MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
| MATHEMATICS | Time Allowed : (3) Hours |
SECTION (A)
(Answer ALL questions)
1.(a) If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x)=b x-57$ where $b$ is a constant and $g(x)=x+7,(f \circ g)(1)=7$, find the value of $b$. (3 marks)
(b) Given that the expression $x^{3}-a x^{2}+b x+c$ leaves the same remainder when divided by $x+1$ or $x-2$, find the relation between $a$ and $b$. (3 marks)
2.(a) Given that the coefficient of $x^{3}$ in the expansion of $(k+3 x)^{6}$ is 20 , find the value of $k$. (3 marks)
(b) If the third term and the tenth term of an A.P. are 11 and 39 respectively, find the first term and the common difference of the A.P. (3 marks)
3.(a) Given that $A=\left(\begin{array}{rr}h & 3 \\ -3 & 2\end{array}\right), B=\left(\begin{array}{ll}2 & -3 \\ 3 & -4\end{array}\right)$ and $A B=I$ where $I$ is the unit matrix of order 2 , find the value of $h$. (3 marks)
(b) If a die is rolled 240 times, find the expected frequency of getting a factor of 30 . (3 marks)
4.(a) In the figure, $O$ is the centre of the circle and ${AOD}$ is a diameter. If $\angle {CBD}=38^{\circ}$, find $\angle {ADC}$. (3 marks)
(b) Find the matrix which rotates through $60^{\circ}$ and find the map of the point $(0,2)$. (3 marks)
5.(a) Find the value of $\cos 165^{\circ}$ in surd form. (3 marks)
(b) Calculate $\displaystyle \lim _{x \rightarrow 4} \frac{x^{2}-16}{x-4}$ and $\displaystyle \lim _{x \rightarrow \infty} \frac{x^{2}-2 x+1}{2+x-x^{2}}$. (3 marks)
(Answer any FOUR questions)
6.(a) Functions $f$ and $g$ are defined by $f(x)=c x+d$, where $c$ and $d$ are constants, $g(x)=\frac{1}{3} x-1$. If $f(2)=g^{-1}(2)$ and $(f \circ g)(-3)=-3$, find the values of $c$ and $d$. (5 marks)
(b) Given that the equation $2 x^{3}+p x^{2}+q x-12=0$ has roots $x=1$ and $x=4$, find the values of $p, q$ and the third root. (5 marks)
7.(a) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ be defined by $x \odot y=x y+x-y$ for $x, y \in R$. Find the values of $(2 \odot 3) \odot 4$ and $2 \odot(3 \odot 4)$. Is this binary operation associative? Why? (5 marks)
(b) Using binomial theorem, find the coefficient of $x^{2}$ in the expansion of $\left(3+2 x-x^{2}\right)^{5}$. (5 marks)
8. (a) Find the solution set in $R$ of the inequation $25 x^{2}-5 x-12 \leq 0$ by graphical method and illustrate it on the number line. (5 marks)
(b) The sum to ${n}$ terms of an A.P. is 18. The common difference is 3 and the sum to $3 n$ terms is 135 . Find the sum of the first 20 terms of the progression. (5 marks)
9.(a) The fourth term of a G.P. exceeds the third term by $\frac{3}{8}$ and the third term exceeds the second term by $\frac{1}{4}$. Find the first term and the sixth term of the G.P. (5 marks)
(b) Given that ${A}=\left(\begin{array}{ll}4 & 1 \\ 7 & 2\end{array}\right)$ and ${B}=\left(\begin{array}{cc}-2 & 3 \\ 6 & 8\end{array}\right)$, find $3 {~A}^{\prime}+{BA}^{-1}$. (5 marks)
10.(a) Find the solution set of the system of equations $x+3 y=7$ and $5 y-2 x=-3$ by matrix method. (5 marks)
(b) A coin is tossed three times. Head or tail is recorded each time. Drawing a tree diagram, find the probability of (i) getting exactly one head, (ii) getting at least one head, and (iii) getting at most one head. (5 marks)
(Answer any THREE questions)
11. (a) Two circles touch externally at $P$ and through $P$ two straight lines $A B, C D$ are drawn meeting one circle at ${A}, {C}$ and the other at ${B}, {D}$ respectively. Prove that ${AC}$ and ${DB}$ are parallel. (5 marks)
(b) In $\triangle {ABC}, {D}$ is the midpoint of ${AC}$. ${E}$ is on ${BC}$ such that ${DE} \| {AB}$. Compare the areas of $\triangle {ABC}$ and $\triangle {CDE}$. If $\alpha(\triangle {ABC})=120$, what is $\alpha({ABED})$ ? (5 marks)
12.(a) In the figure, ${PBX}$ and ${QBY}$ are segments and $\angle {PAB}=\angle {QAB}$. Prove that ${PB} \cdot {BX}={QB} \cdot {BY}$. (5 marks)
(b) Given that $2 \sin (\alpha+\beta)=5 \sin (\alpha-\beta)$, show that $3 \tan \alpha=7 \tan \beta$ and hence show also that $49 \cos ^{2} \alpha-9 \cos ^{2} \beta=40 \cos ^{2} \alpha \cos ^{2} \beta$. (5 marks)
13.(a) Solve $\triangle {ABC}$ with $\angle {A}=25^{\circ}, \angle {C}=55^{\circ}, {AC}=12$. (5 marks)
(b) If $y \sin x=e^{x}$, show that $\frac{d^{2} y}{d x^{2}}+2 \cot x \frac{d y}{d x}-2 y=0$. (5 marks)
14.(a) Position vectors of points $P, Q$ and $R$ relative to an origin $O$ are $2 \hat{i}+7 \hat{j}, 6 \hat{i}+\hat{j}$ and $2 t \hat{i}+t \hat{j}$ respectively. If $P, Q$ and $R$ are collinear, find the value of $t$ and the value of $|\overrightarrow{{PQ}}|$. (5 marks) .
(b) Show that the equation of the normal to the curve $y=(2 x+a)^{3}, a \neq 0$, at the point where $y=a^{3}$ is $x+6 a^{2} y=6 a^{5}$. (5 marks)
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