Myanmar Matriculation 2017 (D)

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2017

MATRICULATION EXAMINATION

DEPARTMENT OF MYANMAR EXAMINATION

MATHEMATICS

Time Allowed: (3) Hours

WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.


SECTIONA

(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) A function $f: R \rightarrow R$ is given by $f(x)=4^{2-x}$. Which element of the domain has 64 as its image?

A. 2

B. 3

C. $-2$

D. 1

E. $-1$

(2) A function $f: R \rightarrow R$ is given by $f(x)=7-k x$ and $f^{-1}(5)=1$. Then $k=$

A. $-1$

B. 2

C. 1

D. $-2$

E. 4

(3) If $x^{3}-3 x^{2}+5$ and $x^{3}+5 x^{2}+p$ have same remainder when divided by $x+1$ then $p=$

A. 7

B. $-7$

C. 13

D. $-3$

E. 3

(4) If $x+2$ is a factor of $f(x)=x^{3}-3 x^{2}-a x+2$, then the value of $a$ is

A. 9

B. $-9$

C. 1

D. $-11$

E. $-1$

(5) In the expansion of $(3-5 z)^{14}$, the coefficient of $z$ is

A. $3^{13}\left(\frac{70}{9}\right)$

B. $3^{14}\left(\frac{-70}{9}\right)$

C. $3^{14}\left(\frac{-70}{3}\right)$

D. $3^{13}\left(\frac{-70}{3}\right)$

E. $3^{14}\left(\frac{70}{9}\right)$

(6) In the expansion of $\left(x^{4}-\frac{2}{x}\right)^{10}$, the term independent of $x$ is

A. 11250

B. 12520

C. $-11520$

D. 11520

E. $-11250$


(7) The solution set in $R$ of $-x^{2}-1 \geq 0$ is

A. $R$

B.$\{1\}$

C. $\varnothing$

D $\{-1,1\}$

E. $\{x \mid x<-1$ or $x<1\}$

(8) The sixth term of an A.P. is 21 , and the sum of the first 17 terms is 0 . The first tem is

A. 42

B. $-7$

C. 7

D. $-56$

E. 56

(9) If $3^{2 x+1}, 9^{x}$ and 243 are three consecutive terms of a $G P$, then $x=$

A. 6

B. 7

C. 4

D. 3

E. 5

(10) If the $(m+n)$ th term and $(m-n)$ th term of a G.P. are $p$ and $q$ respectively, then the $m$ th term of this G.P. is

A. $\sqrt{\frac{q}{p}}$

B. $\frac{3 p q}{2}$

C. $\sqrt{p q}$

D. $\sqrt{\frac{p}{q}}$

E. pq

(11) If $K=\left(\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right)$, then $K^{2017}-K^{2016}+K^{2}$ is

A. $K^{5}$

B. $K^{3}$

C. $K$

D. $K^{2}$

E. $K^{4}$

(12) Let $A=\left(\begin{array}{cc}3^{x-2} & 6 \\ 9 & 5\end{array}\right)$. If $\operatorname{det} A=-9$, then $x=$

A. 2

B. $-4$

C. $-2$

D. 4

E. 0

(13) If a die is rolled 60 times, then the expected frequency of not a prime number is

A. 30

B. 40

C. 10

D. 50

E. 20

(14) A coin is tossed two times. The probability of getting at least one tail is $x-2$, then $x$ is

A. $\frac{3}{4}$

B. $\frac{11}{4}$

C. $\frac{15}{4}$

D. $\frac{9}{4}$

E. $\frac{1}{4}$


(15) In the figure, chord $A B$ bisects chord $C D$ at $E$. If $A E=8$ and $B E=9$, the length of $C E$ is 


A. $3 \sqrt{2}$

B. $6 \sqrt{3}$

C. $3 \sqrt{3}$

D. $6 \sqrt{2}$

E. none of these





(16) A pair of opposite angles of a cyclic quardrilateral are in the ratio $1: 2$. Then the larger angle has degree

A. $120^{\circ}$

B. $100^{\circ}$

C. $72^{\circ}$

D. $60^{\circ}$

E. $36^{\circ}$

(17) The lengths of two corresponding medians of two similar triangles are 6 and 9 respectively. If the area of the smaller triangle is 24 , then the area of the larger triangle is

A. 36

B. 54

C. 72

D. 45

E. 63

(18) If $\overrightarrow{P Q}=-2 \overrightarrow{Q R}$, then which of the following is (are) true?

1. $P Q=-2 Q R \quad$ 2. $P, Q, R$ are collinear 3. $R$ lies between $P$ and $Q$

A. 1 only $\quad$ 

B. 2 only $\quad$ 

C. 2 and 3 only $\quad$ 

D. 1 and 3 only

E. 1,2 and 3

(19) The map of $(-1,4)$ by reflection in the $X$-axis is

A. $(1,4)$

B. $(-1,-4)$

C. $(4,1)$

D. $(-4,-1)$

E. none of these

(20) If $\sin x=\frac{12}{13}$, in the second quadrant, then $\sin 2 x$ is

A. $\frac{120}{169}$

B. $\frac{60}{169}$

C. $\frac{25}{169}$

D. $\frac{-60}{169}$

E. $\frac{-120}{169}$

(21) Which of the following is (are) false?

1. If $f(x)=1+x^{2}, g(x)=\tan x$, then $f(g(x))=\sec ^{2} x$.

2. $\tan \alpha+\cot \alpha=\frac{1}{\sin 2 \alpha}$

3. $\sin (\pi+x)=\sin x$

A. 1 only

B. 2 only

C. 1 and 2 only

D. 2 and 3 only

E. 1 and 3 only

(22) $\sin 120^{\circ}+\sin 60^{\circ}=$

A. $\sqrt{3}$

B. $-\sqrt{3}$

C. 1

D. $-1$

E. none of these


(23) If $f(x)=4 x^{2}+e^{-3 x}$, then $f^{\prime \prime}(0)=$

A. $-17$

B. 8

$C .17$

D. $-8$

E. $-3$

(24) The gradient of the curve $2 x y^{2}-x^{3}=3$ at the point $(1,-2)$ is

A. $\frac{-11}{8}$

B. $\frac{5}{8}$

C. $\frac{-5}{8}$

D. $\frac{-3}{8}$

E. $\frac{11}{8}$

(25) The curve $y=a x^{2}-\frac{b}{x}$ has a stationary point at $(1,3)$, then the values of $a$ and $b$ is

A. 3,6

B. $1,-2$

C. $-1,2$

D. $-3,-6$

E. none of these

SECTIONB

(Answer ALL questions)

2. Let $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=k x-1$, where $k$ is a constant and $g(x)=x+12 .$ Find the value of $k$ for which

$$(g \circ f)(2)=(f \circ g)(2)$$

(OR)

If $x^{3}+p x^{2}-8 q x+5$ and $2 x^{3}-q x^{2}+4 p x-18$ have a common factor $x-2$ find the values of $p$ and $q$.


3. The $p$ th term of an A.P. is $q$ and the $q$ th term of this A.P, is $p$. Show that its $(p+q)$ th term is zero.

(OR)

If the first term of a GP. exceeds the second term by 2 and the sum of infinity is 50 . Find the first term and the common ratio.


4. In the figure, $O$ is the centre of the circle, find $\angle R Q T$.



5. Prove that $\sin x+\sin 2 x+\sin 3 x=\sin 2 x(1+2 \cos x)$.


6. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{x^{3}-8}{\sqrt{x+2}-2}$ and $\displaystyle\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$.



SECTIONC

(Answer any SIX questions)

7.(a) Functions $f$ and $g$ are defined by $f: x \mapsto \frac{x}{x-3}, x \neq 3, g: x \mapsto 3 x+5$. Find the value of $x$, for which $(f \circ g)^{-1}(x)=\frac{5}{3}$

(b) A binary operation $\odot$ on $R$ is defined by $x \odot y=y^{x}+2 x^{y} y^{x}-x^{y}$. Evaluate $(2 \odot 1) \odot 1$


8. (a) If $x+2$ is a factor of $x^{3}-a x-6$, then find the remainder when $2 x^{3}+a x^{2}-6 x+9$ is divided by $x+1$

(b) In the expansion of $(1+x)^{a}+(1+x)^{b}$, the coefficients of $x$ and $x^{2}$ are equal for all positive integers $a$ and $b$, prove that $3(a+b)=a^{2}+b^{2}$.


9. (a) Find the solution set in $R$ of $-7+(2 x+1)^{2} \geq 6 x$ by algebraic method and illustrate it on the number line.

(b) If $b, x, y, c$ are consecutive terms of a G.P. and the A. $M$. between $b$ and $c$ is $a$, then prove that $x^{3}+y^{3}=2 a b c$

10. (a) The product of first three terms of a G.P is 1000 . If we add 6 to its second term, 7 to its third term and its first term is not changed, then three terms form an A.P. Find the first three terms of the G.P.

(b) Given that $A=\left(\begin{array}{lr}4 & 1 \\ -9 & -2\end{array}\right)$ and $B=\left(\begin{array}{cc}-3 & 1 \\ -1 & 2\end{array}\right)$. Solve the equation $A X=2 B-A^{2}$


11. (a) Find the inverse of the matrix $\left(\begin{array}{cc}1 & 1 \\ -1 & 1\end{array}\right)$ and use it to solve the following system of equations, $y-x=1$ and $x+y=3$

(b) Draw a tree diagram to list all possible outcomes for a family which has three children. Find the probability that (i) only the first child is a boy (ii) the last child is a boy (iii) the last two children born are boys.


12. (a) Two unequal circles are tangent internally at $A ; B C$, a chord of the larger circle, is tangent to the smaller circle at $D ;$ prove that $A D$ bisects $\angle B A C$.

(b) $P V$ is a tangent to the circle and $Q T$ is parallel to $P V$

Prove that $Q R S T$ is a cyclic quadrilateral.



13. (a) In $\triangle A B C, A D$ and $B E$ are altitudes to the sides $B C$ and $A C$ respectively. If $\angle A C D=45^{\circ}$, prove that $\alpha(\triangle D E C): \alpha(\triangle A B C)=1: 2$

(b) The coordinates of $P, Q$ and $R$ are $(1,3),(5,4)$ and $(1,9)$ respectively. Find the corrdinates of $S$ if $P Q R S$ is a parallelogram.


14. (a) Prove the identity $\sec 2 \alpha=\frac{1+\tan ^{2} \alpha}{2-\sec ^{2} \alpha}$.

(b) A town $P$ is $50 \mathrm{~km}$ away from a town $Q$ in the direction $N 35^{\circ} E$ and a town $R$ is $68 \mathrm{~km}$ from $Q$ in the direction $N 42^{\circ} 12^{\prime} W$. Calculate the distance and bearing of $P$ from $R$.

15.(a) Find the equation of the tangent line to the curve $x^{3}+y^{3}-9 x y=0$ at the point $(3,2)$.

(b) Find the two positive numbers whose sum is 82 and whose product is as large as possible.


Answers

1)

(1)   E    (2)   B     (3)   D     (4)   A      (5)   C

(6)   D    (7)  C      (8)   E    (9)     D    (10)  C

(11) B    (12) D    (13)   A    (14)   B    (15)  D

(16) A    (17) B    (18)   C    (19)   B    (20)  E

(21) D    (22) A    (23)   C    (24)   -    (25)   B

2) $k=1$ or $\quad p=\frac{3}{4}, \quad q=1$

3) Show or $a=10, r=\frac{4}{5}$

4) $\angle Q R T=62$

5) Prove

6) $48,-\frac{1}{2}$

7) (a) $x=\frac{10}{7}$ (b) 4,(Assume $x \neq 0, y \neq 0)$

8) (a) 20 (b) prove

9) $(a)\left\{x \mid x \leqslant-1\text{ or  }x \geqslant \frac{3}{2}\right\}$

10)(a) $5,10,20$ or $20,10,5 \quad$ (b) $x=\left(\begin{array}{cc}10 & -9 \\ -53 & 36\end{array}\right)$

(11) (a) $x=1, y=2$

(b) (i) $\frac{1}{8}$

(ii) $\frac{1}{2}$ (iii) $\frac{1}{4}$

12) (a) Prove (b) Prove

13 )(a) Prove (b) $S=(-3, 8)$

14 ) (a) Prove (b) $74.95 \mathrm{~km}, $ S $82^{\circ}47^{\prime}$ E

15) (a)(b) $x=41,y=41$


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