# Myanmar Matriculation Exam (2016 D)

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2016 D

MARICULATION 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 $f$ is defined on the set of real numbers by $f: x \mapsto \frac{3}{x-2}$, $x \neq k$. Then the value of $k$ is

A. 3

B. 1

C. 2

D. $-1$

E. $-3$

(2) An operation $\odot$ is defined by $x \odot y=\frac{3 x y}{x+y}$, then the value of $x$ for which $x \odot 2 x=4$ is

A. $-3$

B. 3

C. 1

D. $-1$

E. 2

(3) $x^{3}-3 x^{2}+k x+7$ is divided by $x+3$, the remainder is 1 . Then $k=$

A. 16

B. 15

C. $-16$

D. $-15$

E. 13

(4) If $x-p$ is a factor of $4 x^{3}-(3 p+2) x^{2}-\left(p^{2}-1\right) x+3$, then $p=$

A. $-\frac{1}{2}$ or 3

B. $\frac{1}{2}$ or $-3$

C. $-1$ or $\frac{3}{2}$

D. 1 or $-\frac{3}{2}$

E. $-1$ or $\frac{2}{3}$

(5) In the expansion of $(3+k x)^{9}$, the coefficients of $x^{3}$ and $x^{4}$ are equal. Then $k=$

A. 1

B. 2

C. 3

D. $-1$

E. $-2$

(6) ${ }^{n} C_{0}+{ }^{n} C_{n-1}=$

A. 0

B. 1

C. 2

D. $n+1$

E. $n$

(7) The solution set in $R$ for the inequation $(x+2)^{2}>2 x+7$ is

A. $\{x \mid x>-3\}$

B. $\{x \mid x<1\}$

C. $\varnothing$

D. $R$

E. $\{x \mid x<-3$ or $x>1\}$

(8) If $p^{\text {th }}$ term of an A.P. is $q$, and the $q^{\text {th }}$ term is $p$, then the common difference is

A. 0

B. 1

C. $-1$

D. 2

E. $-2$

(9) Three positive consecutive terms of a G.P. are $x+1, x+5$ and $2 x+4$ Then $x=$

A. 2

B. 7

C. 3

D. 4

E. 1

(10) If $x, y, 2 x$ is an A.P. and $3,9, y$ is a G.P., then $x+y=$

A. 45

B. 54

C. 27

D. 9

E. $-9$

(11) $A=\left(\begin{array}{cc}2 & 0 \\ 1 & 5\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ 2 & k\end{array}\right)$. Then the value of $k$ for which $A B=B A$ is

A. $-1$

B. 1

C. 7

D. $-4$

E. 4

(12) Given that $A$ is a $2 \times 2$ matrix such that

$\left(\begin{array}{cc}2 & -1 \\ 3 & 4\end{array}\right) A+\left(\begin{array}{cc}1 & 1 \\ -3 & -1\end{array}\right) A=\left(\begin{array}{cc}3 & 6 \\ -3 & 9\end{array}\right)$, then the matrix $A$ is

$A \cdot\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)$

B. $\left(\begin{array}{cc}1 & 2 \\ -1 & 3\end{array}\right)$

C. B. $\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$

D. $\left(\begin{array}{cc}2 & -1 \\ -1 & 0\end{array}\right)$

E. $\left(\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right)$

(13) If $A$ is an event such that $P(A)=x$ and $P(\operatorname{not} A)=y$, then $x^{3}+y^{3}=$

A. $3 x y$

B. $1+3 x y$

C. $3 x y-1$

D. $1-3 x y \quad$

E. none of these

(14) In 100 trials, $A$ is an event and the expected frequency of $A$ is 30 , then $P(A)=$

A. $\frac{3}{10}$

B. $\frac{3}{5}$

C. $\frac{3}{20}$

D. $\frac{1}{30}$

E. $\frac{1}{100}$

(15) In $\odot O, D C / / A B$ and $\angle C A B=20^{\circ}$, Then $\angle D A C=$

A, $20^{\circ}$

B. $15^{\circ}$

C,.$50^{\circ}$

D. $30^{\circ}$

E. $40^{\circ}$

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

A. $2 \mathrm{~cm}$

B. $1 \mathrm{~cm}$

C, $3 \mathrm{~cm}$

D.  4 cm

E.  5 cm

(17) In the trapezium $A B C D, A B$ is twice $D C$ and $A B / / D C$. If $A C$ and $B D$ intersect at $O$, then $\alpha(\triangle A O B): \alpha(\triangle C O D)=$

A. $1: 4$

B. $2: 3$

C. $4: 1$

D. $3: 2$

E. none of these

(18) If $\vec{a}, \vec{b}$ are non-parallel and non-zero such that $(3 x+y) \vec{a}+(y-3) \vec{b}=\overrightarrow{0}$, then $x=$

A. 1

B, $-1$

C. 3

D. $-3$

E. none of these

(19) If $P=(3,4), R=(8,2)$ and $O$ is the orgin and $\overrightarrow{O P}=\overrightarrow{O T}-\frac{1}{2} \overrightarrow{O R}$, then the coordinates of the point $T$ is

A. $(1,3)$

B. $(2,4)$

C, $(7,5)$

D. $(4,5)$

E. $(5,7)$

(20) What is the smallest value of $x$ for which $\tan 3 x=-1$ ?

A. $15^{\circ}$

B. $45^{\circ}$

C. $75^{\circ}$

D. $90^{\circ}$

E, $105^{\circ}$

(21) If $A, B, C$ are she angles of a triangle and $\tan A=3$ and $\tan B=2$, then $\tan C=$

A. 1

B. 2

C. 3

D. 4

E. 5

(22) If $\sin 20^{\circ}=p$, then $\sec 70^{\circ}=$

A. $p$

B. $2 p$

C. $-p$

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

E. none of these

(23) If $f(x)=1-\frac{1}{x}$, then $f^{\prime}\left(\frac{1}{2}\right)=$

A. 2

B. 3

C. 4

D. 5

E. 6

(24) If $V=\frac{4}{3} r^{3}-\frac{3}{4} r^{2}+r-5$, then the rate of change of $V$ with respect to $r$ when $r=2$ is

A. 6

B. 7

C. 8

D. 9

E. 14

(25) The gradient of normal line to the curve $y=2 \sqrt{x}$ at the point $x=9$ is

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

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

C. 3

D. $-3$

E. 6

SECTION B

2. The function $f$ is defined, for $x \in R$, by $f(x)=2 x-3$. Find the value of $x$ for which $f(x)=f^{-1}(x)$

(OR)

Find the value of $k$ if $4 x^{7}+5 x^{3}-2 k x^{2}+7 k-4$ has a remainder of 12 when divided by $x+1$

3. The ninth term of an arithmetic progression is 6 . Find the sum of the first 17 terms.

(OR)

A geometric progression is such that the sum of the first 3 terms is $0.973$ times the sum to infinity. Find the common ratio.

4. Given : $\odot O$ with $A B=A D$ and $A C$ is a diameter.

$$\text { Prove : } B C=C D$$

5. Given that $A=B+C$, prove that $\tan A-\tan B-\tan C=\tan A \tan B \tan C$.

6. Differentiate $y=\frac{1}{x}$ with repsect to $x$ from the first principles.

SECTIONC

(Answer any SIX questions)

7. (a) Functions $f$ and $g$ are defined by $f(x)=\frac{x}{2-x}, x \neq 2$ and $g(x)=a x+b$. Given that $g^{-1}(7)=3$ and $(g \circ f)(5)=-7$, calculate the value of $a$ and of $b$.

(b) A binary operation $\odot$ on $R$ is defined by $x \odot y=x^{2}-2 x y+2 y^{2}$. Find $(3 \odot 2) \odot 4$. If $(3 \odot k)-(k \odot 1)=k+1$, find the values of $k$

8. (a) The cubic polynomial $f(x)$ is such that the coefficient of $x^{3}$ is $-1$ and the roots of the equation $f(x)=0$ are 1,2 and $k$. Given that $f(x)$ has a remainder of 8 when divided by $x-3$, find the value of $k$ and the remainder when $f(x)$ is divided by $x+3$

(b) The expansion of $(3+4 x)^{n}$, the coefficients of $x^{4}$ and $x^{5}$ are in the ratio of $5: 16$. Find the value of $n$

9. (a) Find the solution set in $R$ of the inequation $(x-6)^{2}>x$ by graphical method and illustrate it on the number line.

(b) The third term of an A.P. is 9 and the seventh term is $49 .$ Calulate the thirteenth term. Which term of the progression, if any, is $289 ?$

10. (a) The first and second terms of a G.P. are 10 and 11 respectively. Find the least number of terms such that their sum exceeds 8000 .

(b) The matrices $A$ and $B$ are such that $A=\left(B^{-1}\right)^{2}$. Given that $B=\left(\begin{array}{cc}2 & -1 \\ 2 & 1\end{array}\right)$ find the value of the constant $k$ for which $k B^{-1}=4 A+I$, where $I$ is the identity matrix of order 2 .

11. (a) Given that $A=\left(\begin{array}{cc}4 & -1 \\ -3 & 2\end{array}\right)$, use the inverse matrix of $A$ to solve the simultaneous equations $y-4 x+8=0,2 y-3 x+1=0$.

(b) Three tennis players $A, B, C$ play each other only once. The probability that $A$ will beat $B$ is $\frac{2}{7}$, that $B$ will beat $C$ is $\frac{1}{3}$ and that $C$ will beat $A$ is $\frac{2}{5}$. Calculate the probability that $A$ wins both games.

12. (a) Prove that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

(b) $A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ and $C B$ produced at $P$ and $Q$ respectively. Prove that $D C Q P$ is cyclic.

13. (a) In the figure, $A B / / C D$ and $\alpha(\triangle E C D): \alpha(A B D C)=16: 9$. Find the numerical value of $C D: A B$.

Given that $\alpha(\triangle E C D)=24 \mathrm{~cm}^{2}$, calculate $\alpha(\Delta E A B)$.

(b) The position vectors of the points $A,B$ and $C$, relative to an origin $O$, are

$2 \hat{i}+3 \hat{j}, 10 \hat{i}+2 \hat{j}$ and $\lambda(-\hat{i}+5 \hat{j})$ respectively. Given that $|\overrightarrow{A B}|=|\overrightarrow{A C}|$ show that $\lambda^{2}-\lambda-2=0$ and hence find the two possible vectors $\overrightarrow{A C}$

14. (a) If $\cot x+\cos x=p$ and $\cot x-\cos x=q$, show that $\sqrt{p q}=\cos x \cot x$, where $x$ is acute and hence, prove that $p^{2}-q^{2}=4 \sqrt{p q}$

(b) A man travels $10 \mathrm{~km}$ in a direction $\mathrm{N} 70^{\circ} \mathrm{E}$ and then $5 \mathrm{~km}$ in a direction N $40^{\circ} \mathrm{E}$. What is his final distance and bearing from his starting point?

15. (a) If $y=(3+4 x) e^{-2 x}$, then prove that $\frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+4 y=0$.

(b) Find the minimum value of the sum of a positive number and its reciprocal.

1)

1 C

2 E

3 C

4 C

5 B

6 D

7 E

8 C

9 B

10 A

11 C

12 B

13 D

14 A

15 C

16 A

17 C

18 B

19 C

20

21 A

22 D

23 C

24 E

25 D

2) $x=3[O R] \quad k=5$

3) $S_{17}=102[O R] \quad r=0.3$

4) Prove

5) Prove

6) $\frac{d y}{d x}=-\frac{1}{x^{2}}$

7) (a) $a=3, b=-2 \quad$ (b) $k=2$ or 3

8) (a) $k=7, R=200$

(b) $n=16$

9) $(a)\{x \mid x<4$ or $x>9\}$ (b) $u_{13}=109,n=31$

10) $(a) 47$ (b) $k=3$

11) (a) $x=3, y=4$ (b) $\frac{6}{35}$

12) (a)  Prove (b) prove

13)  (a) $\alpha(\Delta E A B)=37.5 \mathrm{~cm}^{2}$

(b) $\overrightarrow{A C}=-4 \hat{i}+7 \hat{j} \text { or } \overrightarrow{A C}=-\hat{i}-8 \hat{j}$

14) (a) Prove (b) $14.55 \mathrm{~km}, \mathrm{~N} 60^{\circ}{6}^{\prime} \mathrm{E}$

15) (a) Prove (b) 2