# Myanmar Matriculation (2018 FC)

2018 (FC)

MATRICULATION EXAMINATION

DEPARTMENT OF MYANMAR EXAMINATION

MATHEMATICS

Time Allowed: (3) Hours

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

(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) If $f(x)=3 x+1$ and $g(x)=2 x^{2}-3$, then $(g \circ f)(2)=$

A. 91

B. 93

C. 95

D. 97

E. 99

(2) An operation $\odot$ is defined by $a \odot b=a^{2}+b^{2} .$ Then $(2 \odot 3)(1 \odot 2)=$

A.8

B. $-8$

C. 18

D. $-18$

E. 65

(3) The remainder when $2 x^{3}+3 x^{2}-6 x-4$ is divided by $2 x+3$ is

A. 2

B. $-2$

C. 4

D. $-5$

E. 5

(4) $x-2$ is a factor of $x^{n+2}+5 x^{n}-10 x-52$, then $n=$

A. 2

B. 3

C. 4

D. 5

E. 6

(5) If $(2+3 x)^{2018}=c_{0}+c_{1} x+c_{2} x^{2}+c_{3} x^{3}+\ldots+c_{2018} x^{2018}$

then $c_{0}-c_{1}+c_{2}-c_{3}+\ldots+c_{2018}=$

A. 2018

B. 2019

C. 1

D. $-1$

E. 0

(6) In the expansion of $(1+a x)^{24}$ where $a>0$, the coefficient of $x^{2}$ is 1104, then $a=$

A. 2

B. $-2$

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

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

E. 4

(7) The solution set of the inequetion $x^{2}-2>7$ is

A. $\{x \mid x>3\}$

B. $\varnothing$

C. R

D. $\{x \mid x<-3$ or $x>3\}$

E. $\{x|-3<x<3\}$

(8) In an A.P., $u_{1}=2, u_{n+1}=u_{n}+\frac{1}{2}$, then $S_{n}=$

A. $\frac{n+1}{2}$

B. $\frac{n+3}{2}$

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

D. $\frac{n^{2}+3 n}{4}$

E. $\frac{n^{2}+7 n}{4}$

(9) The fourth, seventh and tenth terms of a G.P. are $a, b, c$ respectively, then

A. $a^{2}=b^{2}+c^{2}$

B. $b^{2}=a c$

C. $a^{2}=b c$

D. $a=b c$

E. none of these

(10) $\mathrm{G}$ iven that three consecutive term sofa $\mathrm{G} . \mathrm{P}$ are $3^{x+3}, 9^{x}$ and 729 . Then $x=$

A. 4

B. 3

C. 2

D. 1

E. 5

(11) If $A=\left(\begin{array}{ll}1 & a^{2} \\ 2 a+3 & 1\end{array}\right), A=A^{\prime}$ and $A$ is non-singular, then $a=$

A. 3 or $-1$

B. $-3$ or 1

C. 3 only

D. $-1$ only

E. l only

(12) If $\mathrm{P}=\left(\begin{array}{ll}2 & 0 \\ 4 & 2\end{array}\right)$ and $\operatorname{det}\left(x P^{\prime}\right)=16$, then $x=$

A. 2

B. $\pm 2$

C. 4

D. $\pm 4$

E. 16

(13) If any number is chosen at random from the whole numbers 1 to 60 inclusive, the probability of getting a prime number is

A. $\frac{7}{30}$

B. $\frac{1}{4}$

C. $\frac{4}{15}$

D. $\frac{17}{60}$

E. $\frac{3}{10}$

(14) A die is rolled $x$ times. If the expected frequency of a number which is divisible by 3 is 60 , then the éxpected frequency of a number not less than 3 is $\begin{array}{lllll}\text { A. } 120 & \text { B. } 180 & \text { C. } 90 & \text { D. } 60 & \text { E. } 30\end{array}$

(15) Chords $A B$ and $C D$ of a circle intersect et $P$ within the circle. If $A P=x$, $P B=x-2, C P=8$ and $P D=3$, then $x=$

A. 2

B. 3

C. 4

D. 5

E.6

(16) $A B C D$ is a cyclic quadrilateral. If $\angle A=125^{\circ}$ and $\angle B=45^{\circ}$, then $\angle D-2 \angle C=$

A. $25^{\circ}$

B. $30^{\circ}$

C. $45^{\circ}$

D. $50^{\circ}$

E. $55^{\circ}$

(17) In the figure, if $\alpha(\Delta P A B): \alpha(\Delta P C D)=1: 4$ $A C=6$ and $B D=3$, then $A P=$

A. $2.5$

B. $3.5$

C. 2

D. 3

E. 4

(18) The position vectors, realtive to an origin $O$, of the points $A$ and $B$ are $\left(\begin{array}{l}2 \\ 7\end{array}\right)$ and $\left(\begin{array}{l}5 \\ 3\end{array}\right)$ respectively. Then the unit vector parallel to $\overrightarrow{A B}$ is

A. $\left(\begin{array}{r}3 \\ -4\end{array}\right)$

B. $\left(\begin{array}{r}-3 \\ 4\end{array}\right)$

C. $\frac{1}{5}\left(\begin{array}{r}3 \\ -4\end{array}\right) \quad$

D . $\pm \frac{1}{5}\left(\begin{array}{r}3 \\ -4\end{array}\right)$

E. $-\frac{1}{5}(-4$

(19) The map of the point $(3,4)$ by reflection matrix about the $Y$-axis is

A. $(3,4)$

B. $(4,3)$

C. $(-3,4)$

D. $(3,-4)$

E. none of these

(20) $\cos ^{2} \frac{\pi}{3}+\sin ^{2} \frac{2 \pi}{3}+2 \cot ^{2} \frac{3 \pi}{4}=$

A. 2

B. $-2$

C. 3

D. $-3$

E. 4

(21) $\sin 180^{\circ}+\sin 30^{\circ}=$

A. $0.5$

B. $-0.5$

C. 1

D. $1.5$

E. $-1.5$

(22) $\sin (\alpha+\beta)-\sin (\alpha-\beta)=$

A. $2 \sin \alpha \cos \beta$

B. $2 \cos \alpha \sin \beta$

C. $2 \cos \alpha \cos \beta$

D. $2 \sin \alpha \sin \beta$

E. $-2 \sin \alpha \sin \beta$

(23) The gradient of the tangent to the curve $3 x y-y^{2}=x-1$ at the point $(0,-1)$

A. 2

B. $-2$

C. 1

D. $-1$

E. 3

(24) The stationary point of the curve $y=2 x+\frac{1}{x^{2}}$ is at $x=$

A. 1

B. $-1$

C. $2$

D. $-2$

E. $\pm 1$

(25) $\frac{d}{d x}(\sin x-\cos x)^{2}=$

A. $2 \sin 2 x$

B. $-2 \sin 2 x$

C. $2 \cos 2 x$

D. $-2 \cos 2 x$

E. $2 \sin x-2 \cos x$

SECTION (B)

2. If $f(x)=p x^{2}+1$ where $p$ is a constant and $f(3)=28$, find the value of $p$. Find also the formula of $f \circ f$ in simplified form. (3 marks)

(OR) The expression $x^{3}-2 x^{2}-k x+6$ and $x^{3}+x^{2}+(8-k) x+10$ have the same remainder when divided by $x+a$. Show that $3 a^{2}-8 a+4=0$(3 marks)

3. If $x, y, z$ is a G.P., show that $\log x, \log y, \log z$ is an A.P.(3 marks)

(OR) If $\log x, \log y, \log z$ is an A.P., show that $x, y, z$ is a G.P.(3 marks)

4. Given : $A B C D E F$ is an inscribed regular hexagon. $P F$ is a tangent to the circle $O$ at $F$. Prove : $P F$ and $E A$ are parallel. (3 marks)

5. Prove that $\frac{1-\cos 2 x+\sin 2 x}{1+\cos 2 x+\sin 2 x}=\tan x$.(3 marks)

6. Evaluate $\displaystyle\lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{3}-1}$ and $\displaystyle\lim _{x \rightarrow 0} \frac{\frac{1}{x-1}+\frac{1}{x+1}}{x}$(3 marks)

SECTION (C)

7.(a) Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by $f(x)=x+7$ and $g(x)=3 x-1$.  Find $\left(f^{-1}\circ g\right)(x)$ and $\left(g^{-1}\circ f\right)(x).$ What are the values of  $\left(f^{-1}\circ g\right)(3)$ and $\left(g^{-1}\circ f\right)(2).$(5 marks)

(b) A binary operation $\odot$ on $R$ is defined by $x \odot y=(x+2 y)^{2}-3 y^{2}$. Show that the binary operation is commutative. Find the possible values of $k$ such that $(k-3) \odot(k+2)=25$(5 marks)

8.(a) Given $f(x)=x^{3}+p x^{2}-2 x+4 \sqrt{3}$ has a factor $x-2 \sqrt{3}$, find the value of $p$. Show that $x+\sqrt{2}$ is also a factor and solve the equation $f(x)=0$.(5 marks)

(b) Use the binomial theorem to find the value of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$.(5 marks)

9.(a) Find the solution set of the inequation $5 x^{2}<18 x+8$, by graphical method and illustrate it on the number line.(5 marks)

(b) An A.P. contains 25 terms. The last three terms are $\frac{1}{x-4}, \frac{1}{x-1}$ and $\frac{1}{x}$. Calculate the value of $x$ and the sum of all the terms of the progression.(5 marks)

10.(a) If $a, b, c$ be in A.P., and if $u, v$ be the A.M. and G.M. between $a$ and $b, x, y$ the A.M. and G.M. between $b$ and $c$, then prove that $u^{2}-v^{2}=x^{2}-y^{2}$.(5 marks)

(b) Solve the matrix equation $\left(\begin{array}{lr}-2 & 3 \\ 1 & -4\end{array}\right) X=\left(\begin{array}{cc}-3 & -5 \\ -16 & -20\end{array}\right)$, Hence find $x$ and $y$, if $X=\left(\begin{array}{lr}x+2 y & 16 \\ 7 & 2 x-y\end{array}\right)$(5 marks)

11.(a) Find the inverse of the matrix $\left(\begin{array}{ll}4 & 3 \\ 7 & 6\end{array}\right)$, and use it to solve the system of equations \begin{aligned}&7 x+6 y+16=0 \\&3 y+4 x+7=0\end{aligned}(5 marks)

(b) How many 3 -digit numerals can you form from $3,0,1$ and 6 without repeating any digit? Find the probability of an even number and find the probability that a numeral which is divisible $3 .$(5 marks)

12.(a) If $A, B, C$ are three points on the circumference of a circle such that the chord $A B$ is equal to the chord $A C$, prove that the tangent at $A$ bisects the exterior angle between $A B$ and $A C$.(5 marks)

(b) Two circles intersect at $A$ and $B$. $A$ point $P$ is taken on one so that $P A$ and $P B$ cut the other at $Q$ and $P$ respectively. The tangents at $Q$ and $R$ meet the tangent at $P$ in $S$ and $T$ respectively. Prove that $\angle T P R=\angle B R Q$ and $P B Q$. is cyclic.(5 marks)

13.(a) In the figure $\angle A P Q=\angle C, A P: P B=3: 1$ and $A Q: Q C=1: 2$. If $A Q=2$, find the length of $\mathrm{AP}$ and the ratios of $\alpha(\triangle A P Q): \alpha(\triangle A B C)$ and $\alpha(\triangle A P Q): \alpha(B C Q P)$.(5 marks)

(b) Points $A$ and $B$ have position vectors $\left(\begin{array}{l}5 \\ 1\end{array}\right)$ and $\left(\begin{array}{l}3 \\ 4\end{array}\right)$ respectively, relative to an origin $O$. Given that point $C$ with position vector $\left(\begin{array}{l}0 \\ k\end{array}\right)$ lies on $A B$ produced, calculate the vlaue of $k$ and the value of $|2 \overrightarrow{A B}+\overrightarrow{O C}|$(5 marks)

14.(a) If $\alpha+\beta+\gamma=180^{\circ}$, show that $$\cos \frac{\alpha}{2}+\cos \frac{\beta}{2}+\cos \frac{\gamma}{2}=4 \cos \frac{\beta+\gamma}{4} \cos \frac{\gamma+\alpha}{4} \cos \frac{\alpha+\beta}{4}$$(5 marks)

(b) In $\triangle A B C, c=10, b=6$ and $a=5$. Check whether $\angle A C B$ is acute or obtuse and find its magnitude.(5 marks)

15.(a) Given that $y=\sin (\sin x)$, prove that $\frac{d^{2} y}{d x^{2}}+\tan x \frac{d y}{d x}+y \cos ^{2} x=0$.(5 marks)

(b) Show that the point $(0, \pi)$ lies on the cirve $x^{2} \cos ^{2} y=\sin y$. Then find the equations of tangent and normal at $(0,\pi)$(5 marks)

1 (1)C        (2)E        (3)E     (4)B      (5)C

(6)A        (7)D      (8)E      (9)B    (10)B

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

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

(21)A    (22)B    (23)A    (24)A    (25)D

2) $p=3,27 x^{4}+18 x^{2}+4$

(O R) show

3) Show

(OR) Show

4) Prove

5) Prove

6) $\frac{4}{3},-2$

7) (a) $\left(f^{-1} \circ g\right)(x)=3 x-8,1,\left(g^{-1} \circ f\right)(x)=\frac{x+8}{3}, \frac{10}{3}$

(b) $k=3$ or $-2$

8) (a) $x=2 \sqrt{3}, \sqrt{2},-\sqrt{2}$

(b) $64 x^{6}-96 x^{4}+36 x^{2}-2$

9)(a) $\left\{x \mid-\frac{2}{5}<x<4\right\}$

(b) $x=-2,37.5$

10)(a) Prove

(b) $x=\left(\begin{array}{cc}12 & 16 \\ 7 & 9\end{array}\right), x=6, y=3$

11)(a) $\left(\begin{array}{cc}2 & -1 \\ -\frac{7}{3} & \frac{4}{3}\end{array}\right), x=2, y=-5$

(b) $\frac{5}{9}, \frac{2}{9}$

12) (a) Prove

(b) Prove

13) (a) $A P=3, \frac{1}{4}, \frac{1}{3}$

(b) $k=\frac{17}{2}, \frac{\sqrt{905}}{2}$

14) (a) Prove

\begin{aligned}&\text { If } \alpha+\beta+\gamma=180^{\circ} \text {, show that }\\&\cos \frac{\alpha}{2}+\cos \frac{\beta}{2}+\cos \frac{\gamma}{2}=4 \cos \frac{\beta+\gamma}{4} \cos \frac{\gamma+\alpha}{4} \cos \frac{\alpha+\beta}{4}\\&\begin{aligned}&\text { Proof: } \\&R H S=\left(2 \cos \frac{\beta+\gamma}{4} \cos \frac{\gamma+\alpha}{4}\right)\left(2 \cos \frac{\alpha+\beta}{4}\right)\end{aligned}\\&=\left(\cos \left(\frac{\beta+\gamma}{4}+\frac{\gamma+\alpha}{4}\right)+\cos \left(\frac{\beta+\gamma}{4}-\frac{\gamma+\alpha}{4}\right)\right)\left(2 \cos \frac{\alpha+\beta}{4}\right)\\&=2 \cos \left(\frac{\beta+\gamma+\gamma+\alpha}{4}\right) \cos \left(\frac{\alpha+\beta}{4}\right)+2 \cos \left(\frac{\beta+\gamma-\gamma-\alpha}{4}\right) \cos \frac{\alpha+\beta}{4}\\&=\left[\cos \left(\frac{\beta+\gamma+\gamma+\alpha+\alpha+\beta}{4}\right)+\cos \left(\frac{\beta+\gamma+\gamma+\alpha-(\alpha+\beta)}{4}\right)\right]\\&+\left[\cos \frac{\beta+\gamma-\gamma-\alpha+\alpha+\beta}{4}+\cos \left(\frac{\beta+\gamma-\gamma-\alpha-\alpha-\beta}{4}\right)\right]\\&=\cos \left(\frac{360}{4}\right)+\cos \left(\frac{2 \gamma}{4}\right)+\cos \left(\frac{2 \beta}{4}\right)+\cos \left(-\frac{2 \alpha}{4}\right)\\&=\cos \left(90^{\circ}\right)+\cos \left(\frac{\gamma}{2}\right)+\cos \left(\frac{\beta}{2}\right)+\cos \left(-\frac{\alpha}{2}\right)\\&=0+\cos \left(\frac{\gamma}{2}\right)+\cos \left(\frac{\beta}{2}\right)+\cos \left(\frac{\alpha}{2}\right)\\&=\cos \left(\frac{\alpha}{2}\right)+\cos \left(\frac{\beta}{2}\right)+\cos \left(\frac{\gamma}{2}\right)\\&=\text { LHS }\end{aligned}

(b) Obtuse, $130^{\circ} 33^{\prime}$

15) (a) Prove

(b) $y=\pi, x=0$