$\newcommand{\iixii}[4]{\left(\begin{array}{cc}#1\\#3 \end{array}\right)}$
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MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
| MATHEMATICS | Time Allowed : (3) Hours |
SECTION (A)
(Answer ALL questions. Choose the correct or the most appropriate answer for each question. Write the letter of the correct or the most appropriate answer.)
1. (1) If $f(x)=3 x-1$ and $g(x)=2 x^{2}+3$, then $(g \circ f)(2)=$
E. 59
A. 51
B. 53
C. 55
D. 57
(2) An operation $\odot$ is defined by $a \odot b=a^{2}-b^{2}$. Then $(2 \odot 3)(1 \odot 2)=$
A. 4
B. $-4$
C. 8
D. $-8$
E. 15
(3) When $(2 x+k)^{2017}+(x-1)^{2}$ is divided by $x+1$, the remainder is 5 , then $k=$
A. $-1$
B. 1
C. $-3$
D. 6
E. 3
(4) If $x-2$ is a factor of $x^{n+1}+5 x^{n}-10 x-36$, then $n=$
A. 2
B. 3
C. 4
D. 5
E. 6
(5) The term independent of $\mathrm{x}$ in the expansion of $\left(\mathrm{x}^{2}+\frac{\mathrm{a}}{\mathrm{x}}\right)^{6}$ is 15 , then $\mathrm{a}=$
A. $\pm 1$
B. 1 only
C. $-1$ only
D. 4 only
E. 6 only
(6) The sum of all the coefficients of the terms in the expansion of $(3 x-1)^{10}$ is
A. 256
B. 512
C. 1024
D. 2048
E. none of these
(7) The solution set of the inequation $\mathrm{kx}^{2} \leq 0$ is $\mathrm{R}$ if
A. $\mathrm{k} \leq 1$
B. $\mathrm{k}>1$
C. $\mathrm{k}<0$
D. $\mathrm{k}>0$
E. $\mathrm{k}>-1$
(8) In an A.P., $\mathrm{S}_{15}=240$. Then $\mathrm{u}_{7}+\mathrm{u}_{8}+\mathrm{u}_{9}=$
A. 36
B. 48
C. 54
D. 60
E. 72
(9) If the A.M between $x$ and $y$ is 3 , then $x^{3}+y^{3}+18 x y=$
A. 216
B. 125
C. 64
D. 27
E. 8
(10) In a G.P., each term is positive, the third term is 18 and the fifth term is 162 , then the
common ratio is
A. $\pm 3$
B. $-3$ only
C. 3 only
D. 6 only
E. $\pm 6$
(11) Let $\mathrm{A}=\left(\begin{array}{ll}1 & 2 \\ 0 & 4\end{array}\right)$ be a matrix and given that $\operatorname{det}(\mathrm{x} \mathrm{A})=4$. Then $\mathrm{x}=$
A. $\pm 2$
B. 4
C. 3
D. $\pm 1$
E. 0
(12) Given that $\mathrm{A}$ is $2 \times 2$ matrix such that $\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right) \mathrm{A}=\left(\begin{array}{rr}2 & -4 \\ -2 & 4\end{array}\right)$, then the matrix $\mathrm{A}$ is
A. $\left(\begin{array}{rr}2 & 1 \\ -1 & 2\end{array}\right)$
B. $\left(\begin{array}{rr}-1 & 2 \\ 1 & -2\end{array}\right)$
C. $\left(\begin{array}{rr}1 & -2 \\ -1 & 2\end{array}\right)$
D. $\left(\begin{array}{ll}2 & -1 \\ 1 & -2\end{array}\right)$
E. $\left(\begin{array}{rr}-2 & 1 \\ 1 & 2\end{array}\right)$
(13) If $\mathrm{A}$ is an event such that $6(\mathrm{P}(\mathrm{A}))^{2}=\mathrm{P}(\operatorname{not} \mathrm{A})$, then $\mathrm{P}(\mathrm{A})=$
$\begin{array}{llll}\text { A. } \frac{1}{2} & \text { B. } \frac{1}{3} & \text { C. } \frac{1}{6} & \text { D. } \frac{2}{3}\end{array}$ E. $\frac{5}{6}$
(14) A die is rolled $x$ times. If the expected frequency of a number which is a multiple of 3 is 30 , then the expected frequency of a number not greater than 3 is
$\begin{array}{llll}\text { A. } 90 & \text { B. } 80 & \text { C. } 75 & \text { D. } 60\end{array}$ E. 45
(15) ABCDEF is a regular hexagon inscribed in a circle $\mathrm{AG}$ is a tangent at $\mathrm{A}$, then $\angle \mathrm{FAG}=$ 
A. $30^{\circ}$
B. $60^{\circ}$
C. $150^{\circ}$
D. $120^{\circ}$
E. none of these
(16) In circle $\mathrm{O}, \mathrm{AC}$ is a diameter, $\mathrm{PA}$ is a tangent at $\mathrm{A}$, and $\mathrm{PBC}$ is a secant meeting the circle at $\mathrm{B}$ and $\mathrm{C}$. If the radius of the circle is $2 \mathrm{~cm}$ and $\angle \mathrm{APB}=30^{\circ}$, then the length of the segment $\mathrm{PB}$, in $\mathrm{cm}$, is
A. 2 $\begin{array}{lll}\text { B. } 4 & \text { C. } 6 & \text { D. } 8\end{array}$ E. 9
(17) If $\triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}, \alpha(\triangle \mathrm{ABC})+\alpha(\Delta \mathrm{PQR})=75 \mbox{ cm}^{2}, \mathrm{AB}$ and $\mathrm{PQ}$ are corresponding
sides and $A B: P Q=4: 3$, then $\alpha(\Delta A B C)$, in $\mathrm{cm}^{2}$, is
$\begin{array}{lllll}\text { A. } 25 & \text { B. } 27 & \text { C. } 36 & \text { D. } 48 & \text { E. } 50\end{array}$
(18) Given that $\overrightarrow{O P}=\left(\begin{array}{l}p \\ 4\end{array}\right)$ and $\overrightarrow{O Q}=\left(\begin{array}{l}2 \\ 3\end{array}\right)$. If $\overrightarrow{P Q}$ is a unit vector, the possible value of $p$ is
$\begin{array}{lllll}\text { A. } 1 & \text { B. } 2 & \text { C. } 3 & \text { D. } 4 & \text { E. } 5\end{array}$
(19) The map of the point $(2,-3)$ by the reflection matrix in $\mathrm{Y}$-axis is
A. $(2,3)$
B. $(2,-3)$
C. $(-2,-3)$
D. $(-2,3) \quad$
E. $(4,6)$
(20) If $\log (\cos x)=k$, then $\log \left(\sec ^{3} x\right)=$
$\begin{array}{ll}\text { A. } \frac{1}{3} k & \text { B. }-\frac{1}{3} k\end{array}$
$\begin{array}{lll}\text { C. } 3 \mathrm{k} & \text { D. }-3 \mathrm{k} & \text { E. none of these }\end{array}$
(21) If $180^{\circ}<\theta<360^{\circ}$ and $\tan \theta=-\sqrt{3}$, then $\theta=$
A. $225^{\circ}$
B. $240^{\circ}$
C. $ 210^{\circ} $
D. $ 300^{\circ}$
E. $330^{\circ}$
(22) In $\triangle \mathrm{ABC}, \angle \mathrm{B}=\angle \mathrm{A}+15^{\circ}$, and $\angle \mathrm{B}=\angle \mathrm{C}-15^{\circ}$. Then $\mathrm{BC}: \mathrm{AC}=$
A. $1: \sqrt{2}$
B. $ 1: \sqrt{3} $
C. $ 1: 2$
D. $ \sqrt{2}: \sqrt{3}$
E. $ 2: \sqrt{3}$
(23) The gradient of the curve $3 x^{2}-y^{2}=11$ at the point $(3,4)$ is
A. $\frac{4}{9}$
B. $\frac{9}{4}$
C. $-\frac{9}{4}$
D. $-\frac{4}{9}$
E. $-\frac{4}{3}$
(24) Given that $y=\sin 4 x+\cos ^{4} x$, then the value of $\frac{d y}{d x}$ when $x=\frac{\pi}{4}$ is
$\begin{array}{ll}\text { A. }-5 & \text { B. } 5\end{array}$ C. $-4$ $\begin{array}{ll}\text { D. } 4 & \text { E. } 6\end{array}$
(25) A stationary point of the curve $y=x+\ln (\cos x)$ is at $x=$
A. 0
B. $\frac{\pi}{4}$
C. $\frac{\pi}{3}$
D. $\frac{\pi}{2}$
E. $\frac{2 \pi}{3}$
Section B
(Answer ALL Questions)
2. If $\mathrm{f}$ and $\mathrm{g}$ are functions such that $\mathrm{f}(\mathrm{x})=2 \mathrm{x}-1$ and $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})=4 \mathrm{x}^{2}-2 \mathrm{x}-3$, find the formula of g in simplified form.
(OR) $x^{2}-1$ is a factor of $x^{3}+a x^{2}-x+b$. When the expression is divided by $x-2$ the remainder is 15 . Find the values of a and b. $(3$ marks)
3. Given that $\sin ^{2} x, \cos ^{2} x$ and $5 \cos ^{2} x-3 \sin ^{2} x$ are in A.P., find the value of $\sin ^{2} x$. (3 marks)
(OR) If $\log _{2} 3, \log _{5} \mathrm{x}, \log _{3} 16$ is a G.P., then find the possible values of $\mathrm{x}$. (3 marks)
4. In circle $\mathrm{O}, \mathrm{PS}$ is a diameter and $\angle \mathrm{POQ}=60^{\circ}, \angle \mathrm{ROS}=70^{\circ}$. find $\angle \mathrm{PTQ} .$ 
5. Prove that $\frac{1+\cos x+\cos 2 x}{\sin x+\sin 2 x}=\cot x$. (3 marks)
6. Find $\displaystyle\lim _{x \rightarrow 2} \frac{2^{2 x}-5\left(2^{x}\right)+4}{2^{x}-4}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{x^{2}-16}{x^{4}-4 x^{3}}$. $(3 \mathrm{marks})$
SECTION (C)
(Answer any SIX questions)
7.(a) Let $\mathrm{f}$ and $\mathrm{g}$ be two functions defined by $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+1$ and $\mathrm{f}(\mathrm{g}(\mathrm{x}))=3 \mathrm{x}-1$. Find the formula of $(\mathrm{f} \circ \mathrm{g})^{-1}$ and hence find $(\mathrm{f} \circ \mathrm{g})^{-1}(8) . \quad(5 \mathrm{marks})$
(b) Let $\mathrm{R}$ be the set of real numbers and a binary operation $\odot$ on $\mathrm{R}$ be defined by $a \odot b=2 a b-a+4 b$ for $a, b \in R$. Find the values of $3 \odot(2 \odot 4)$ and $(3 \odot 2) \odot 4$. If $x \odot y=2$ and $x \neq-2$, find the numerical value of $y\odot y$. (5 marks)
8.(a) Given that $4 \mathrm{x}^{4}-9 \mathrm{a}^{2} \mathrm{x}^{2}+2\left(\mathrm{a}^{2}-7\right) \mathrm{x}-18$ is exactly divisible by $2 \mathrm{x}-3 \mathrm{a}$, show that $\begin{array}{ll}a^{3}-7 a-6=0 \text { and hence find the possible values of a. } & \text { (5 marks) }\end{array}$
(b) The first four terms in the binomial expression of $(a+b)^{n}$, in descending powers of $a$, are $w, x, y$ and $z$ respectively. Show that $(n-2) x y=3 n w z$. (5 marks)
9.(a) Find the solution set of the inequation $12-25 x+12 x^{2} \leq 0$ by graphical method and illustrate it on the number line. $\quad$ (5 marks)
(b) Let a and $b$ be two numbers, $x$ be the single arithmetic mean of $a$ and $b$. Show that the sum of $\mathrm{n}$ arithmetic means between a and $\mathrm{b}$ is $\mathrm{nx}$. $\quad$ (5 marks)
10.(a) The three numbers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ between 2 and 18 are such that their sum is 25 , the numbers $2, \mathrm{a}, \mathrm{b}$ are consecutive terms of an arithmetic progression, and the numbers $\mathrm{b}, \mathrm{c}, 18$ are consecutive terms of a geometric progression. Find the three numbers. ( 5 marks)
(b) If $\mathrm{ps} \neq \mathrm{qr}$, find the $2 \times 2$ matrix $\mathrm{X}$ such that $\left(\begin{array}{ll}\mathrm{p} & \mathrm{q} \\ \mathrm{r} & \mathrm{s}\end{array}\right) \mathrm{X}=\left(\begin{array}{ll}\mathrm{q} & \mathrm{p} \\ \mathrm{s} & \mathrm{r}\end{array}\right)$. Find also $\mathrm{X}^{-1}$, if it exists. (5 marks)
11.(a) Find the inverse of the matrix $\left(\begin{array}{ll}7 & 4 \\ 3 & 2\end{array}\right)$, and use it to find the solution set of the system of equations $\quad 7 x+4 y=16$ ,$2 y+3 x=6 . \quad(5$ marks $)$
(b) $\mathrm{X}$ and $\mathrm{Y}$ are two independent events. The probability that the event $\mathrm{X}$ will occur is twice the probability that the event $\mathrm{Y}$ will occur and the probability that $\mathrm{Y}$ will not occur is four times the probability that $\mathrm{X}$ will not occur. Then find the probability that both $X$ and $Y$ will not occur. $\quad(5$ marks)
12.(a) In the figure, $O$ is the centre of the circle, $\mathrm{AFG} / / \mathrm{OB}, \angle \mathrm{AOB}=120^{\circ}$ and $\angle \mathrm{EAG}=80^{\circ}$.
Find $\angle \mathrm{BFG}$ and $\angle \mathrm{EBO}$. (5 marks)
(b) In $\triangle \mathrm{ABC}, \mathrm{AB}=\mathrm{AC} . \mathrm{P}$ is any point on $\mathrm{BC}$, and $\mathrm{Y}$ any point on $\mathrm{AP}$. The circle $BPY $ and $CPY$ cut $\mathrm{AB}$ and $\mathrm{AC}$ respectively at $\mathrm{X}$ and $\mathrm{Z}$. Prove $\mathrm{XZ} / / \mathrm{BC} . \quad$ ( 5 marks)
13.(a) In trapezium $\mathrm{ABCD}$ the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ intersect at $\mathrm{O}$. If $\mathrm{AB} / / \mathrm{DC}$ and $16 \alpha(\triangle \mathrm{AOB})=25 \alpha(\Delta \mathrm{COD})$, find the ratios $\mathrm{AB}: \mathrm{CD}$ and $\alpha(\triangle \mathrm{BOC}): \alpha(\Delta \mathrm{COD})$ $(5 \mathrm{marks})$
(b) The position vectors of $\mathrm{A}$ and $\mathrm{B}$ relative to an origin $\mathrm{O}$ are $\left(\begin{array}{c}5 \\ 15\end{array}\right)$ and $\left(\begin{array}{c}13 \\ 3\end{array}\right)$ respectively. Given that $C$ lies on $\mathrm{AB}$ and has position vector $\left(\begin{array}{c}2 t+1 \\ t+1\end{array}\right)$, find the value of $t$ and the ratio $AC : CB.$ (5 marks)
14.(a) Solve the equation $ \sin x+\sin \frac{x}{2}=0 \text { for } 0 \leq x \leq 2 \pi.$ (5 marks)
(b) $\mathrm{A}$ and $\mathrm{B}$ are two points on one bank of a straight river, distant from one another $649 \mathrm{~m} . \mathrm{C}$ is on the other bank and the measures of the angles $\mathrm{CAB}, \mathrm{CBA}$ are respectively $48^{\circ} 31^{\prime}$ and $75^{\circ} 25^{\prime} .$ Find the width of the river. $\quad(5$ marks $)$
15.(a) Show that the point $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$ lies on the curve $x \sin 2 y=y \cos 2 x$. Then find the equations of tangent and normal to the curve at the point $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$. (5 marks)
(b) If $y=A \cos \left(\ln \frac{x}{2}\right)+B \sin \left(\ln \frac{x}{2}\right)$, where $A$ and $B$ are constants, show that $\mathrm{x}^{2} \mathrm{y}^{\prime \prime}+\mathrm{x} \mathrm{y}^{\prime}+\mathrm{y}=0$.
Answers
1. $\begin{array}[t]{lllll}1\text{ B }&6\text{ C }&11\text{ D }&16\text{ C }&21\text{ D }\\2\text{ E }&7\text{ C }&12\text{ C }&17\text{ D }&22\text{ D }\\3\text{ E }&8\text{ B }&13\text{ B }&18\text{ B }&23\text{ B }\\4\text{ B }&9\text{ A }&14\text{ E }&19\text{ C }&24\text{ A }\\5\text{ A }&10\text{ C }&15\text{ C }&20\text{ D }&25\text{ B }\end{array}$
2. $g(x)=x^2+x-3$ OR $a=3,b=-3$
3. $\dfrac{3}{5}$ OR $x=25$ or $\dfrac{1}{25}$
4. $65^{\circ}$
5. Prove
6. 3,0
7(a) $(f\circ g)^{-1}(x)=\dfrac{x+1}{3}, 3$ (b) 279,135,2
8(a) $a=-1,3,-2$ (b) Prove
9(a) $\{x|\dfrac{3}{4}\le x\le \dfrac{4}{3}\}$ (b) Prove
10(a) 5,8,12 (b) $X=\iixii 0110, X^{-1}=\iixii 0110$
11(a) $\iixii{1}{-2}{-\dfrac 32}{\dfrac 72}$ $(4,-3)$ (b) $\dfrac{4}{19}$
12(a) $60^{\circ},40^{\circ}$ (b) Prove
13(a) $\dfrac{5}{4},5:4$ (b) $t=5,AC:CB=3:1$
14(a) $0,2\pi,\dfrac{4\pi}{3}$ (b) 567
15(a) $y-2x=0,8y+4x=5\pi$ (b) Prove
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