Myanmar Matriculation Examination (2016 F)

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2016
MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
MATHEMATICS Time Allowed : (3) Hours
WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.


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) Functions $f$ and $g$ are given by $f(3)=-1$ and $g(-1)=5$. Then $(g \circ f)^{-1}(5)=$

A. $-1$

B. 5

C. 3

D. 4

E. 0

(2) Given that $a \odot b$ means "add 4 to $a$ and multiply the result by $b$ ", then the value of, $(2 \odot 1) \odot 3$ is

A. 55

B. 50

C. 45

D. 40

E. 30

(3) If $x$ is a factor of $x^{3}-4 x^{2}+15 x+a^{2}-2 a$, where $a$ is a constant, then $a=$

A. 0 only

B. $-2$ only

C. 2 only

D. 0 or 2

E. 0 or $-2$

(4) If $a x^{2}+3 x-1$ has remainder $3 a+14$ when divided by $x-3$, then $a=$

A. 1

B. 2

C. 3

D. $-1$

E. $-2$

(5) The coefficient of $x y^{4}$ in the expansion of $(x-2 y)^{5}$ is

A. 80

B. 40

C. $-10$

D. $-32$

E. $-80$

(6) The middle term in the expansion of $(x+y)^{2 k}$ is

A. $k^{\text {th }}$ term

B. $(k+1)^{\text {th }}$ term

C. $(k+2)^{\text {th }}$ term

D. $(k+3)^{\text {th }}$ term

E. $(2 k+1)^{\text {th }}$ term


(7) The solution set in $R$ for the inequation $-4-3 x^{2} \geq 0$ is

A. $\{x \mid x \geq 0\}$

B. $\left\{x \mid-2 \leq x \leq \frac{2}{3}\right\}$ 

C. $\left\{x \mid-\frac{2}{3} \leq x \leq 2\right\}$

D. $R \quad$ 

E. $\varnothing$

(8) The three angles of a triangle form an A.P.. If the largest angle is twice the smallest angle, then the smallest angle is

A. $20^{\circ}$

B. $40^{\circ}$

C. $60^{\circ}$

D. $80^{\circ}$

E. $90^{\circ}$

(9) If $1+2+2^{2}+2^{3}+\ldots+2^{n}=1023$, then $n$ is

A. 9

B. 10

C. 11

D. 12

E. 18

(10) For two numbers $a$ and $b$, the A.M. between $a$ and $b$ is 5 and the G.M. between $a$ and $b$ is 4 . If $a>b$, then $a^{b}=$

A. 256

B. 64

C. 16

D. 10

E. None of these

(11) Given that $A=\left(\begin{array}{cc}k & 3 \\ -3 & 2\end{array}\right)$ and $B=\left(\begin{array}{rr}2 & -3 \\ 3 & -4\end{array}\right)$ and $A B=B A$, then $k=$

A. $-4$

B. 2

C. 3

D. 4

E. None of these

(12) If the determinant of the matrix $\left(\begin{array}{cc}2 x+1 & 2 x \\ 3 & 2\end{array}\right)$ is $-2$, then $x=$

A. 2

B. 3

C. 4

D. 5

E. 6

(13) The number of possible outcomes for tossing five fair coins is

A. 64

B. 32

C. 18

D. 16

E. 8

(14) Two dice are thrown 180 times. The expected frequency of obtaining total score 6 is

A. 60

B. 50

C. 40

D. 30

E. 25

(15) In a cyclic quadrilateral $A B C D, \angle A=25^{\circ}, \angle B=60^{\circ}$. Then $\angle C-\angle D=$

A. $35^{\circ}$

B. $40^{\circ}$

C. $45^{\circ}$

D. $50^{\circ}$

E. $55^{\circ}$

(16) In circle $O, P Q$ is tangent at $Q$. If $P Q=4 \mathrm{~cm}$, . he length of the diameter is $6 \mathrm{~cm}$, the length of $P R$ is 


A. $6 \mathrm{~cm}$

B. $5 \mathrm{~cm}$

C. $4 \mathrm{~cm}$

D. $3 \mathrm{~cm}$

E. $2 \mathrm{~cm}$



(17) If the ratio of areas of two similar triangles is $16: 225$, then the ratio of the lengths of corresponding angle bisectors is

A. $16: 25$

B. $4: 15$

C. $25: 4$

D. $6: 25$

E. $13: 25$

(18) The position vectors of $A$ and $B$, relative to an origin $O$ are $3 \vec{p}+2 \vec{q},-5 \vec{p}-3 \vec{q}$ respectively. If $M$ is the mid-point of $A B$, then the position vector of $M$ is

A. $\vec{p}+\frac{1}{2} \vec{q}$

B. $-\vec{p}+\frac{1}{2} \vec{q}$

C. $-\vec{p}-\frac{1}{2} \vec{q}$

D. $\vec{p}-\frac{1}{2} \vec{q}$

E. $\frac{1}{2} \vec{p}+\vec{q}$

(19) Given that $\vec{a}=3 \hat{i}+4 \hat{j}$. Then the vector with magnitude 20 units and in the direction of $\vec{a}$ is

A. $9 \hat{i}+12 \hat{j}$

B. $60 \hat{i}+120 \hat{j}$

C. $-12 \hat{i}-16 \hat{j}$

D. $12 \hat{i}+16 \hat{j}$

E. $21 \hat{i}+28 \hat{j}$

(20) If $\tan \theta=2, \tan \phi=1$, then $\cot (\theta-\phi)=$

A. $-3$

B. 3

C. $-\frac{1}{3}$

D. $\frac{1}{3}$

E. 1

(21) $\cos \left(-45^{\circ}\right)=$

A. $\frac{2}{\sqrt{2}}$

B. $\frac{-2}{\sqrt{2}}$

C. $\frac{\sqrt{2}}{2}$

D. $\frac{-\sqrt{2}}{2}$

E. 1

(22) In $\triangle A B C, B C: C A: A B=3: 4: \sqrt{37} .$ Then $\angle C=$

A. $60^{\circ}$

B. $75^{\circ}$

C. $105^{\circ}$

D. $120^{\circ}$

E. $150^{\circ}$

(23) $\lim _{t \rightarrow \infty}\left(\sqrt{t^{2}+2 t+1}-t\right)=$

A. 0

B. 1

C. 2

D. $\infty$

E.none of these

(24) The second derivative of $y=\cos x+x^{2}$ is

A. $-\cos x-2$

B. $2-\cos x$

C. $\sin x+2 x$

D. $\cos x-2 \quad$ 

E. $\sin x-2 x$


(25) The coordinates of the point on the curve $y=x^{3}-6 x$ at which the gradient of tangent $-3$ is

A. $(1,-5)$

B. $(-1,5)$

C. $(1,-5)$ and $(-1,5)$ 

D. $(2,-4)$

E. $(2,4)$



SECTION B

(Answer ALL questions)

2. A function $f: x \mapsto \frac{b}{x-a}, x \neq a$ and $a>0$ is such that $(f \circ f)(x)=x$. Show that $x^{2}-a x-b=0$

(OR)

Given that $x^{3}-2 x^{2}-3 x-11$ and $x^{3}-x^{2}-9$ have the same remainder when divided by $x+a$, determine the values of $a$.


3. If $29, a-b, a+b, 95$ is an A.P., find the values of $a$ and $b$.

(OR)

In an G.P, the ratio of the sum of the first three terms to the sum to infinity of the G.P. is $19: 27$. Find the common ratio.


4. $A$ and $B$ are two points on a circle $3 \mathrm{~cm}$ apart. The chord $A B$ is produced to $C$ making $B C=1 \mathrm{~cm}$. Find the length of the tangent from $C$ to the circle.


5. If $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$, prove that $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$.


6. Find the value of $a$ and $b$ for which $\frac{d}{d x}\left[\frac{\sin x}{2+\cos x}\right]=\frac{3 a+b \cos x}{(2+\cos x)^{2}}$.



SECTION C

(Answer any SIX questions)

7. (a) Functions $f$ and $g$ are defined by $f: \mathrm{x} \mapsto 2 x+1$ and $g: x \mapsto \frac{2 x+5}{3-x}, x \neq 3$.

Find the values of $x$ for which $\left(f \circ g^{-1}\right)(x)=x-4$

(b) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ is defined by $x \odot y=x y-x-y$ for all $x, y$ in $R$. Show that the operation $\odot$ :s commutative. Solve the equation $(2 \odot 3) \odot x=(x \odot x) \odot 5$.


8. (a) The expression $a x^{3}-(a+3 b) x^{2}+2 b x+c$ is exactly divisible by $x^{2}-2 x$. When the expression is divided by $x-1$, the remainder is 8 more than when it is divided by $x+1$. Find the values of $a, b$ and $c$, hence factorize the expression completely.

(b) Write down and simplify the first four terms in the binomial expansion of $(1-2 x)^{7}$. Use it to find the value of $(0.98)^{7}$, correct to four decimal places.


9. (a) Use a sketch graph to obtain the solution set of $\frac{15-4 x}{4} \leq x^{2}$ and illustrate it on the number line.

(b) Find the sum of all two-digit natural numbers which are not divisible by $3 .$


10. (a) Find the sum of $(b+2)+\left(b^{2}+5\right)+\left(b^{3}+8\right)+\ldots$ to 18 terms in terms of $b$ where $b \neq 1$

(b) Find the inverse of the matrix $M=\left(\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right)$ and investigate whether or not the squares of $M$ and $M^{1}$ are inverses of each other.


11. (a) Find the inverse of the matrix $\left(\begin{array}{ll}4 & 3 \\ 7 & 6\end{array}\right)$, and use it to solve the system of equations $3 y+4 x+7=0$ and $14 x+12 y+32=0$

(b) A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the scores obtained in these 360 trials are added together, what is the expected total score?


12. (a) In the figure, $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C, \angle C P B$ and $\angle Q P C$ in terms of $x$.


(b) Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic.


13. (a) $A, B, C$ and $D$ are four points in order on a circle $O$, so that $A B$ is a diameter and $\angle C O D=90^{\circ}.$ If $A D$ produced and $B C$ produced meet at $E$, prove that $\alpha(\Delta E C D)=\alpha(A B C D)$

(b) The coordinates of points $P, Q$ and $R$ are $(1,2),(7,3)$ and $(4,7)$ respectively. If $P Q S R$ is a parallelogram, find the coordinates of $S$ by vector method. If $P S$ and $Q R$ meet at $T$, find the coordinate of $T$ by using vectors.


14. (a) Given that $\frac{\cos (\alpha-\beta)}{\cos (\alpha+\beta)}=\frac{7}{5}$, prove that $\cos \alpha \cos \beta=6 \sin \alpha \sin \beta$ and deduce a relationship between $\tan \alpha$ and $\tan \beta .$ Given further that $\alpha+\beta=45^{\circ}$, calculate the value of $\tan \alpha+\tan \beta$.

(b) A town $P$ is 25 miles away from the town $Q$ in the direction $N 35^{\circ} \mathrm{E}$ and a town $R$ is 10 miles from $Q$ in the direction $42^{\circ}$ W. Calculate the distance and bearing of $P$ from $R$.


15.(a) Find the coordinates of the points on the curve $x^{2}-y^{2}=3 x y-39$ at which the tangents are (i) parallel (ii) perpendicular to the line $x+y=1$.

(b) Find the stationary points on the curve $y=27+12 x+3 x^{2}-2 x^{3}$ and determine the nature of these points.


Answer

$\def\myrow#1#2#3#4#5{\mbox{#1 }\quad&\mbox{#2 }\quad&\mbox{#3 }\quad&\mbox{#4 }\quad&\mbox{#5 }}\def\mytab#1{\begin{array}{rrrrr}#1\end{array}}$ $\mytab{\myrow{1 C}{2 E}{3 D}{4 A}{5 A}\\ \myrow{6 B}{7 E}{8 B}{9 A}{10 B}\\\myrow{11 A}{12 A}{13 B}{14 E}{15 A}\\\myrow{16 E}{17 B}{18 C}{19 D}{20 B}\\\myrow{21 C}{22 D}{23 B}{24 B}{25 C}}$

2) Show (OR) $a=1$ or 2

3) $a=62, b=11$

(OR) $r=\frac{2}{3}$

4) $2 \mathrm{~cm}$

5) Prove

6) $a=\frac{1}{3}, b=2$

7) (a) $x=0$ or 9 (b) $x=1 \pm \sqrt{2}$

8) (a) $a=2, b=1, c=0, f(x)=x(x-2)(2 x-1)$

(b) $1-14 x+84 x^{2}-280 x^{3}+\cdots, 0.8681$

9) (a) $\left\{x \mid x \leqslant-\frac{5}{2}\right.$ (or) $\left.x \geqslant \frac{3}{2}\right\}$ (b) 3240

10) (a) $495+\frac{b\left(1-b^{18}\right)}{1-b}$

(b) $M^{-1}=\left(\begin{array}{cc}2 & -5 \\ -1 & 3\end{array}\right)$ (Yes)

11) $(a)\left(\begin{array}{cc}2 & -1 \\ -\frac{7}{3} & \frac{4}{3}\end{array}\right), x=2, y=-5$

(b) $240,180,1260$

12) (a) $\angle D P C=\frac{90-x}{2}=\angle C P B, \angle Q P C=\frac{270-x}{2}(b)$ Prove

13)(a) Prove (b) $T=\left(\frac{11}{2}, 5\right)$

14) (a) $\frac{5}{6} \quad$ (b) $612,5, N 58^{\circ} 15^{\prime} $ E

15) (a) (i) $(1,5),(-1,-5)$ (ii) No tangent (b) $(2,47) \max ;(-1,20) \min$



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