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


2019 Matriculation Examination
Sample Paper (2)
Mathematics                              Time allowed : 3 hours
WRITE YOUR ANSWER IN THE ANSWER BOOKLET.
Section (A)
Answer ALL Questions.

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)
Answer any FOUR Questions.

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)
Answer any THREE Questions.

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)