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MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
| MATHEMATICS | Time Allowed : (3) Hours |
Click for Solution
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) Click for Solution 1(a) |
| (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)
Click for Solution 1(b) |
| 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 $)$ Click for Solution 2(a) |
| (b) | How many terms of the arithmetic progession $9,7,5, \ldots .$ add up to $24 ? \quad$ (3 marks) Click for Solution 2(b) |
| 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) Click for Solution 3(a) |
| (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) Click for Solution 3(b) |
| 4(a) | Given: $\mathrm{XY}$ is the tangent at C. Prove: $\mathrm{XY} / / \mathrm{DE}$ (3 marks) Click for Solution 4(a) |
| (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) Click for Solution 4(b) |
| 5(a) | $\begin{array}[t]{ll}\text { Solve the equation } \cos ^{2} x=2+\cos x \text { for } 0^{*} \leq x \leq 360^{\circ} . & \text { }\end{array}$. (3 marks) Click for Solution 5(a) |
| (b) | Differentiate $\mathrm{x}^{2}-3 \mathrm{x}$ with respect to $\mathrm{x}$ from the first principles. (3 marks) Click for Solution 5(b) |
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 $\left(f^{-1} \circ g\right)(x)=\left(g^{-1} \circ f\right)(x)-4 .$ (5 marks) Click for Solution 6(a) |
| (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) Click for Solution 6(b) |
| 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) Click for Solution 7(a) |
| (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) Click for Solution 7(b) |
| 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) Click for Solution 8(a) |
| (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) Click for Solution 8(b) |
| 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 $\mathrm{n}$ " term in terms of n. (5 marks ) Click for Solution 9(a) |
| (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 ) Click for Solution 9(b) |
| 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 intersection of the lines $3 x+4 y=18$ and $6 y+2 x=22$. (5 marks) Click for Solution 10(a) |
| (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) Click for Solution 10(b) |
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) Click for Solution 11(a) |
| (b) | Given : $\triangle P Q R$ with two medians $P M$ and $\mathrm{QN}$ intersecting at $\mathrm{K}$. Prove $: \alpha(\triangle \mathrm{PNK})=\alpha(\Delta \mathrm{QMK})$ (5 marks) Click for Solution 11(b)  |
| 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 $\mathrm{A}$, 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) Click for Solution 12(a) |
| (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) Click for Solution 12(b) |
| 13 (a) | In $\Delta A B C$, if $\angle A: \angle B: \angle C=1: 3: 8$ and $A B=9$, find $A C$. (5 marks) Click for Solution 13(a) |
| (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) Click for Solution 13(b) |
| 14 (a) | By using geometric vectors, show that the diagonals of a parallelogram bisect each other. (5 marks) Click for Solution 14(a) |
| (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) Click for Solution 14(b) |
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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