Myanmar Matriculation 2019 Mathematics Question (Foreign)

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2019 International 

MATRICULATION EXAMINATION DEPARTMENT OF MYANMAR EXAMINATION

$\begin{array}{lll}\text { MATHEMATICS } & \text { Time Allowed : (3) Hours }\end{array}$

WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.

SECTION (A) (Answer ALL questions) 


1. (a) Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-1$, and $\mathrm{g}(\mathrm{x})=3 \mathrm{x}+1$. Find the values of $\mathrm{x}$ which satisfy the equation $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})=7 \mathrm{x}-4, \quad$ (3 marks)

(b) When $\mathrm{f}(\mathrm{x})=(\mathrm{x}-1)^{3}+6(\mathrm{px}+4)^{2}$ is divided by $\mathrm{x}+2$, the remainder is $-3$. Find the values of p. (3 marks)


2. (a) Find and simplify the coefficient of $x^{6}$ in the expansion of $\left(x-\frac{3}{x}\right)^{14}, x \neq 0 .$ (3 marks )

(b) How many terms of the arithmetic progession $9,7,5, \ldots .$ add up to $24 ? \quad$ (3 marks)


3. (a) Given that $2\left(\begin{array}{rr}1 & 4 \\ -6 & 3\end{array}\right)+\left(\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right)\left(\begin{array}{rr}3 & 4 \\ 5 & -4\end{array}\right)=\left(\begin{array}{cc}a & b \\ 0 & c\end{array}\right)$, find the values of $a, b$ and $c$.(3 marks)

(b) A die is thrown. If the probability of getting a number less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. ( 3 marks)


4. (a) Given: $\mathrm{XY}$ is the tangent at C. 

          Prove: $\mathrm{XY} / / \mathrm{DE}$ (3 marks)


(b) The coordinates of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $(1,0),(4,2)$ and $(5,4)$ respectively. Use vector method to determine the coordinates of $D$ if $ACBD$ is a parallelogram.  (3 marks) 


5 (a) $\begin{array}[t]{ll}\text { Solve the equation } \cos ^{2} x=2+\cos x \text { for } 0^{\circ} \leq x \leq 360^{\circ} . & \text { (3 marks) }\end{array}$

(b) Differentiate $\mathrm{x}^{2}-3 \mathrm{x}$ with respect to $\mathrm{x}$ from the first principles. (3 marks)


SECTION (B) (Answer any FOUR questions)

6. (a) Functions $\mathrm{f}: \mathrm{R} \mapsto \mathrm{R}$ and $\mathrm{g}: \mathrm{R} \mapsto \mathrm{R}$ are defined by $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-1$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}+2$. Find the value of $x$ for which $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)-4 . \quad(5$ marks $)$

(b) The polynomial $a x^{3}+b x^{2}-5 x+2 a$ is exactly divisible by $x^{2}-3 x-4$. Find the values of a and $b$. What is the remainder when it is divided by $x+2 ?$  (5 marks)


7. (a) A binary operation $\odot$ on the.set $\mathrm{R}$ of real numbers is defined by $\mathrm{x} \odot \mathrm{y}=\mathrm{x}^{2}+\mathrm{y}^{2}$. Evaluate $[(1 \odot 3) \odot 2]+[1 \odot(3 \odot 2)]$. Show that $x \odot(y \odot x)=(x \odot y) \odot x .(5$ marks $)$

(b) If the coefficients of $x^{r}$ and $x^{r-2}$ in the expansion of $(1+x)^{2 n}$ are equal, show that $r=n-1$ (5 marks)


8. (a) Find the solution set in $\mathrm{R}$ of the inequation $\mathrm{x}^{2}-2 \mathrm{x} \leq 0$ by algebraic method and illustrate it on the number line. (5 marks)

(b) The sum of the first six terms of an A.P. is 96 . The sum of the first ten terms is one-third of the sum of the first twenty terms. Calculate the first term and the tenth term. ( 5 marks)


9. (a) The sum of the first n terms of a certain sequence is given by $\mathrm{S}_{\mathrm{n}}=\frac{1}{2}\left(3^{\mathrm{n}}-1\right).$  Find the first 3 terms of the sequence and express the $n^{th}$ " term in terms of n. (5 marks )

(b) Using the definition of inverse matrix, find the inverse of the matrix $\left(\begin{array}{ll}3 & 5 \\ 2 & 2\end{array}\right)$. (5 marks ) 


10. (a) Find the inverse of the matrix $\left(\begin{array}{ll}3 & 4 \\ 2 & 6\end{array}\right)$. Use it to determine the coordinates of the point of intersection of the lines $3 x+4 y=18$ and $6 y+2 x=22$. (5 marks)

(b) Construct the table of outcomes for rolling two dice, a blue die and a black die. Find the probability that the score on the blue die is less than that on the black die. Find also the probability that the score on the blue die is prime and the score on the black die is even. (5 marks)


SECTION (C) (Answer any THREE questions)

11. (a) $\mathrm{ABC}$ is an acute-angled triangle inscribed in a circle whose centre is $\mathrm{O}$, and $\mathrm{OD}$ is the perpendicular drawn from $\mathrm{O}$ to $\mathrm{BC}$. Prove $\angle \mathrm{BOD}=\angle \mathrm{BAC}$. (5 marks)

(b) Given : $\triangle P Q R$ with two medians $P M$ and $\mathrm{QN}$ intersecting at  $K$. 

Prove: $\alpha(\triangle \mathrm{PNK})=\alpha(\Delta \mathrm{QMK})$   (5 marks) 



12. (a) Two circles cut at $\mathrm{C}, \mathrm{D}$ and through $\mathrm{C}$ any line $\mathrm{ACB}$ is drawn to meet the circles at $A, B.$  $AD$ and $BD$ are joined and produced to meet the circles again at $E, F. $ If $AF, BE$ produced meet at $\mathrm{G}$, prove that $\mathrm{D}, \mathrm{F}, \mathrm{G}, \mathrm{E}$ are concyclic. (5 marks)

(b) If $\alpha+\beta+\gamma=180^{\circ}$, prove that $\sin 2 \alpha+\sin 2 \beta+\sin 2 \gamma=4 \sin \alpha \sin \beta \sin \gamma$. (5 marks)


13. (a) In $\Delta A B C$, if $\angle A: \angle B: \angle C=1: 3: 8$ and $A B=9$, find $A C$

(b) If $y=\operatorname{In}\left(\sin ^{3} 2 x\right)$, then prove that $3 \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+36=0$. (5 marks)


14. (a) By using geometric vectors, show that the diagonals of a parallelogram bisect each other.

(b) Show that the point $\left(1, \frac{\pi}{2}\right)$ lies on the curve $2 \mathrm{xy}+\pi \sin \mathrm{y}=2 \pi$. Then find the equations of tangent and normal to the curve at the point $\left(1, \frac{\pi}{2}\right)$. (5 marks)


Answers

1(a) $\dfrac 13$ or 2 (b) 1 or 3

2(a) 81081 (b) 4 or 6

3(a) $a=15,b=4,c=-10$ (b) $4<x\le 5$

4(a) Prove (b) $D=(0,-2)$

5(a) $180^{\circ}$ (b) $2x-3$

6(a) $x=3$ (b) $a=2,b=-7, -30$

7(a) 274 (b) Prove

8(a) {$x|0\le x\le 2$} (b) $a=11,u_{10}=29$

9(a) $1,3,9,\ldots$ $u_n=3^{n-1}$

(b) $A^{-1}=\left(\begin{array}{ll}-\dfrac 12 & \dfrac 54\\ \dfrac 12& -\dfrac 34\end{array}\right)$

10 (a) $A^{-1}=\left(\begin{array}{ll}\dfrac 35 & -\dfrac 25\\ -\dfrac 15& \dfrac{3}{10}\end{array}\right)$ , (2,3)
(b) $\dfrac{5}{12},\dfrac 14$
11(a) Prove (b) Prove
12(a) Prove (b) Prove
13 (a) $3\sqrt 6$ (b) Prove
14(a) Prove 
(b) Tangent equation $y-\dfrac{\pi}{2}=-\dfrac{\pi}{2}(x-1)$
      Normal equation $y-\dfrac{\pi}{2}=\dfrac{2}{\pi}(x-1)$



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