MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
| MATHEMATICS | Time Allowed : (3) Hours |
SECTION (A)
(Answer ALL questions)
1.(a) If $f: R \rightarrow R$ is defined by $f(x)=x^{2}+3$, find the function $g$ such that $(f \circ g)(x)=9 x^{2}-12 x+7$ (3 marks) (b) When $f(x)=2 x^{3}+x^{2}-k x+k$ is divided by $2 x+1$, the remainder is 6 . Find the value of $k$ and hence show that $x+2$ is a factor of $f(x)$. (3 marks)
2.(a) Find the coefficient of $x^{2}$ in the expansion of $(1-2 x)^{5}(1+x)^{4}$. (3 marks)
(b) The sum of an infinite geometric progression is 10 and its first term is $6 .$ Find the first 3 terms of the G.P. (3 marks)
3,(a) If $\mathrm{A}=\left(\begin{array}{cc}1 & -2 \\ -3 & 6\end{array}\right)$ and $\mathrm{B}=\left(\begin{array}{cc}2 & -2 \\ 1 & 1\end{array}\right)$, find $\mathrm{AB}+\mathrm{BA}$. (3 marks)
(b) A blue die and a black die are rolled. Find the probability of getting a score which includes a 2 on the blue die or a 5 on the black dic. (3 marks)
4.(a) In the figure, $\mathrm{O}$ is the centre of the circle. If $\angle O C B=68^{\circ}$ and $\angle A O C=112^{\circ}$, find $\angle \mathrm{OAB}$. (3 marks)
(b) In $\triangle \mathrm{ABC}, \mathrm{D}$ is the point on $\mathrm{BC}$ such that $\mathrm{CD}=\frac{1}{4} \mathrm{CB}$. If $\overrightarrow{\mathrm{AB}}=12 \overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{AC}}=4\overrightarrow{\mathrm{q}},$ find $\overrightarrow{\mathrm{AD}}$ in terms of $ \overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$. (3 marks)
5.(a) Solve the equation $2 \cos ^{2} x+\cos x-1=0$ for $0^{\circ} \leq x \leq 180^{\circ}$. (3 marks)
(b) Differentiate $f(x)=x^{2}-1$ with respect to $x$ at $x=1$ from the first principles. (3 marks) SECTION (B) (Answer any FOUR questions)
6. (a) The functions $f$ and $g$ are defined by $f(x)=3 x+1$ and $g(x)=2 x-3$. Show that $(f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)$. (5 marks)
(b) The expression $x^{3}+a x^{2}+b x+9$ is exactly divisible by $x+3$ but it leaves a remainder of 1 when divided by $x+4$. What is the remainder when it is divided by $x+1$ ? (5 marks)
7.(a) Show that the binary operation $\odot$ on the set $R$ of real numbers defined by $a \odot b=a+b+a b$ is associative. (5 marks)
(b) In the expansion of $\left(\frac{a}{x}-x^{2}\right)^{8}, a \neq 0$, the coefficient of $x^{7}$ is four times the coefficient of $x^{10}$. Find the value of $a$. (5 marks)
8. (a) Sketch the graph of $y=9-3 x-2 x^{2}$ and use it to find the solution set of $9-3 x-2 x^{2} \geq 0$. Illustrate the solution set on the number line. (5 marks)
(b) The sum of the first 4 terms of an A.P. is 10 and the sum of their squares is 70 . Find the first 4 terms. (5 marks)
9. (a) Solve the equation $1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+x^{7}+x^{8}+x^{9}=x+5+\frac{x^{10}}{x-1}$. (5 marks) (b) $M=\left(\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right)$. Find $M^{2}$ and $\left(M^{-1}\right)^{2}$. Show that $M^{2}\left(M^{-1}\right)^{2}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$. (5 marks)
10,(a) Given that $A=\left(\begin{array}{ll}2 & 1 \\ 3 & 1\end{array}\right)$ and $B=\left(\begin{array}{cc}2 & 5 \\ -1 & -3\end{array}\right)$, write down the inverse matrix of $A$. Use your result to find the matrix $\mathrm{Q}$ such that $\mathrm{QA}=\mathrm{B}$. (5 marks)
(b) What is the probability of scoring 11 when throwing two dice at once? If such experiment is repeated 720 times, what would you expect if the score not being 11 ? (5 marks) SECTION (C) (Answer any THREE questions)
11 (a) - $\mathrm{ABC}$ is a triangle inscribed in a circle. The tangent at $\mathrm{A}$ meets $\mathrm{BC}$ produced at $\mathrm{P}$, and $\mathrm{AR}$ bisecting $\angle \mathrm{CAB}$ meets side $\mathrm{CB}$ at $\mathrm{R}$. Prove $\triangle \mathrm{PAR}$ isosceles. (5 marks)
(b) In a given circle AP is a tangent segment and $A Q R$ is a secant segment. Prove that $\frac{\mathrm{PQ}^{2}}{\mathrm{RP}^{2}}=\frac{\mathrm{AQ}}{\mathrm{AR}} . (5 marks)
12. (a) $\mathrm{ABCD}$ is a parallelogram. A circle through $\mathrm{A}, \mathrm{B}$ cuts the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ at $\mathrm{P}$ and $\mathrm{Q}$ respectively. Prove that $\mathrm{C}, \mathrm{D}, \mathrm{Q}, \mathrm{P}$ are concyclic. (5 marks)
(b) Show that $\sin \frac{7 \pi}{12} \cos \frac{\pi}{12}=\frac{2+\sqrt{3}}{4}$. (5 marks)
13. (a) Solve $\triangle \mathrm{ABC}$, with $\mathrm{AB}=5, \mathrm{AC}=7, \angle \mathrm{BAC}=32^{\circ}$. (5 marks)
(b) Find the equation of the tangent line to the curve $2 x^{2}-3 y^{2}=2 x y+1$ at the point $(2,1)$. (5 marks)
14.(a) In the diagram, $M$ is the midpoint of $O P$ and $\mathrm{Q}$ is the midpoint of OY. $\overrightarrow{\mathrm{OP}}=4 \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{OQ}}=\overrightarrow{\mathrm{b}}$. If $\overrightarrow{X Q}=\lambda \overrightarrow{P Q}$ and $\overrightarrow{X Y}=\mu \overrightarrow{M Y}$, evaluate $\lambda$ and $\mu$. (5 marks)
(b) If a piece of string of length $32 \mathrm{~cm}$ is made to enclose a rectangle, show that the enclosed area is the greatest when the rectangle is a square. (5 marks)
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