# Sample Question for 2020 Matriculation Examination

2020
MATRICULATION EXAMINATION
Sample Question Set (3)
MATHEMATICS                        Time allowed: 3hours
SECTION (A)

1 (a).     Let $\displaystyle f:R\backslash \{\pm 2\}\to R$ be a function defined by $\displaystyle f(x)=\frac{{3x}}{{{{x}^{2}}-4}}$.Find the positive value of $z$ such that $f(z) = 1$.
(3 marks)

(b).     If the polynomial $x^3 - 3x^2 + ax - b$ is divided by $(x - 2 )$ and $(x + 2)$, the remainders are $21$ and $1$ respectively. Find the values of $a$ and $b$.

(3 marks)

2(a).     Find the middle term in the expansion of $(x^2 - 2y)^{10}$.
(3 marks)

(b).     In a sequence if $u_1=1$ and $u_{n+1}=u_n+3(n+1)$, find $u_5$ .
(3 marks)

3(a).     If $\displaystyle P=\left( {\begin{array}{*{20}{c}} x & {-4} \\ {8-y} & {-9} \end{array}} \right)$ and $\displaystyle {{P}^{{-1}}}=\left( {\begin{array}{*{20}{c}} {-3x} & 4 \\ {-7y} & 3 \end{array}} \right)$, find the values of $x$ and $y$.
(3 marks)

(b).     A bag contains tickets, numbered $11, 12, 13, ...., 30$. A ticket is taken out from the bag at random. Find the probability that the number on the drawn ticket is

(i) a multiple of $7$

(ii) greater than $15$ and a multiple of $5$.

(3 marks)

4(a).     Draw a circle and a tangent $TAS$ meeting it at $A$. Draw a chord $AB$ making $\displaystyle \angle TAB=\text{ }60{}^\circ$ and another chord $BC \parallel TS$. Prove that $\triangle ABC$ is equilateral.

(3 marks)

(b).     If $\displaystyle 3~\overrightarrow{{OA}}-2\overrightarrow{{OB}}-\overrightarrow{{OC}}~=\vec{0}$, show that the points $A, B$ and $C$ are collinear.
(3 marks)

5(a).     Solve the equation $2 \sin x \cos x -\cos x + 2\sin x - 1 = 0$ for $\displaystyle 0{}^\circ \le x\le \text{ }360{}^\circ$.
(3 marks)

(b).     Differentiate $\displaystyle y=\frac{1}{{\sqrt[3]{x}}}$ from the first principles.
(3 marks)

SECTION (B)

6 (a).    Given that Given $\displaystyle A=\{x\in R|\ x\ne -\frac{1}{2},x\ne \frac{3}{2}\}$. If $f:A\to A$ and $g:A\to A$ are defined by $f(x)=\displaystyle \frac{3x-5}{2x+1}$ and $g(x)=\displaystyle \frac{x+5}{3-2x}$, show that $f$ and $g$ are inverse of each other.
(5 marks)

(b).     Given that $x^5 + ax^3 + bx^2 - 3 = (x^2 - 1) Q(x) - x - 2$, where $Q(x)$ is a polynomial. State the degree of $Q (x)$ and find the values of $a$ and $b$. Find also the remainder when $Q (x)$ is divided by $x + 2$.

(5 marks)

7 (a).     The binary operation $\odot$ on $R$ be defined by $x\odot y=x+y+10xy$ Show that the binary operation is commutative. Find the values $b$ such that $(1\odot b)\odot b=485$.

(5 marks)

(b).     If, in the expansion of $(1 + x)^m (1 – x)^n$, the coefficient of $x$ and $x^2$ are $-5$ and $7$ respectively, then find the value of $m$ and $n$.

(5 marks)

8 (a).     Find the solution set in $R$ for the inequation $2x (x + 2)\ge (x + 1) (x + 3)$ and illustrate it on the number line.

(5 marks)

(b).     If the ${{m}^{{\text{th}}}}$ term of an A.P. is $\displaystyle \frac{1}{n}$ and ${{n}^{{\text{th}}}}$ term is $\displaystyle \frac{1}{m}$ where $m\ne n$, then show that $u_{mn} = 1$.

(5 marks)

9 (a).     The sum of the first two terms of a geometric progression is $12$ and the sum of the first four terms is $120$. Calculate the two possible values of the fourth term in the progression.

(5 marks)

(b).     Given that $A=\left( {\begin{array}{*{20}{c}} {\cos \theta } & {-\sin \theta } \\ {\sin \theta } & {\cos \theta } \end{array}} \right)$. If $A + A' = I$ where $I$ is a unit matrix of order $2$, find the value of $\theta$ for $\displaystyle 0{}^\circ <\theta <90{}^\circ$.

(5 marks)

10 (a).   The matrix $A$ is given by $\displaystyle A=\left( {\begin{array}{*{20}{c}} 2 & 3 \\ 4 & 5 \end{array}} \right)$.
(i) Prove that $A^2 = 7A + 2I$ where $I$ is the unit matrix of order $2$.
(ii) Hence, show that $\displaystyle {{A}^{{-1}}}=\frac{1}{2}~\left( {A-7I} \right)$.

(5 marks)

(b).   Draw a tree diagram to list all possible outcomes when four fair coins are tossed simultaneously. Hence determine the probability of getting:
(ii) two heads and two tails,
(iv) at least one tail,

(5 marks)

SECTION (C)

11 (a).   $PQR$ is a triangle inscribed in a circle. The tangent at $P$ meet $RQ$ produced at $T$,and $PC$ bisecting $\angle RPQ$ meets side $RQ$ at $C$. Prove $\triangle TPC$ is isosceles.

(5 marks)

(b).   In $\triangle ABC$, $D$ is a point of $AC$ such that $AD = 2CD$. $E$ is on $BC$ such that $DE \parallel AB$. Compare the areas of $\triangle CDE$ and $\triangle ABC$. If $\alpha (ABED) = 40$, what is $\alpha(ΔABC)$?

(5 marks)

12 (a).   If $L, M, N,$ are the middle points of the sides of the $\triangle ABC$, and $P$ is the foot of perpendicular from $A$ to $BC$. Prove that $L, N, P, M$ are concyclic.

(5 marks)

(b).   Solve the equation $\displaystyle \sqrt{3}\cos \theta +\sin \theta =\sqrt{2}$ for $\displaystyle 0{}^\circ \le \theta \le 360{}^\circ$.

(5 marks)

13 (a).   In $\triangle ABC, AB = x, BC = x + 2$, $AC = x – 2$ where $x > 4$, prove that $\displaystyle \cos A=\frac{{x-8}}{{2(x-2)}}$. Find the integral values of $x$ for which $A$ is obtuse.

(5 marks)

(b).   The sum of the perimeters of a circle and square is $k$, where $k$ is some constant. Using calculus, prove that the sum of their areas is least, when the side of the square is double the radius of the circle.

(5 marks)

14 (a).  The vector $\overrightarrow{{OA}}$ has magnitude $39$ units and has the same direction as $\displaystyle 5\hat{i}+12\hat{j}$. The vector $\overrightarrow{{OB}}$ has magnitude $25$ units and has the same direction as $\displaystyle -3\hat{i}+4\hat{j}$. Express $\overrightarrow{{OA}}$ and $\overrightarrow{{OB}}$ in terms of $\hat{i}$ and $\hat{j}$ and find the magnitude of $\overrightarrow{{AB}}.$

(5 marks)

(b).  Find the coordinates of the stationary points of the curve $y = x\ln x - 2x$. Determine whether it is a maximum or a minimum point.

(5 marks)

Target Mathematics ၏ အစဉ်အလာအတိုင်း တက္ကသိုလ်ဝင်တန်း စာမေးပွဲကို ဝင်ရောက်ဖြေဆိုကြမည့် ကျောင်းသား/သူတို့အတွက် လေ့ကျင့်ရန် မေးခွန်းတစ်စုံ တင်ပြလိုက်ပါသည်။ လေ့ကျင့်ဖြေဆိုကြစေလိုပါသည်။ အဖြေများကို နောက်ရက်တွင် ဆက်လက် ဖော်ပြပေးပါမည်။