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2019
2019
MATRICULATION EXAMINATION
DEPARTMENT OF MYANMAR EXAMINATION
| MATHEMATICS | Time Allowed : (3) Hours |
WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.
SECTION A
(Answer ALL questions)
| 1 (a) |
2 (a) Find and simplify the coefficient of $\mathrm{x}^{7}$ in the expansion of $\left(\mathrm{x}^{2}+\frac{2}{\mathrm{x}}\right)^{8}, \mathrm{x} \neq 0 .$ (3 marks) (b) Find the sum of all even numbers between 69 and 149 . (3 marks) Click for Solution
3 (a) The matrices $\mathrm{A}=\left(\begin{array}{ll}2 & 0 \\ 0 & 5\end{array}\right)$ and $\left(\begin{array}{ll}\mathrm{x} & \mathrm{y} \\ 0 & \mathrm{z}\end{array}\right)$ are such that $\mathrm{AB}=\mathrm{A}+\mathrm{B}$. Find the values of $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ (b) A die is thrown. If the probability of getting a number not less than $\mathrm{x}$ is $\frac{2}{3}$, find $\mathrm{x}$. (3 marks) Click for Solution
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4 (a) AT and BT are tangents to the circle $A B C$ at $\mathrm{A}$ and $\mathrm{B}$. Prove that $\angle \mathrm{BTX}=2 \angle \mathrm{ACB}$. (3 marks)
(b) The coordinates of $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are $(1,0),(4,2)$ and $(5,4)$ respectively. Use vector method to determine the coordinates of $\mathrm{D}$ if $\mathrm{ABCD}$ is a parallelogram. (3 marks) Click for Solution5 (a) Solve the equation $2 \cos x \sin x=\sin x$ for $0^{\circ} \leq x \leq 360^{\circ}$. (3. marks) (b) Differentiate $\mathrm{x}^{3}+2 \mathrm{x}$ with respect to $\mathrm{x}$ from the first principle. (3 marks) Click for Solution
SECTION B
(Answer any FOUR questions)
| 6 (a) |
7 (a) A binary operation $\odot$ on R is defined by $x\odot y = (3y−x)^2 −8y^2.$ Show that the binary operation is commutative. Find the possible values of $k$ such that $2 \odot \mathrm{k}=-31.(5 \mathrm{marks}$ (b) If the coefficients of $\mathrm{x}^{\mathrm{r}}$ and $\mathrm{x}^{\mathrm{r}+2}$ in the expansion of $(1+\mathrm{x})^{2 \mathrm{~m}}$ are equal, show that $r=n-1 .$ $(5$ marks) Click for Solution
8 (a) Find the solution set in $\mathrm{R}$ of the inequation $\mathrm{x}^{2}-3 \mathrm{x}+2 \leq 0$ by algebraic method and illastrate it on the number line. $\quad$ (5 marks) (b) Find the sum of the first 12 terms of the A.P. 44,$40 ; 36, \ldots \ldots .$ Find also the sum of the terms between the $12^{\text {th }}$ term and the $26^{\text {th }}$ term of that A.P. (5 marks) Click for Solution
9 (a) The sum of the first n terms of a certain sequence is given by $\mathrm{S}_{\mathrm{n}}=2^{\mathrm{n}}-1$. Find the first: 3 terms of the sequence and express the $\mathrm{n}^{\text {th }}$ term in terms of n. $\quad$ (5 marks) (b) Using the definition of inverse matrix, find the inverse of the matrix $\left(\begin{array}{ll}3 & 1 \\ 2 & 1\end{array}\right) .(5$ marks $)$ Click for Solution
10 (a) Find the inverse of matrix $\left(\begin{array}{ll}5 & 6 \\ 7 & 8\end{array}\right)$. Use it to determine the coordinates of the point of intersection of the lines $5 x+6 y=7$ and $8 y+7 x=10 . \quad$ (5 marks) (b) Construct the table of outcomes for rolling two die. Find the probability of an outcome in which the score on the first die is less than that on the second die. Find also the probability that the score on first die is prime and the score on the second is even. ( 5 marks) Click for Solution
SECTION (C)
(Answer any THREE questions)
Click Here for Solution of Section C
| 11 (a) |
12 (a) Prove that the quadrilateral formed by producing the bisectors of the interior angles of any quadrilateral is cyclic. ( 5 marks)
(b) If $\alpha+\beta+\gamma=180^{\circ}$, prove that $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}+\tan \frac{\beta}{2} \tan \frac{\gamma}{2}+\tan \frac{\alpha}{2} \tan \frac{\gamma}{2}=1$. (5marks) Click for Solution13 (a) In $\triangle \mathrm{ABC}$, if $\angle \mathrm{B}=\angle \mathrm{A}+15^{\circ}, \angle \mathrm{C}=\angle \mathrm{B}+15^{\circ}$ and $\mathrm{BC}=6$, find $\mathrm{AC}$. $\quad$ (5 marks) Click for Solution(b) If $y=\operatorname{In}\left(\sin ^{3} 2 x\right)$, then prove that $3 \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+36=0 . \quad$ (5 marks)
14 (a) In the quadrilateral $\mathrm{ABCD}, \mathrm{M}$ and $\mathrm{N}$ are the midpoints of $\mathrm{AC}$ and $\mathrm{BD}$ respectively. Prove that $\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{CB}}+\overrightarrow{\mathrm{AD}}+\overrightarrow{\mathrm{CD}}=\overrightarrow{\mathrm{MN}} \quad$ (5 marks) Click for Solution(b) Find the normals to the curve $x y+2 x-y=0$ that are parallel to the line $2 x+y=0$. (5 marks)
Answers
1. (a) $x=\frac 32$ or $x=1$
(b) $p=\frac 23$ or $p=3$
2(a) 448 (b) 4360
3(a) $x=2,y=0,z=\frac 54$
(b) $2<x\le 3$
4(a) proof (b) $D=(2,2)$
5(a) $x=0^{\circ},60^{\circ}, 180^{\circ}, 300^{\circ}, 360^{\circ}$
(b) $3x^2+2$
6(a) $(g\circ f)(x)=8x-1,g^{-1}(x)=\frac{x-3}{4}$
(b) $a=3,b=2,23$
7(a) $k=3$ or $k=7$
(b) Proof
8(a) $\{x|1\le x\le 2\}$
(b) $S_{12}=264,-364$.
9(a) 1,2,4, $u_n=2^{n-1}$
(b) $A^{-1}=\left(\begin{array}{cc}1&-1\\-2&3\end{array}\right)$
10 (a) $A^{-1}=\left(\begin{array}{cc}-4&3\\3.5&-2.5\end{array}\right), (2,-0.5)$
(b) $\dfrac{5}{12},\dfrac 14$
11(a) Proof (b) 12, 16
12 (a) Proof (b) Proof
13(a) $3\sqrt 6$ (b) Proof
14(a) Proof (b) $2x+y=\pm 3$
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