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 defined by $f(x)=x+1$, then the function $g$ such that $(g \circ f)^{-1}(x)=(x-3)$ satisfied $g(x)=$.
$\begin{array}{lllll}\text { A. } x-1 & \text { B. } x-2 & \text { C. } x+2 & \text { D. } x+1 & \text { E. } x+5\end{array}$
(2) $\odot$ is defined on the set of real numbers by $(a-b) \odot(a+b)=k a^{2}+b$. If $6 \odot 3=50$, then $k=$
A. 2 B.1 $\begin{array}{ll}\text { C. } 0 & \text { D: }-1 & \text { E. }-2\end{array}$
(3) If $n$ is an integer, then the remainder when $2 x^{2 n+1}-4 x^{2 n}+5 x^{2 n-1}+3$ is divided by $x+1$ is
$\begin{array}{lllll}\text { A. }-8 & \text { B. }-4 & \text { C. } 0 & \text { D. } 4 & \text { E. } 8\end{array}$
(4) If $x-3$ is a factor of $x^{3}-6 x^{2}+a x-6$, then $a+4$ is
A. 22
B. 15
C. 12
D. 11
E. 5
(5) If ${ }^{n} C_{2}=66$, then $n=$
A. 9
B. 10
C. 11
D. 12
E. 13
(6) The coefficient of the middle tetm in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{6}$ is
A. $-120$
B. 125
C. 240
D. $-240$
E. 160
(7) The parabola $y=12 x^{2}-25 x+12 \mathrm{c}$. s the $X$ axis at $A$ and $B$. The distance between $A$ and $B$ is
A. 2
B. $\frac{7}{4}$
C. $\frac{7}{12}$
D. $-\frac{7}{12}$
E. $-\frac{7}{4}$
(8) Given that $7, a, b, c,-5$ in an A.P., then the mean of $a, b, c$ is
A. $-2$
B. 1
C. $\frac{3}{2}$
D. 3
E. 4
(9) In a G.P. each term is positive, the 4th term is 54 and the 6 th term is 486 , then the common ratio is
A. 3 or $-3$
B. $-3$ only
C. 3 only
D. 6 only $\quad$
E. 6 or $-6$
(10) The product of the A.M. and G.M. between 4 and 16 is
A. 40
B. 60
C. 70
D. 80
E. 160
(11) The matrix $M=\left(\begin{array}{ll}a & 4 \\ 16 & b\end{array}\right)$ is singular and $a, b$ are positive integers. Then $a+b$ cannot be
A. 16
B. 20
C. 34
D. 48
E. 65
(12) If the multiplicative inverse of the matrix $\left(\begin{array}{cc}1 & 3 \\ x & 1\end{array}\right)$ is $-\frac{1}{5}\left(\begin{array}{rc}1 & -3 \\ -2 & 1\end{array}\right)$, then $x=$
A. 1
B. $-1$ -
C. 3
D. 2
E. $-2$
(13) Acletter is chosen from the letters of the word MATRICULATION. The probability that it will not be a vowel is
A. $\frac{10}{3}$
B. $\frac{7}{13}$
C. $\frac{6}{13}$
D. $\frac{4}{13}$
E. $\frac{3}{13}$
(14) Two table-tennis players $P$ and $Q$ played 25 games. From those games, the probability that $P$ will win $Q$ is $0.6$. Therefore, $P$ did not win $Q$ in
A. 15 games$\begin{array}{llll}\text { B. } 12 \text { games } & \text { C. 11 games } & \text { D. 10 games } & \text { E. } 8 \text { games }\end{array}$
(15) The opposite angles of a cyclic quadrilateral are in the ratio $3: 7$. The difference of their degree measure is
A. $72^{\circ}$
B. $54^{\circ}$
C $36^{\circ}$
D. $18^{\circ}$
E. none of these
(16) The arc forms part of a circle whose radius is
A. 9
B: 10
C. 16
D. 18
E. 20
(17) Two corresponding altitudes of two similar triangles are $9 \mathrm{~cm}$ and $12 \mathrm{~cm}$. Then $\alpha($ the smaller $\Delta): \alpha($ the larger $\Delta)=$
A. $4: 5$
B. $3: 4$
C. $9: 25$
D. $9: 16$
E. $16: 25$
(18) The vector of magnitude of 65 which is parallel to the vector $5 \hat{i}-12 \hat{j}$ is
A. $5(5 \hat{i}-12 \hat{j})$
B. $7(5 \hat{i}-12 \hat{j})$,
C. $13(5 \hat{i}-12 \hat{j})$
D. $12 \hat{i}-5 \hat{j}$
E. $6(5 \hat{i}-12 \hat{j})$
(19) If $\overrightarrow{A B}=\left(\begin{array}{r}8 \\ -6\end{array}\right)$ and $\overrightarrow{C D}=\frac{3}{2} \overrightarrow{A B}$; then the $|\overrightarrow{C D}|=$
A. 10
B. 15
C. 25
D. 30
E. 35
(20) In any $A B C$, if $A+B+C=180^{\circ}$, then $\sin (A+B)=$
A. $-\sin C$
B. $\sec \left(90^{\circ}+\mathrm{C}\right)$
C. $\sin \left(90^{\circ}-\mathrm{C}\right)$
D. $\cdot \cos \left(90^{\circ}-C\right) $
E. $-\cos C$
(21) $\cos ^{2} 60^{\circ}+\sin ^{2} 120^{\circ}+2 \cot ^{2} 135^{\circ}=$
A. 4
B. 3
C. 2
D. $-2$
E. $-1$
(22) If $\theta$ is an acute angle and $\sin \theta=k$, then $\sin 2 \theta=$
A. $2 k \sqrt{1-k^{2}}$
B. $k \sqrt{1-k^{2}}$
C. $2 k \sqrt{k^{2}-1}$
D. $k \sqrt{k^{2}-1}$
E. $\sqrt{k^{2}-1}$
(23) If $\lim _{x \rightarrow 1} \frac{a(x-1)}{x^{2}-1}=2$ where $a$ is a constant, then $a=$
A. 0
B. 1
C: 2
D. 3
E. 4
(24) The rate of change of the function $f: x \mapsto \frac{4}{3} x^{3}-\frac{3}{4} x^{2}+x-5$ at $x=2$ is
A. $-14$
B. 20
C. 14
D. 12
E. 6
(25) The gradient of the tangent to the curve $y=x^{2}-a x+6$ at the point where $x=2$ is $-1$, then the value of $a$ is
A. $-5$
B. $-3$
C. 5
D. 4
E. 3 (25 marks)
SECTION B (Answer ALL questions)
2. A function $f$ is defined by $f(2 x+1)=x^{2}-3$. Find $a \in R$ such that $f(5)=a^{2}-8 .$(3 Marks)
(OR) Given that the expression $x^{3}-p x^{2}+q x+\dot{r}$ leaves the same remainder when divided $s y x+1$ or $x-2$. Find $p$ in terms of $q$ (3 Marks)
3. Given that $\sin ^{2} x, \cos ^{2} x$ and $5 \cos ^{2} x-3 \sin ^{2} x$ are in A.P., find the value of $\sin ^{2} x .$(3 Marks)
(OR) Write down the next two terms of the sequence $\sqrt{2}, \sqrt{10}, 5 \sqrt{2}, 5 \sqrt{10}, \cdots$ and defermine the $n^{\text {th }}$ term of the sequence. (3 Marks)
4. The position vectors of $A, B$ and $C$ are $2 \vec{p}-\vec{q}, k \vec{p}+\vec{q}$ and $12 \vec{p}+4 \vec{q}$ respectively. Calculate the value of $k$ if $A, B$ and $C$ are collinear with $\vec{p} \neq \overrightarrow{0}, \vec{q} \neq \overrightarrow{0}, \vec{p}$ and $\vec{q}$ are not parallel. (3 Marks)
5. Prove the identity $\cos 3 \theta-\cos \theta=-4 \sin ^{2} \theta \cos \theta .$ (3 Marks)
6. Find $\displaystyle\lim _{x \rightarrow 5} \frac{x^{3}-125}{5-x}$ and $\displaystyle\lim _{x \rightarrow \infty} \frac{4 x^{2}-10 x+15}{2 x^{2}-3 x-5}, \quad$ (3 Marks)
SECTION (C) (Answe any SIX questions)
7.(a) Let $f(x)=2 x-1, g(x)=\frac{2 x+3}{x-1}, x \neq 1$. Find the formula for $(g \circ f)^{-1}$ and state the domain of $(g \circ f)^{-1} . $ (5 Marks)
(b) Show that the mapping $\odot$ defined by $x \odot y=x y+x^{2}+y^{2}$ is a binary operation on the set $R$ and verify that it is commutative and but not associative. (5 Marks)
8.(a) Given that $f(x)=x^{3}+p x^{2}-2 x+4 \sqrt{3}$ has a factor $x+\sqrt{2}$, find the value of $p$. Show that $x-2 \sqrt{3}$ is also a factor and solve the equation $f(x)=0$. (5 Marks)
(b) Given that $\left(p-\frac{1}{2} x\right)^{6}=r-96 x+s x^{2}+\cdots$, find $p, r, s .$(5 Marks)
9.(a) Find the solution set of the inequation $3 x^{2}<x^{2}-x+3$, by graphical method . and illustrate it on the number line.(5 Marks)
(b) The sum of four consecutive numbers in an A.P. is 28 . The product of the second and third numbers exceeds that of the first and last numbers by 18 . Find the numbers. (5 Marks)
10.(a) A geometric progression has three terms $a ; b, c$ whose sum"is 42 . If 6 is added to each of the first two terms. and 3 to the third, a new G.P. results whose first term is the same as $b$. Find $a, b$ and $c .$ (5 Marks)
(b) Given that $A=\left(\begin{array}{cc}4 & 3 \\ 1 & 1\end{array}\right)$ and $B=\left(\begin{array}{rr}4 & 2 \\ -5 & -3\end{array}\right)$, write down the inverse matrix $A^{-1}$ and find the matrices $P$ and $Q$ such that $P A=2 I$ and $A Q=2 B$. (5 Marks)
11.(a) Find the solution set of the system of equations $2 x-5 y=1$ , $3 x-7 y=2$ by matrix method. (5 Marks)
(b) The probability of an event $A$ happening is $\frac{2}{3}$ and the probability that an event $B$ happening is $\frac{3}{8}$ : Given that $A$ and $B$ are independent, calculate the probability thai neither event happens and just one of the two events happens.(5 Marks)
12.(a) In the figure, $A B C D E$ is a semicircle at centre $O$, the segment $A E$ is the diameter and $B, C, D$ are any points on the arc. Prove that $\angle A B C+\angle C D E=270^{\circ}$. (5 Marks)
(b) Given: $\angle A B E=\angle A D E$ and $\angle D A C=\angle D E C$.
Prove: $A, B, C, D$ and $E$ all lie on one circle.(5 Marks)
13.(a) $A B C$ is a triangle. If $B P C, C Q A, A R B$ are equilateral triangles and $\alpha(\triangle B P C)+\alpha(\Delta C Q A)=\alpha(\triangle A R B)$, then prove that $A B C$ is a right triangle. (5 Marks)
(b) In a qưadrilateral $O L N M, O M \| L N$, where $\overrightarrow{O L}=\vec{a}, \overrightarrow{O M}=\vec{b}$ and $\overrightarrow{L N}=k \vec{b}, O P$ is drawn parallel to $M N$ to meet the diagonal $M L$ at $P$. If $L P=\frac{1}{4} L M .$ Find the value of $k . \quad$ (5 Marks)
14.(a) Show that $\sin (\alpha+\beta) \cdot \sin (\alpha-\beta)=\sin ^{2} \alpha-\sin ^{2} \beta \quad$ (5 Marks)
(b) Given that $\sin A=\frac{2}{\sqrt{5}}, \cos B=-\frac{\sqrt{2}}{3}$ and that both $A$ and $B$ are in the same quadrant, calculate tbe value of each of the following:
$\begin{array}{lll}\text { (i) } \cos (A+B) & \text { (ii) } \cos (2 A-B) & \end{array}$(5 Marks)
15.(a) If $y=A \cos (\ln x)+B \sin (\ln x)$, where $A$ and $B$ are constants, show that $x^{2} y^{\prime \prime}+x y^{\prime} =0 . $ (5 Marks)
(b) Given that $x+y=5$; calculate the minimum value of $x^{2}+x y+y^{2}$. (5 Marks)
Answers
(1)C
(2)B
(3)A
(4)B
(5)D
(6)E
(7)C
(8)B
(9)C
(10)D
(11)D
(12)D
(13)B
(14)D
(15)A
(16)B
(17)D
(18)A
(19)B
(20)D
(21)B
(22)A
(23)E
(24)C
(25)C
2 $a=\pm 3$ or $p=q+3$
3 $\frac 35$ or $25\sqrt 2,25\sqrt{10},u_n=(\sqrt 5)^{n-1}\sqrt 2$
4 $k=6$
5 Prove
6 $-75,2$
7(a) $(g \circ f)^{-1}=\frac{1+2x}{2x-4},x\not=2$
domain of $(g \circ f)^{-1} =\{x|x\not=2,x\in R\}$
(b) $(1\odot 0)\odot 2\not=1\odot (0\odot 2)$
8(a) $p=-2\sqrt 3,x=2\sqrt 3,\pm \sqrt 2$
(b) $p=2,r=64,s=60$
9(a) $\{x|-\frac 32<x<1\}$
(b) $11\frac 12,8\frac 12,5\frac 12,2\frac 12$
10(a) $a=6,b=12,c=24$
(b) $A^{-1}=\left(\begin{array}{cc}1 & -3 \\ -1 & 4\end{array}\right)$
$P=\left(\begin{array}{cc}2 & -6 \\ -2 & 8\end{array}\right)$
$\left(\begin{array}{cc}38 & 22 \\ -48 & -28\end{array}\right)$
11(a) $\{(3,1)\}$
(b) $\frac{5}{24},\frac{13}{24}$
12(a) Prove (b) Prove
13(a) Prove (b) $\frac 43$
14(a) Prove (b) $\cos(A+B)=\frac{\sqrt{10}-2\sqrt{35}}{15}$
$\cos(2A-B)=\frac{3\sqrt 2-4\sqrt 7}{15}$
15(a) Prove (b) $\frac{75}{4}$
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