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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) 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
$$(g \circ f)(2)=(f \circ g)(2)$$
Q(2) Solution
(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 $q$.
Q2(OR) Solution
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.
Q(3) Solution
(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.
Q3(OR) Solution
4. In the figure, $O$ is the centre of the circle, find $\angle R Q T$.
Q(4) Solution
5. Prove that $\sin x+\sin 2 x+\sin 3 x=\sin 2 x(1+2 \cos x)$.
Q(5) Solution
6. Calculate $\displaystyle\lim _{x \rightarrow 2} \frac{x^{3}-8}{\sqrt{x+2}-2}$ and $\displaystyle\lim _{x \rightarrow 0} \frac{\cos x-1}{\sin ^{2} x}$.
Q(6) Solution
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 $(f \circ g)^{-1}(x)=\frac{5}{3}$
Q7(a) Solution
(b) A binary operation $\odot$ on $R$ is defined by $x \odot y=y^{x}+2 x^{y} y^{x}-x^{y}$. Evaluate $(2 \odot 1) \odot 1.$
Q7(b) Solution
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 $x+1$
Q8(a) Solution
(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 $3(a+b)=a^{2}+b^{2}$.
Q8(b) Solution
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.
Q9(a) Solution
(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 $x^{3}+y^{3}=2 a b c.$
Q9(b) Solution
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.
Q10(a) Solution
(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 $A X=2 B-A^{2}$
Q10(b) Solution
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 $x+y=3$
Q11(a) Solution
(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.
Q11(b) Solution
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 $\angle B A C$.
Q12(a) Solution
(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.
Q12(b) Solution
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 $\alpha(\triangle D E C): \alpha(\triangle A B C)=1: 2$
Q13(a) Solution
(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.
Q13(b) Solution
14. (a) Prove the identity $\sec 2 \alpha=\frac{1+\tan ^{2} \alpha}{2-\sec ^{2} \alpha}$.
Q14(a) Solution
(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 $R$.
Q14(b) Solution
15.(a) Find the equation of the tangent line to the curve $x^{3}+y^{3}-9 x y=0$ at the point $(3,2)$.
Q15(a) Solution
(b) Find the two positive numbers whose sum is 82 and whose product is as large as possible.
Q15(b) Solution
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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