Myanmar Matriculation 2015 Math (Foreign)

2015(FC)
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)

2. A function $f$ is defined by $f(2 x+1)=x^{2}-3$. Find $a \in R$ such that $f(5)=a^{2}-8 . \quad(3$ 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 $\sin ^{2} x . \quad(3$ 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. $\quad$ (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 $(g \circ f)^{-1} . \quad(5$ 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. . $\quad$ (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 $p, r, s . \quad$ (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. $\quad(5$ 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 $c . \quad(5$ 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. $\quad(5$ 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.

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 $\angle A B C+\angle C D E=270^{\circ}$.

(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:

$\begin{array}{lll}\text { (i) } \cos (A+B) & \text { (ii) } \cos (2 A-B) & (5 \text { marks })\end{array}$

15.(a) If $y=A \cos (\ln x)+B \sin (\ln x)$, where $A$ and $B$ are constants, show that $x^{2} y^{\prime \prime}+x y^{\prime} =0 . \quad(5, m a r k s)$

(b) Given that $x+y=5$; calculate the minimum value of $x^{2}+x y+y^{2}$. (5 marks)

(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) $\frac{75}{4}$