2018 (D)

MATRICULATION EXAMINATION

DEPARTMENT OF MYANMAR EXAMINATION

MATHEMATICS

Time Allowed : (3) Hours

WRITE YOURANSWERS IN THE ANSWER BOOKLET.

SECTION (A) $\def\frac{\dfrac}$

(AnswerALL 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 $f$ and $g$ be functions given by $f(x)=2 x$ and $g(x)=x+3$. If $(g \circ f)^{-1}(t)=1$ then $t=$

A. 3

B. $-3$

C. 5

D. $-5$

E. 2

(2) An operation $\odot$ is defined by $x \odot y$ is the remainder when $x$ is divided by $y$, then $(61 \odot 7)-(5 \odot 5)=$

A. $-4$

B. 4

C. 3

D. 5

E. 6

(3) It is given that the remainder is 178 when $x^{n}-5 x^{2}-20$ is divided by $x-3$, then the value of $n$ is

A. $-4$

B. 4

C. 3

D. $-3$

E. 5

(4) If $x-2$ is a factor of $x^{2 n+1}+3 x^{2}-44$, then the value of $n$ is

A. 2

B. $-2$

C. 4

D. 1

E. $-1$

(5) The term independent of $x$ in the expansion of $(\tan x+\cot x)^{6}$ is

A. 6

B. 15

C. 12

D. 18

E. 20

(6) The coefficient of $x^{n}$ in the expansion of $(1+x)^{2 n}$ is

A. ${ }^{2 n} \mathrm{C}_{n-1}$

B. ${ }^{2 n} \mathrm{C}_{n}$

C ${ }^{n} \mathrm{C}_{n}$

D. ${ }^{2 n} \mathrm{C}_{n+1}$

E. none of thes

(7) The solution set of the inequation $\frac{1}{3 x^{2}+2} \geq 0$ is

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

B. $\varnothing$

C. $\{x \mid x<0\}$

D. $\{x \mid x>0\}$

E. $R$

(8) In a certain series, if $S_{n}=(n-1)(n-2)$, then $u_{n}=$

A. $4(n-1)$

B. $4(n-2)$

C. $2(2 \cdots n)$

D. $2(n-2)$

E. $2(n-1)$

(9) In an A.P., the fourth term is 6 . Then $S_{7}=$

A. 35

B. 40

C. 42

D. 45

E. 50

(10) If the A.M. between 3 and $x$ is 6 , then the positive G.M. between $x$ and $x+7$ is

A. 9

B. 12

C. 15

D. 18

E. $2 i$

(11) If $A$ is $2 \times 2$ matrix such that det $A=k$ and $p$ is real number then $\operatorname{det}(p A)=$

A. $p k$

B. $p k^{2}$

C. $p^{2} k$

D. $p^{2} k^{2}$

E. $p$

(12) The matrix $M=\left(\begin{array}{ll}a & 4 \\ 16 & \mathrm{~b}\end{array}\right)$ is singular and $a, b$ are positive integers. Then $a+b$ cannot be

A. 16

B. 20

C. 34

D. 48

E. 65

(13) A box contains 5 cards numbered as $2,3,4,5$ and 9 . If a card is chosen, then the probability of getting not a prime number is

A. $\frac{2}{3}$

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

C. $\frac{2}{5}$

D. $\frac{1}{5}$

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

(14) A die is rolled 120 times. The expected frequency of a prime number is

A. 40

B. 60

C. 80

D. 20

E. 120

(15) In the figure, $A T$ is a tangent at $A$ and $\angle A B C=125^{\circ}$

then $\angle T$

A. $140 ^{\circ}$

B. $125^{\circ}$

C. $70^{\circ}$

D. $80^{\circ}$

E. $55^{\circ}$

(16) In the figure, $P T$ is a tangent at $P . P T=6, P Q=4$ and $P R=3$ then $R T=$

A. 3

B. 4

C. 5

D. 6

E. 7

(17) In $\triangle A B C, D$ is a point on $A B$ such that $A D=3 D B \cdot E$ is a point on $A C$ such that $D E / / B C$. If $\alpha(\triangle A D E)=36$, then $\alpha(\triangle A B C)=$

A. 56

B. 64

C. 72

D. 80

E. 96

(18) If the position vectors of $P$ and $Q$ with respect to origin $O$ are $-2 \hat{i}+7 \hat{j}$ and $4 \hat{i}-5 \hat{j}$ respectively and $P R: R Q=2: 1$, then $\overrightarrow{R Q}=$

A. $2 \hat{i}-4 \hat{j}$

B. $-2 \hat{i}+4 \hat{j}$

C. $3 \hat{i}-\hat{j}$

D. $-3 \hat{i}+\hat{j}$

E. $\hat{i}-2 \hat{j}$

(19) The map of the point $(4,0)$ by a rotation through an angle $180^{\circ}$ about $O$ in clockwise direction is

A. $(4,-4)$

B. $(0,-4)$

C. $(0,4)$

D. $(-4,0)$

E. $(-4,4)$

(20) $\sin ^{2} x+\tan ^{2} x \sin ^{2} x=$

A. $\sin ^{2} x$

B. $\cos ^{2} x$

C. $\tan ^{2} x$

D. $\cot ^{2} x$

E. $\sec ^{2} x$

(21) If $\theta$ is an acute angle and $\sin \theta=x$, then $\sin \left(270^{\circ}-2 \theta\right)$ is

A. $2 x \sqrt{1-x^{2}}$

B. $-2 x \sqrt{1-x^{2}}$

C. $2 x^{2}-1$

D. $1-2 x^{2}$

E. none of these

(22) The exact value of $\sin \frac{7 \pi}{9} \cos \frac{4 \pi}{9}-\cos \frac{7 \pi}{9} \sin \frac{4 \pi}{9}=$

A. 1

B. 0

C. $\frac{1}{2}$

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

E. $\frac{\sqrt{3}}{2}$

(23) If $n$ is a rational number, then $\lim _{h \rightarrow 0} \frac{(x+h)^{n}-x^{n}}{h}=$

A. $\boldsymbol{n}$

B. $x^{n}$

C. $nx^{n-1}$

D. $(n-1)x^{n} \quad $

E $x^{n-1}$

(24) The gradient of the curve $x y=1$ at $(-1,-1)$ is

A. 0

B. $-1$

C. 1

D. 2

E. 3

(25) It is given that $y=2 x^{3}$ and $x$ is changed from 2 to $1.995$. Then the approximate change in $y$ is

A. $-0.18$

B. $-0.27$

C. $-0.12$

D. $-0.24$

E. $-0.36$

SECTION (B)

(AnswerALL questions)

2. If the function $f: R \rightarrow R$ is a one-to-one correspondence, then verify that $\left(f \circ f^{-1}\right)(y)=y$ and $\left(f^{-1} \circ f\right)(x)=x$

(OR) The remainder when $2 x^{3}+k x^{2}+7$ is divided by $x-2$ is half the remainder when the same expression is divided by $2 x-1$. Find the value of $k$.

Click for Solution3. In an A.P. whose first term is $-27$, the tenth term is equal to the sum of the first nine terms. Calculate the common difference.

(OR) The first term of a G.P. is $a$ and the common ratio is $r$. Given that $a=12 r$ and that the sum to infinity is 4 , find the third term.

Click for Solution4. In the figure, $\angle B C D=125^{\circ}$ find $x$ and $y$.

Click for Solution

5. If $A+B=45^{\circ}$, show that $\tan A+\tan B+\tan A \tan B=1$.

Click for Solution6. Differentiate $f(x)=1-2 x^{2}$ with respect to $x$ at $x=2$ from the first principles.

SECTION (C)

(Answer any SIX questions)

7.(a) The functions $f$ and $g$ are defined for real $x$ by $f(x)=2 x-1$ and $g(x)=2 x+3 .$ Evaluate $\left(g^{-1} \circ f^{-1}\right)(2)$

Click for Solution(b) The binary operation $\odot$ on $R$ is defined by $x \odot y=\frac{x^{2}+y^{2}}{2}-x y$, for all real numbers $x$ and $y$. Show that the operation is commutative, and find the possible values of $a$ such that $a \odot 2=a+2$

Click for Solution8.(a) The expression $p x^{3}-5 x^{2}+q x+10$ has factor $2 x-1$ but leaves a remainder of $-20$ when divided by $x+2$. Find the values of $p$ and $q$ and factorize the expression completely.

Click for Solution(b) In the expansion of $(1-2 x)^{n}$, the sum of the coefficients of $x$ and $x^{2}$ is 16 Given that $n$ is positive, find the value of $n$ and the coefficient of $x^{3}$.

Click for Solution9.(a) Use a graphical method to find the solution set of the inequation $2 x(x-1)<3-x$ and illustrate it on the number line.

Click for Solution(b) An A.P. is such that the 5 th term is three times the 2 nd term. Given further that the sum of the $5 \mathrm{th}, 6$ th, 7 th and 8 th terms is 240 , calculate the value of the first term.

Click for Solution10. (a) A G.P. of positive terms and an A.P. have the same first term. The sum of their first terms is 1 , the sum of their second terms is $\frac{1}{2}$ and the sum of their third terms is 2 . Calcualte the sum of their fourth terms.

Click for Solution(b) Given that $A=\left(\begin{array}{cc}5 & 1 \\ a+1 & a\end{array}\right)$ and $\operatorname{det} A=7$, find the value of $a$ and then calculate the values of $x$ and $y$ such that $A^{2}-x A^{-1}-y I=O$, where $I$ is the unit matrix order 2 .

Click for Solution11.(a) Find the inverse of the matrix $\left(\begin{array}{rr}7 & -4 \\ -3 & 2\end{array}\right)$ and use it to solve the following systems: $7 x-4 y=13,2 y-3 x=-5$

Click for Solution(b) How many 3 digit numbers less than 400 can you form by using $1,2,3$ and 4 without repeating any digit? If one of these numbers is chosen at random, find the probability that it is divisible by 3 but not divisible by 4 . Find also the probability that a number which is not divisible by 3 .

Click for Solution12.(a) Through the points of intersection of two circles, two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C / / B D$.

Click for Solution(b) $A B C$ is a triangle inscribed in a circle and $D E$ the tangent at $A$. A line drawn parallel to $D E$ meets $A B, A C$ at $F, G$ respectively. Prove that $B F G C$ is a cyclic quadrilateral.

Click for Solution13.(a) $A B C$ is a right triangle with $\angle A$ the right angle. $E$ and $D$ are points on opposiie side of $A C$, with $E$ on the same side of $A C$ as $B$, such that $\triangle A C D$ and $\Delta B C E$ are both equilateral. If $\alpha(\triangle B C E)=2 \alpha(\triangle A C D)$, Prove that $A B C$ is an isosceles right triangle.

Click for Solution(b) Relative to an origin $O$ the position vectors of the points $P$ and $Q$ are $3 \hat{i}+\hat{j}$ and $7 \hat{i}-15 \hat{j}$ respectively. Given that $R$ is the point such that $3 \overrightarrow{P R}=\overrightarrow{R Q}$, find a unit vector in the direction $\overrightarrow{O R}$.

Click for Solution14.(a) Two acute angles, $\alpha$ and $\beta$, are such that $\tan \alpha=\frac{4}{3}$ and $\tan (\alpha+\beta)=-1$. Without evaluating $\alpha$ or $\beta$, show that $\tan \beta=7$, evaluate $\sin \alpha$ and $\sin \beta$.

Click for Solution(b) A ship is $5 \mathrm{~km}$ away from a boat in a direction $N 37^{\circ} \mathrm{W}$ and a lighthouse is $12 \mathrm{~km}$ away from the boat in a direction $S 53^{\circ} W .$ Calcualate the distance and direction of the ship from the lighthouse.

Click for Solution15.(a) Given that $x y=\sin x$, prove that $\frac{d^{2} y}{d x^{2}}+\frac{2}{x} \frac{d y}{d x}+y=0$.

Click for Solution(b) Find the approximate change in the volume of a sphere when its radius increases fron $2 \mathrm{~cm}$ to $2.05 \mathrm{~cm}$.

Answer

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

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

(11)C (12)D (13)C (14)B (15)E

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

(21)C (22)E (23)C (24)B (25)C

2) Verify

(OR) $k=-5$

3) $d=8 \quad$

(OR) $u_{3}=\frac{3}{16}$

4) $x=35, y=55$

5) Show

6) $f^{\prime}(2)=-8$

7) (a)$-\frac{3}{4}$

(b) $a=0 $ or $6$

8) (a) $p=6, q=-19, f(x)=(2 x-1)(3 x+5)(x-2)$

(b) $n=4$, $-32$

9) (a) $\left\{x \mid-1<x<\frac{3}{2}\right\}$

b) $a=5$

10) (a) $\frac{19}{2}$

(b) $a=2, x=-49, y=42$

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

(b) $\frac{7}{18}, \frac{4}{9}$

12) (a) Prove

(b) Prove

13 (a) Prove

(b) $\frac{4}{5} \hat{i}-\frac{3}{5} \hat{j}$

14) (a) $\sin \alpha=\frac{4}{5}, \sin \beta=\frac{7 \sqrt{2}}{10}$

(b) $13 \mathrm{~km}$, $N 30^{\circ} 23^{\prime} E$

15) (a) Prove

(b) $0.8 \pi $

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