname{gh}(x)=12$

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