2020 MEB Foreign Question

$\let\frac=\dfrac$ $\let\frac=\dfrac$
2020 Foreign
MATHEMATICS Time Allowed : (3) Hours

(Answer ALL questions)

1. (a) Functions $f$ and $g$ are defined by $f(x)=3 x+4, g(x)=x^{2}+6$. Find the values of $x$ for which $(f \circ g)(x)=(g \circ f)(x)$. (3 marks)

(b) $x$ and $x+2$ are factors of $p x^{2}-6 x+q$. Find the values of $p$ and $q$. (3 marks)

2.(a) Find and simplify the coefficient of $x^{3}$ in the expansion of $(2+x)^{5}+(1-2 x)^{6}$. (3 marks) (

b) How many terms of an A.P. $24,20,16, \ldots$ give a sum of 0 ? (3 marks)

3. (a) The square of the matrix $\left(\begin{array}{ll}x & 1 \\ 0 & 1\end{array}\right)$ is $\left(\begin{array}{cc}4 & -1 \\ 0 & 1\end{array}\right)$. Find $x$. (3 marks)

(b) If a die is rolled $x$ times, the expected frequency of a prime number turns up is 50 . Find the value of $x$. (3 marks)

4.(a) In the given figure, $A B=B C$ find $x$ and $y$. (3 marks)

(b) If $P$ is a point inside a parallelogram $A B C D$, then prove that $\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PC}}=\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PD}}$. (3 marks)

5.(a) Show that $\frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^{2} \theta$. (3 marks)

(b) Given that $f(x)=(3 x-2)^{2}$, find $f^{\prime}(x)$ and $f^{\prime}(-1)$. (3 marks)

(Answer any FOUR questions)

6.(a) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=3 x-1$ and $g(x)=x+7$. Find $\left(f^{-1} \circ g\right)(x)$ and $\left(g^{-1} \circ f\right)(x)$. (3 marks)

(b) $x^{3}+a x^{2}-x+b$ and $x^{3}+b x^{2}-5 x+3 a$ have a common factor $x+2$. Find the values of $a$ and $b$. Find also the other a common factor. (5 marks)

7.(a) A binary operation $\odot$ on the set of real numbers $R$ is defined by $x \odot y=(4 x+y)^{2}-15 x^{2}$. Is the binary operation commutative? Why? Find also the values of $k$ such that $(\mathrm{k}+1) \odot(\mathrm{k}-2)=109 .$ (5 marks)

(b) If the coefficient of $x^{4}$ in the expansion of $(3+2 x)^{6}$ is equal to the coefficient of $x^{4}$ in the expansion of $(k+3 x)^{6}$, find $k$ (5 marks)

8.(a) Find the solution set of the inequation $(2 x+1)(3 x-1)<14$ by graphical method and illustrate it on the number line. (5 marks)

(b) For a certain A.P. $S_{n}=\frac{n}{2}(3 n-17)$. Find the first 4 terms of the corresponding sequence and a formula for the $\mathrm{n}^{\text {th }}$ term. (5 marks)

9.(a) A G.P. has first term 5 and last term 2560 . If the sum of all its terms is 5115 , how many terms are there? (5 marks)

(b) It is given that $\mathrm{A}=\left(\begin{array}{ll}3 & 1 \\ 5 & \mathrm{p}\end{array}\right)$ and that $\mathrm{A}+\mathrm{A}^{-1}=\mathrm{kI}$, where $\mathrm{p}$ and $\mathrm{k}$ are constants and $\mathrm{I}$ is the identity matrix. Find the values of $p$ and $k$. (5 marks)

10.(a) Given that $A=\left(\begin{array}{cc}4 & -1 \\ -3 & 2\end{array}\right)$, use the inverse matrix of $A$ to solve the simultaneous equations $y-4 x+8=0,2 y-3 x+1=0$ (5 marks)

(b) How many 2-digit numbers less than 30 can you form by using the digits $0,1,2$ and 3 if the repetition of any digit is allowed? If one of these numbers is chosen at random, find the probability that it is a multiple of 3 . Find also the probability that it is a prime number. (5 marks)

(Answer any THREE questions)

11.(a) Two circles intersect at $\mathrm{A}$ and $\mathrm{B}$. At $\mathrm{A}$ a tangent is drawn to each circle meeting the circles again at $\mathrm{P}$ and $\mathrm{Q}$ respectively. Prove that $\angle \mathrm{ABP}=\angle \mathrm{ABQ}$ and $\mathrm{AB}^{2}=\mathrm{BP} \cdot \mathrm{BQ}$. (5 marks)

(b) In trapezium $\mathrm{ABCD}, \mathrm{AB}=3 \mathrm{DC}$ and $\mathrm{AB} / / \mathrm{DC}$. $\mathrm{AC}$ and $\mathrm{BD}$ intersect at $\mathrm{O}$. Prove that $\alpha(\triangle \mathrm{AOB})=9 \alpha(\triangle \mathrm{COD})$. (5 marks)

12.(a) In $\triangle \mathrm{ABC}, \mathrm{AB}=\mathrm{AC}$. $\mathrm{P}$ is a point inside the triangle such that $\angle \mathrm{PAB}=\angle \mathrm{PBC}$. $\mathrm{Q}$ is the point on $\mathrm{BP}$ produced such that $\mathrm{AQ}=\mathrm{AP}$. Prove that $\mathrm{ABCQ}$ is cyclic. (5 marks)

(b) Given that $0^{\circ}<\alpha, \beta<360^{\circ}, \operatorname{cosec} \alpha=\frac{17}{13}, \tan \beta=-\frac{4}{3}$ and $\alpha, \beta$ are in the same quadrant, calculate the values of $\sin 2 \alpha, \cos \frac{1}{2} \beta$ and $\cot (\alpha+\beta)$. (5 marks)

13. (a) Find the matrix which will rotate $30^{\circ}$ and then reflect in the line OY. What is the map of the point $(-1,0)$ ? (5 marks)

(b) If $y=3 e^{\cos x}$, prove that $\frac{d^{2} y}{d x^{2}}=(\cot x-\sin x) \frac{d y}{d x}$. (5 marks)

14.(a) $A$ and $B$ are two points on the level ground which lie on the opposite sides of a tower CD. The distance between $A$ and $B$ is $649 \mathrm{ft}$, and the angles of elevation of the top of the tower $\mathrm{C}$ from $\mathrm{A}$ and $\mathrm{B}$ are $48^{\circ}$ and $75^{\circ}$ respectively. Find the height of the tower $\mathrm{CD}$. (5 marks)

(b) Find the equations of the tangent and normal lines to the curve $y=x^{2}-x-2$ at the point where it meets the positive X-axis. (5 marks)


1.(a) $x=0$ or $x=-4$ (b) $p=-3,q=0$
2.(a) $-120$ (b) $13$
3.(a) $x=-2$ (b) $x=100$
4.(a) $x=72^{\circ},y=144^{\circ}$ (b) Prove
5.(a) Show (b) $f^{\prime}(x)=6(3x-2), f^{\prime}(-1)=-30$
6.(a) $(f^{-1}\circ g)(x)=\frac{x+8}{3}, (g^{-1}\circ f)(x)=3x-8$ (b) $a=2,b=-2$
7.(a) Yes, $k=4,-3$ (b) $k=\pm\frac 43$
8.(a) $-\frac 53 < x < \frac 32$

(b) $-7,-4,-1,2; u_n=3n-10$
9.(a) 10 (b) $p=2,k=5$
10.(a) $x=3,y=4$ (b) 8; $\frac 14,\frac 38$
11.(a) Prove (b) Prove
12.(a) Prove (b) $\sin(2\alpha)=-\frac{2\sqrt{30}}{17},$ $\cos\frac 12\beta=\frac{1}{\sqrt 5},$ $\cot (\alpha+\beta)=\frac{52-6\sqrt{30}}{29+8\sqrt{30}}$
13.(a) $\left(\begin{array}{cc}-\frac{\sqrt 3}{2}&\frac 12\\\frac 12&\frac{\sqrt 3}{2}\end{array}\right),$ $\left(\frac{\sqrt 3}{2},-\frac 12\right)$ (b) Prove
14.(a) 555.5 (b) tangent equation: $y=3(x-2),$ normal equation: $y=-\frac 13(x-2)$

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