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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) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x^{2}-1$ and $g(x)=2^{x}$ If $(g \circ f)(k)=8$, then $k=$
A. 2
B. $-2$
C. $\pm 2$
D. 5
E. $\pm 8$
(2) If a binary operation $\odot$ is defined by $p \odot q=$ the remainder when $2 p+q$ is divided by 10, then $6 \odot 4=$
A. 3
B. 4 $=$
E. 7
C. 5
D. 6
(3) Given $n$ is an integer, the remainder when $x^{2 n}-4 x+7$ is divided by $x+1$ is
A. 14
B. 12 .
C. 10
D. $-14$
E. $-12$
(4) If $x-2$ is a factor of $x^{n}+9 x^{2}-68$, then $n=$
A. 3
B. 4
C. 1
D. 2
E. 5
(5) In the expansion of $\left(\frac{1}{2 x^{2}}-x\right)^{9}$, the term independence of $x$ is
A. $4^{\text {th }}$ term
B. $5^{\text {th }}$ term
C. $6^{\text {th }}$ term
D. $7^{\text {th }}$ term
E. $8^{\text {th }}$ term
(6) ${ }^{n} C_{r}+{ }^{n} C_{n-r}=$
A.. $2 \stackrel{n}{C_{r}}$
B. ${ }^{2 n} C_{r}$
C. ${ }^{n} C_{2 n-1}$
D. ${ }^{n} C_{n-2 r}$
E. none of these
(7) The solution set in $R$ for the inequation $x^{2}+12<3$ is
A. $\{x \mid x<\pm 3\}$
B. $\{x \mid x>3$ or $x<-3\}$
C. $\{x \mid-3<x<3\}$
D. $\varnothing$
E. $R$
(8) In a certain sequence if $u_{1}=3, u_{n+1}=\frac{u_{n}}{n}$, then $u_{4}=$
A. 3
B. 2
C. $\frac{9}{2}$
D. $\frac{1}{2}$
E. $\frac{1}{4}$
(9) Given that $3 , x, y, z, 23, \ldots$ is an arithmetic sequence, then $z=$
A. 13
B. 18
C. 21
D. 22
E. none of these
(10) If the arithmetic mean between 1 and $x$ is 4 and the geometric mean between 2 and $y$ is 6 , then $x+y=$
A. 7
B. 10
C. 25
D. 11
E. none of these
(11) If $P=\left(\begin{array}{c}1+2 x \\ 10\end{array}\right), Q=\left(\begin{array}{c}2 \\ 1-y\end{array}\right)$ and, $P+2 Q=\left(\begin{array}{c}3 \\ 2 y\end{array}\right)$, then $\frac{y}{x}=$
A. 3
B. 2
C. $-3$
D. $-2$
E. $-4$
(12) If $M=\left(\begin{array}{cc}2^{x} & -3 \\ -9 & 7\end{array}\right)$ and det $M=1$, then $M^{-1}=$
A. $\left(\begin{array}{rr}4 & -3 \\ -9 & 7\end{array}\right)$.
B. $\left(\begin{array}{rr}4 & -9 \\ -3 & 7\end{array}\right)$ C. $\left(\begin{array}{rr}7 & -3 \\ -9 & 4\end{array}\right) D \cdot\left(\begin{array}{cc}7 & 3 \\ .9 & 4\end{array}\right)$
E. $\left(\begin{array}{rr}-7 & 3 \\ 9 & -4\end{array}\right)$
(13) A bag contains 12 balls of 2 red, 4 blue, and 6 white. If a draw is made, then the probability of getting blue or white is
A. $\frac{1}{4}$
B. $\frac{1}{6}$
$\mathrm{C} \frac{5}{6}$
D. $\frac{1}{2}$
E. 0
(14) A coin is tossed 2 times. the probability of getting at least one tail is
A. $ \frac{1}{2}$
B. $\frac{1}{4}$
C. $ \frac{3}{4}$
D. 1
E. 0
(15) In figure, $A D$ and $B D$ are tangents to the circle whose centre is $O$. If $\angle A D B=40^{\circ}$, then $\angle A C B$ is
A. $140^{\circ}$
B. $ 70^{\circ}$
C. $35^{\circ}$
D. $55^{\circ}$
E. $65^{\circ}$
(16) In circle $O, P Q$ is a tangent at $Q$. If $P Q=4 \mathrm{~cm}, P R=2 \mathrm{~cm}$, then the length of the diameter is
A. $6 \mathrm{~cm}$
B. $10 \mathrm{~cm} \quad$
C. $12 \mathrm{~cm}$
D. $16 \mathrm{~cm}$.
E. $24 \mathrm{~cm}$
(17) In $\triangle A B C, P$ and $Q$ are two points on the sides $A B$ and $A C$ respectively:
If $P Q / / B C$ and $\alpha(\triangle A P Q): \alpha(B C Q P)=9: 16$, then $A P: P B$ is
A. $3: 4$
B. $4: 3$
C. $3: 5$
D. $5: 3$
E. $3: 2$
(18) The position vector of $A, B, C$ are $\vec{a}, \vec{b}$ and $\vec{c}$ respectively. If $\overrightarrow{A C}=-2 \overrightarrow{C B}$ then $\vec{c}$ is
A. $-\vec{a}+2 \vec{b}$
B. $\vec{a}-2 \vec{b}$
C. $2 \vec{a}+\vec{b}$
D. $2 \vec{a}-\vec{b}$
E. $-2 \vec{a}+\vec{b}$
(19). The map of the point $(2,0)$ which rotates through an angle of $90^{\circ}$ about $O$ in clockwise direction is
A. $(2,-2)$
B. $(0,-2)$
C. $(0,2)$
D. $(-2 ; 0)$
E. $(-2,2)$.
(20) $\sin 270^{\circ}+\cos 720^{\circ}$ is
A. 0 .
B. $-1$
C. 1
D. $\sqrt{2}$
E. none of these
(21) If $\theta$ is an acute angle and $\sin \theta=k$, then $\sin 2 \theta$ is
A. $2 k \sqrt{1-k^{2}}$
B. $k \sqrt{1-k^{2}}$
C. $2 k \sqrt{k^{2}-1}$
D. $k \sqrt{k^{2}-1}$
E, $\sqrt{k^{2}-1}$
(22) In triangle $A B C$ if $\alpha=30^{\circ}, \gamma=105^{\circ}$ and $b=8$, then $a=$
A. $8 \sqrt{3}$
B. $8 \sqrt{2}$
C. $6 \sqrt{2}$.
D. $4 \sqrt{3}$
E. $4 \sqrt{2}$
(23) The gradient of the tangent to the curve $y=a x^{2}-4 x+3$ at the point $x=1$ is $-2$, The value of $a$ is
A. 3
B. 2
C. 1
D. $-3$.
$\mathrm{E} .4$
(24) The stationary point of the curve $y=x^{2}-4 x$ is
A. $(2,4)$
B. $(2,0)$
C. $(-2,12) \quad$
D. $(0,4)$
E. $(2,-4)$
(25) Given that $y=\frac{\ln x^{2}}{3 x}$, the value of $\frac{d y}{d x}$ when $x=1$ is
A. 1
B. $\frac{2}{3}$
C. $\frac{1}{3}$
D. $-1$
E. $-2$
(25 marks)
SECTION B (Answer ALL questions)
2. If $f: R \rightarrow R$ is defined by $f(x)= x^{2}+3$, find the function $g$ such that $(g \circ f)(x)=2 x^{2}+3$ (3 marks)
(OR) Given that the expression $2 x^{3}+a x^{2}+b x+\mathrm{c}$ leaves the same remainder when divided by $x-2$ or by $x+1$, prove that $a+b=-6$. (3 marks)
3. The four angles of a quadrilateral are in A.P. Given that the value of the largest angle is three times the value of the smallest angle, find the values of all four angles. (3 marks)
(OR) The second term of a G.P. is 64 and fifth term is 27 . Find the first 6 terms of the G.P. $(3$ marks)
4. In $\triangle A B C, \overrightarrow{B P}=\overrightarrow{P C}$ and $\overrightarrow{C Q}=\frac{1}{3} \overrightarrow{C A}$. Prove that $2 \overrightarrow{B C}+\overrightarrow{C A}+\overrightarrow{B A}=6 \overrightarrow{P Q}$ (3 marks).
5. If $\alpha+\beta+\gamma= 180^{\circ}$, prove that $\sin \frac{\alpha+\beta}{2}=\sin \left(90^{\circ}+\frac{\gamma}{2}\right)$. (3 marks)
6. Evaluate $\displaystyle\lim _{x \rightarrow 2} \dfrac{x^{3}-8}{x^{2}+3 x-10}$ and $\displaystyle\lim _{x \rightarrow \infty} \dfrac{\sqrt{2 x}-\sqrt{a}}{\sqrt{2 x}+\sqrt{a}}$. $\quad(3$ marks $)$
SECTION C (Answe any SIX questions)
7.(a) The functions $f$ and $g$ are defined for real $x$ by $f(x)=2 x-1$ and $g(x)=\frac{2 x+3}{x-1}, x \neq 1 .$ Evaluate $\left(\dot{g}^{-1} \circ f^{-1}\right)(2) . \quad(5$ marks $)$
(b) Let $J^{+}$be the set of all positive integers. Is the function $\odot$ defined by $x \odot y=x+3 y$ a binary operation on $J^{+} ?$ If it is a binary operation, solve the equation $(k \odot$ 5) $-(3 \odot k)=2 k+13$ (5 marks)
8.(a) The expression $x^{3}+a x^{2}+b x+3$ is exactly divisible by $x+3$ but it leaves a remainder of 91 when divided by $x-4$. What is the remainder when it is divided by $x+2 ?$ (5 marks)
(b) If the $2^{\text {nd }}$ and the $3^{\text {rd }}$ term in $(a+b)^{n}$ are in the same ratio as the $3^{\text {rd }}$ and $4^{\text {th }}$ $\begin{array}{ll}\text { in }(a+b)^{n+3}, \text { then find'n. } & \text { (5 marks) }\end{array}$
9.(a) Use a graphical method to find the solution set of $x^{2} \leq \frac{4}{5}(x+3)$, and illustrate it on the number line. (5 marks)
(b) The fourth term of an A.P. is 1 and the sum of the first 8 terms is 24 . Find the sum of the first three terms of the progression. (5 marks)
10.(a) The sum of the first three terms of a G.P. is 27 and the sum of the fourth, fifth and sixth terms is $-1$. Find the common ratio and the sum to infinity of the G.P. (5 marks )
(b) Given that the matrix $A=\left(\begin{array}{rr}2 & -1 \\ -3 & 4\end{array}\right)$ and that $A^{2}-k A+5 I=O$, find the value of $k$. (5 marks)
11.(a) Find the solution set of the systems of equations $\left.\begin{array}{l}5 x+6 y=25 \\ 3 x+4 y=17\end{array}\right\}$ by matrix method, the variables are on the set of real numbers.
(b) The probabilities of students $A, B, C$ to pass an examination are $\frac{3}{4} ; \frac{4}{5}$ and $\frac{5}{6}$ respectively. Find the probability that at least one of them will pass the examination. (5 marks)
12. Two circles cut at $A, B$. The tangent to the first at $A$ meets the second again at $C$; and the tangent to the second at $B$ meets the first again at $D$. Prove that $A D$ and $C B$ are parallel. (5 marks)
(b) $A B C$ is a triangle in which $A B=A C$. $P$ is a point inside the triangle such that $\angle P A B=\angle P B C $. $Q$ is the point on $B P$ such that $A Q=A P$. Prove that $A B C Q$ is cyclic. (5 marks)
13.(a) In $\triangle A B C, A D$ and $B E$ are the altitudes. If $\alpha(\triangle D E C)=\frac{3}{4} \alpha (\triangle ABC)$, prove that $\angle A C B=30^{\circ}, \quad(5$ marks $)$
(b) Find the matrix which will rotate $30^{\circ}$. and then reflect in the line $O Y$. What is the map of the point $(1 ; 0) ? \quad (5$ marks $)$
14.(a) Without the use of table evaluate $\tan (\alpha+\beta+\gamma)$, given that $\tan \alpha=\frac{1}{2}$; $\tan \beta=\frac{1}{3}$ and $\tan \gamma=\frac{1}{4}$. $\quad$ (5 marks)
(b) $A, B, C$ are three towns, $B$ is 10 miles from $A$ in a direction $N 47^{\circ} E . C$ is 17 miles away from $B$ in a direction $N 70^{\circ} W$. Calculate the distance and direction of $A$ from $C . \quad(5$ marks $)$
15.(a) If $y=\ln (\cos 2 x)$, prove that $\frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+4=0. $ (5 marks )
(b) Determine the turning point on the curve $y=2 {x}^{3}+3 x^{2}-12 x+7$ and state whether it is a maximum or a minimum. Then sketch the graph of the curve. (5 marks)
Answers
1
(1)C
(2)D
(3)B
(4)E
(5)D
(6)A
(7)D
(8)D
(9)B
(10)C
(11)C
(12)D
(13)C
(14)C
(15)B
(16)A
(17)E
(18)A
(19)B
(20)A
(21)A
(22)E
(23)C
(24)E
(25)B
2 $g(x)=2x-3$
3 45,75,105,135
or 256/3,64,48,36,27,81/4
4 Prove
5 Prove
6 $\frac{12}{7},1$
7(a) $-9$ (b) No solution
8(a) 7 (b) $n=5$
9(a) $\{x|-\frac 65\le x\le 2\}$ (b) $S_3=-21$
10(a) $r=-\frac 13, S=\frac{729}{28}$ (b) $k=6$
11(a) $\{(-1,5)\}$ (b) $\frac{119}{120}$
12(a) Prove (b) Prove
13(a) Prove (b) $\left(\begin{array}{cc}\frac{\sqrt 3}{2}&-\frac 12\\ \frac 12&\frac{\sqrt 3}{2}\end{array} \right),\left(-\frac{\sqrt 3}{2},\frac 12\right)$
14(a) $5/3$ (b) 15.32, S $34^{\circ}26'$ E
15(a) Prove
(b) $(-2,27)$ maximum point, (1,0) minimum point
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