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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 $f$ 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. $\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$
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
(Answer ALL questions)
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.
Answer
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
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