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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) Functions $f$ and $g$ are given by $f(3)=-1$ and $g(-1)=5$. Then $(g \circ f)^{-1}(5)=$
A. $-1$
B. 5
C. 3
D. 4
E. 0
(2) Given that $a \odot b$ means "add 4 to $a$ and multiply the result by $b$ ", then the value of, $(2 \odot 1) \odot 3$ is
A. 55
B. 50
C. 45
D. 40
E. 30
(3) If $x$ is a factor of $x^{3}-4 x^{2}+15 x+a^{2}-2 a$, where $a$ is a constant, then $a=$
A. 0 only
B. $-2$ only
C. 2 only
D. 0 or 2
E. 0 or $-2$
(4) If $a x^{2}+3 x-1$ has remainder $3 a+14$ when divided by $x-3$, then $a=$
A. 1
B. 2
C. 3
D. $-1$
E. $-2$
(5) The coefficient of $x y^{4}$ in the expansion of $(x-2 y)^{5}$ is
A. 80
B. 40
C. $-10$
D. $-32$
E. $-80$
(6) The middle term in the expansion of $(x+y)^{2 k}$ is
A. $k^{\text {th }}$ term
B. $(k+1)^{\text {th }}$ term
C. $(k+2)^{\text {th }}$ term
D. $(k+3)^{\text {th }}$ term
E. $(2 k+1)^{\text {th }}$ term
(7) The solution set in $R$ for the inequation $-4-3 x^{2} \geq 0$ is
A. $\{x \mid x \geq 0\}$
B. $\left\{x \mid-2 \leq x \leq \frac{2}{3}\right\}$
C. $\left\{x \mid-\frac{2}{3} \leq x \leq 2\right\}$
D. $R \quad$
E. $\varnothing$
(8) The three angles of a triangle form an A.P.. If the largest angle is twice the smallest angle, then the smallest angle is
A. $20^{\circ}$
B. $40^{\circ}$
C. $60^{\circ}$
D. $80^{\circ}$
E. $90^{\circ}$
(9) If $1+2+2^{2}+2^{3}+\ldots+2^{n}=1023$, then $n$ is
A. 9
B. 10
C. 11
D. 12
E. 18
(10) For two numbers $a$ and $b$, the A.M. between $a$ and $b$ is 5 and the G.M. between $a$ and $b$ is 4 . If $a>b$, then $a^{b}=$
A. 256
B. 64
C. 16
D. 10
E. None of these
(11) Given that $A=\left(\begin{array}{cc}k & 3 \\ -3 & 2\end{array}\right)$ and $B=\left(\begin{array}{rr}2 & -3 \\ 3 & -4\end{array}\right)$ and $A B=B A$, then $k=$
A. $-4$
B. 2
C. 3
D. 4
E. None of these
(12) If the determinant of the matrix $\left(\begin{array}{cc}2 x+1 & 2 x \\ 3 & 2\end{array}\right)$ is $-2$, then $x=$
A. 2
B. 3
C. 4
D. 5
E. 6
(13) The number of possible outcomes for tossing five fair coins is
A. 64
B. 32
C. 18
D. 16
E. 8
(14) Two dice are thrown 180 times. The expected frequency of obtaining total score 6 is
A. 60
B. 50
C. 40
D. 30
E. 25
(15) In a cyclic quadrilateral $A B C D, \angle A=25^{\circ}, \angle B=60^{\circ}$. Then $\angle C-\angle D=$
A. $35^{\circ}$
B. $40^{\circ}$
C. $45^{\circ}$
D. $50^{\circ}$
E. $55^{\circ}$
(16) In circle $O, P Q$ is tangent at $Q$. If $P Q=4 \mathrm{~cm}$, . he length of the diameter is $6 \mathrm{~cm}$, the length of $P R$ is
A. $6 \mathrm{~cm}$
B. $5 \mathrm{~cm}$
C. $4 \mathrm{~cm}$
D. $3 \mathrm{~cm}$
E. $2 \mathrm{~cm}$
(17) If the ratio of areas of two similar triangles is $16: 225$, then the ratio of the lengths of corresponding angle bisectors is
A. $16: 25$
B. $4: 15$
C. $25: 4$
D. $6: 25$
E. $13: 25$
(18) The position vectors of $A$ and $B$, relative to an origin $O$ are $3 \vec{p}+2 \vec{q},-5 \vec{p}-3 \vec{q}$ respectively. If $M$ is the mid-point of $A B$, then the position vector of $M$ is
A. $\vec{p}+\frac{1}{2} \vec{q}$
B. $-\vec{p}+\frac{1}{2} \vec{q}$
C. $-\vec{p}-\frac{1}{2} \vec{q}$
D. $\vec{p}-\frac{1}{2} \vec{q}$
E. $\frac{1}{2} \vec{p}+\vec{q}$
(19) Given that $\vec{a}=3 \hat{i}+4 \hat{j}$. Then the vector with magnitude 20 units and in the direction of $\vec{a}$ is
A. $9 \hat{i}+12 \hat{j}$
B. $60 \hat{i}+120 \hat{j}$
C. $-12 \hat{i}-16 \hat{j}$
D. $12 \hat{i}+16 \hat{j}$
E. $21 \hat{i}+28 \hat{j}$
(20) If $\tan \theta=2, \tan \phi=1$, then $\cot (\theta-\phi)=$
A. $-3$
B. 3
C. $-\frac{1}{3}$
D. $\frac{1}{3}$
E. 1
(21) $\cos \left(-45^{\circ}\right)=$
A. $\frac{2}{\sqrt{2}}$
B. $\frac{-2}{\sqrt{2}}$
C. $\frac{\sqrt{2}}{2}$
D. $\frac{-\sqrt{2}}{2}$
E. 1
(22) In $\triangle A B C, B C: C A: A B=3: 4: \sqrt{37} .$ Then $\angle C=$
A. $60^{\circ}$
B. $75^{\circ}$
C. $105^{\circ}$
D. $120^{\circ}$
E. $150^{\circ}$
(23) $\lim _{t \rightarrow \infty}\left(\sqrt{t^{2}+2 t+1}-t\right)=$
A. 0
B. 1
C. 2
D. $\infty$
E.none of these
(24) The second derivative of $y=\cos x+x^{2}$ is
A. $-\cos x-2$
B. $2-\cos x$
C. $\sin x+2 x$
D. $\cos x-2 \quad$
E. $\sin x-2 x$
(25) The coordinates of the point on the curve $y=x^{3}-6 x$ at which the gradient of tangent $-3$ is
A. $(1,-5)$
B. $(-1,5)$
C. $(1,-5)$ and $(-1,5)$
D. $(2,-4)$
E. $(2,4)$
SECTION B
(Answer ALL questions)
2. A function $f: x \mapsto \frac{b}{x-a}, x \neq a$ and $a>0$ is such that $(f \circ f)(x)=x$. Show that $x^{2}-a x-b=0$
(OR)
Given that $x^{3}-2 x^{2}-3 x-11$ and $x^{3}-x^{2}-9$ have the same remainder when divided by $x+a$, determine the values of $a$.
3. If $29, a-b, a+b, 95$ is an A.P., find the values of $a$ and $b$.
(OR)
In an G.P, the ratio of the sum of the first three terms to the sum to infinity of the G.P. is $19: 27$. Find the common ratio.
4. $A$ and $B$ are two points on a circle $3 \mathrm{~cm}$ apart. The chord $A B$ is produced to $C$ making $B C=1 \mathrm{~cm}$. Find the length of the tangent from $C$ to the circle.
5. If $\cos \theta-\sin \theta=\sqrt{2} \sin \theta$, prove that $\cos \theta+\sin \theta=\sqrt{2} \cos \theta$.
6. Find the value of $a$ and $b$ for which $\frac{d}{d x}\left[\frac{\sin x}{2+\cos x}\right]=\frac{3 a+b \cos x}{(2+\cos x)^{2}}$.
SECTION C
(Answer any SIX questions)
7. (a) Functions $f$ and $g$ are defined by $f: \mathrm{x} \mapsto 2 x+1$ and $g: x \mapsto \frac{2 x+5}{3-x}, x \neq 3$.
Find the values of $x$ for which $\left(f \circ g^{-1}\right)(x)=x-4$
(b) Let $R$ be the set of real numbers and a binary operation $\odot$ on $R$ is defined by $x \odot y=x y-x-y$ for all $x, y$ in $R$. Show that the operation $\odot$ :s commutative. Solve the equation $(2 \odot 3) \odot x=(x \odot x) \odot 5$.
8. (a) The expression $a x^{3}-(a+3 b) x^{2}+2 b x+c$ is exactly divisible by $x^{2}-2 x$. When the expression is divided by $x-1$, the remainder is 8 more than when it is divided by $x+1$. Find the values of $a, b$ and $c$, hence factorize the expression completely.
(b) Write down and simplify the first four terms in the binomial expansion of $(1-2 x)^{7}$. Use it to find the value of $(0.98)^{7}$, correct to four decimal places.
9. (a) Use a sketch graph to obtain the solution set of $\frac{15-4 x}{4} \leq x^{2}$ and illustrate it on the number line.
(b) Find the sum of all two-digit natural numbers which are not divisible by $3 .$
10. (a) Find the sum of $(b+2)+\left(b^{2}+5\right)+\left(b^{3}+8\right)+\ldots$ to 18 terms in terms of $b$ where $b \neq 1$
(b) Find the inverse of the matrix $M=\left(\begin{array}{ll}3 & 5 \\ 1 & 2\end{array}\right)$ and investigate whether or not the squares of $M$ and $M^{1}$ are inverses of each other.
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 $3 y+4 x+7=0$ and $14 x+12 y+32=0$
(b) A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the scores obtained in these 360 trials are added together, what is the expected total score?
12. (a) In the figure, $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C, \angle C P B$ and $\angle Q P C$ in terms of $x$.
(b) Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic.
13. (a) $A, B, C$ and $D$ are four points in order on a circle $O$, so that $A B$ is a diameter and $\angle C O D=90^{\circ}.$ If $A D$ produced and $B C$ produced meet at $E$, prove that $\alpha(\Delta E C D)=\alpha(A B C D)$
(b) The coordinates of points $P, Q$ and $R$ are $(1,2),(7,3)$ and $(4,7)$ respectively. If $P Q S R$ is a parallelogram, find the coordinates of $S$ by vector method. If $P S$ and $Q R$ meet at $T$, find the coordinate of $T$ by using vectors.
14. (a) Given that $\frac{\cos (\alpha-\beta)}{\cos (\alpha+\beta)}=\frac{7}{5}$, prove that $\cos \alpha \cos \beta=6 \sin \alpha \sin \beta$ and deduce a relationship between $\tan \alpha$ and $\tan \beta .$ Given further that $\alpha+\beta=45^{\circ}$, calculate the value of $\tan \alpha+\tan \beta$.
(b) A town $P$ is 25 miles away from the town $Q$ in the direction $N 35^{\circ} \mathrm{E}$ and a town $R$ is 10 miles from $Q$ in the direction $42^{\circ}$ W. Calculate the distance and bearing of $P$ from $R$.
15.(a) Find the coordinates of the points on the curve $x^{2}-y^{2}=3 x y-39$ at which the tangents are (i) parallel (ii) perpendicular to the line $x+y=1$.
(b) Find the stationary points on the curve $y=27+12 x+3 x^{2}-2 x^{3}$ and determine the nature of these points.
Answer
$\def\myrow#1#2#3#4#5{\mbox{#1 }\quad&\mbox{#2 }\quad&\mbox{#3 }\quad&\mbox{#4 }\quad&\mbox{#5 }}\def\mytab#1{\begin{array}{rrrrr}#1\end{array}}$ $\mytab{\myrow{1 C}{2 E}{3 D}{4 A}{5 A}\\ \myrow{6 B}{7 E}{8 B}{9 A}{10 B}\\\myrow{11 A}{12 A}{13 B}{14 E}{15 A}\\\myrow{16 E}{17 B}{18 C}{19 D}{20 B}\\\myrow{21 C}{22 D}{23 B}{24 B}{25 C}}$
2) Show (OR) $a=1$ or 2
3) $a=62, b=11$
(OR) $r=\frac{2}{3}$
4) $2 \mathrm{~cm}$
5) Prove
6) $a=\frac{1}{3}, b=2$
7) (a) $x=0$ or 9 (b) $x=1 \pm \sqrt{2}$
8) (a) $a=2, b=1, c=0, f(x)=x(x-2)(2 x-1)$
(b) $1-14 x+84 x^{2}-280 x^{3}+\cdots, 0.8681$
9) (a) $\left\{x \mid x \leqslant-\frac{5}{2}\right.$ (or) $\left.x \geqslant \frac{3}{2}\right\}$ (b) 3240
10) (a) $495+\frac{b\left(1-b^{18}\right)}{1-b}$
(b) $M^{-1}=\left(\begin{array}{cc}2 & -5 \\ -1 & 3\end{array}\right)$ (Yes)
11) $(a)\left(\begin{array}{cc}2 & -1 \\ -\frac{7}{3} & \frac{4}{3}\end{array}\right), x=2, y=-5$
(b) $240,180,1260$
12) (a) $\angle D P C=\frac{90-x}{2}=\angle C P B, \angle Q P C=\frac{270-x}{2}(b)$ Prove
13)(a) Prove (b) $T=\left(\frac{11}{2}, 5\right)$
14) (a) $\frac{5}{6} \quad$ (b) $612,5, N 58^{\circ} 15^{\prime} $ E
15) (a) (i) $(1,5),(-1,-5)$ (ii) No tangent (b) $(2,47) \max ;(-1,20) \min$
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