# Sample Math Paper - Set (2) - for 2019 Matriculation Examination

2019 Matriculation Examination
Sample Paper (2)
Mathematics                              Time allowed : 3 hours
Section (A)

1.  (a) The function $\displaystyle g : N \to N$ is defined as $\displaystyle g : x\mapsto$ smallest prime factor of $\displaystyle x.$ (i) Find values for $\displaystyle g(10), g (20)$ and $\displaystyle g (81).$ (ii) Does $\displaystyle g$ have an inverse? Give reasons for your answer.
(3 marks)

(b) If $\displaystyle 2x-1$ is a factor of $\displaystyle 2x^3-x^2-8x+k,$ find $\displaystyle k$ and the other factors.
(3 marks)

2.  (a) Find the term independent of $\displaystyle x$ in the expansion of $\displaystyle {{\left( {x-\frac{2}{{{{x}^{2}}}}} \right)}^{9}}.$
(3 marks)

(b) If the sum of n terms of a certain sequence is $\displaystyle 2n + 3n^2,$ find the $\displaystyle {n}^{\text{th}}$ term.
(3 marks)

3.  (a) If $\displaystyle X=\left( {\begin{array}{*{20}{c}} 1 & 0 \\ 2 & 3 \end{array}} \right)$ and $\displaystyle X-kI$ is singular, where $\displaystyle I$ is a unit matrix of order $\displaystyle 2,$ find $\displaystyle k.$
(3 marks)

(b) A number $\displaystyle x$ is chosen at random from the numbers $\displaystyle -4, -3, -2, -1, 0, 1, 2, 3, 4.$ What is the probability that $\displaystyle |x| \le 2?$
(3 marks)

4.  (a) $\displaystyle TA$ is the tangent to the circle at$\displaystyle A, AB = BC, ∠BAC = 41°$ and $\displaystyle ∠ACT = 46°.$ Find $\displaystyle ∠ATC.$
(3 marks)

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

5.  (a) If $\displaystyle \tan \alpha =x+1$ and $\displaystyle \tan \beta =x-1$, find $\displaystyle \cot (\alpha -\beta )$ in terms of $\displaystyle x.$
(3 marks)

(b) Evaluate $\displaystyle \underset{{x\to 1}}{\mathop{{\lim }}}\,\frac{{(2x-3)(\sqrt{x}-1)}}{{2{{x}^{2}}+x-3}}$ and $\displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{3{{{\sin }}^{2}}x-2\sin {{x}^{2}}}}{{3{{x}^{2}}}}.$
(3 marks)

Section (B)

6.  (a) Given that $\displaystyle f(x) =2x^2-1$ and $\displaystyle g(x) = \cos x$ where $\displaystyle x\in A=\{x|0\le x\le \frac{\pi}{2}\}.$ Solve the equation $\displaystyle (f∘g)(x)=0,$ where $\displaystyle x\in A.$
(5 marks)

(b) The curve of the polynomial $\displaystyle f(x)=-x^3+2x^2+ax-10$ cuts the $\displaystyle x$-axis at $\displaystyle x=p, x=2$ and $\displaystyle x=q$. Find the value of $\displaystyle p$ and $\displaystyle q.$ Hence show that $\displaystyle a=5.$
(5 marks)

7.  (a) If $\displaystyle f(x+y,x-y)=xy$ where $\displaystyle x,y\in R$, show that $\displaystyle f(x,y)+f(y,x) =0$.
(5 marks)

(b) If the coefficients of $\displaystyle (2p + 4)^{\text{th}}$ and $\displaystyle (p - 2)^{\text{th}}$ terms in the expansion of $\displaystyle (1 + x)^{18}$ are equal, find the value of $\displaystyle p.$
(5 marks)

8.  (a) Find the solution set of the inequation $\displaystyle 3(x-\frac{3}{2})^2>2x^2-4x+\frac{3}{4}$ and illustrate it on the number line.
(5 marks)

(b) Find three numbers in A.P. whose sum is $\displaystyle 21$ and whose product is $\displaystyle 315.$
(5 marks)

9.  (a) If $\displaystyle S_1, S_2,$ and $\displaystyle S_3$ are the sums of $\displaystyle n, 2n$ and $\displaystyle 3n$ terms of a G.P., show that $\displaystyle S_1(S_3- S_2) = (S_2-S_1)^2.$
(5 marks)

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

10. (a) Given that $\displaystyle A=\left( {\begin{array}{*{20}{c}} {\cos \theta } & {-\sin \theta } \\ {\sin \theta } & {\cos \theta } \end{array}} \right).$ Determine whether $\displaystyle {{A}^{{-1}}}$ exists or not, if exists find $\displaystyle {{A}^{{-1}}}.$ Hence solve the system of equations $\displaystyle x\cos \theta -y\sin \theta =2$ and $\displaystyle x\sin \theta +y\cos \theta =2\sqrt{3}$ when $\displaystyle \theta=30°.$
(5 marks)

(b) A set of cards bearing the number from $\displaystyle 200$ to $\displaystyle 299$ is used in a game. If a card is drawn at random, what is the probability that it is divisible by $\displaystyle 3?$
(5 marks)

Section (C)

11. (a) In the diagram, two circles are tangent at $\displaystyle A$ and have a common tangent touching them $\displaystyle B$ and $\displaystyle C$ respectively. If $\displaystyle BA$ is produced to meet the second circle at $\displaystyle D,$ show that $\displaystyle CD$ is a diameter.
(5 marks)

(b) $\displaystyle ABC$ is a right triangle with $\displaystyle A$ the right angle. $\displaystyle E$ and $\displaystyle D$ are points on opposite side of $\displaystyle AC,$ with $\displaystyle E$ on the same side of $\displaystyle AC$ as $\displaystyle B,$ such that $\displaystyle ΔACD$ and $\displaystyle ΔBCE$ are both equilateral. If $\displaystyle α (ΔBCE) = 2 α (ΔACD),$ prove that $\displaystyle ABC$ is an isosceles right triangle.
(5 marks)

12. (a) Two circles are drawn intersecting at $\displaystyle A, B$ and so that the circumference of each passes through the centre of the another. Through $\displaystyle A,$ a line is drawn meeting the circumference at $\displaystyle C, D$ respectively. Prove that $\displaystyle \vartriangle BCD$ is equilateral.
(5 marks)

(b) Given that $\displaystyle \sin \alpha =\frac{3}{5}$ and $\displaystyle \cos \beta =\frac{{12}}{{13}}$, where $\displaystyle α$ is obtuse and $\displaystyle β$ is acute, find the exact values of $\displaystyle \cos (α+β)$ and $\displaystyle \cot (α- β).$
(5 marks)

13. (a) Solve $\displaystyle ΔABC$ with $\displaystyle b=12.5, c=23$ and $\displaystyle α=38°20′.$
(5 marks)

(b) Find the stationary points on the curve $\displaystyle y=x^4(x^2-6)$ and determine their natures.
(5 marks)

14. (a) Show that the tangent to the curve $\displaystyle y=e^{-2x}-3x$ at the point $\displaystyle (a,0)$ meets the $\displaystyle y$-axis at the point whose $\displaystyle y$-coordinate is $\displaystyle 2ae^{-2a} +3a.$
(5 marks)

(b) Points $\displaystyle A$ and $\displaystyle B$ have position vectors $\displaystyle {\vec{a}}$ and $\displaystyle {\vec{b}}$ respectively, relative to an origin $\displaystyle O.$ The point $\displaystyle C$ lies on $\displaystyle OA$ produced such that $\displaystyle OC = 3OA,$ and $\displaystyle D$ lies on $\displaystyle OB$ such that $\displaystyle OD = \frac {1}{4}OB.$ Express $\displaystyle \overrightarrow{{AB}}$ and $\displaystyle \overrightarrow{{CD}}$ in terms of $\displaystyle {\vec{a}}$ and $\displaystyle {\vec{b}}$. The line segments $\displaystyle AB$ and $\displaystyle CD$ intersect at $\displaystyle P.$ If $\displaystyle CP = hCD$ and $\displaystyle AP = kAB,$ calculate the values of $\displaystyle h$ and $\displaystyle k.$
(5 marks)