# Sample Math Paper for 2019 Matriculation Examination

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

1. (a) Let the function $\displaystyle f(x)=\frac{{1-2x}}{{1+x}},x\ne 1$. If $\displaystyle {{g}^{{-1}}}(x)={{f}^{{-1}}}(x+1)$, evaluate $\displaystyle g(2)$.
(3 marks)
(b) The expression $\displaystyle (x + 4)^3 + ax + b$ has a factor $\displaystyle x + 1$ but leaves a remainder of $\displaystyle 8$ when divided by $\displaystyle x + 5$. Find the values of $\displaystyle a$ and $\displaystyle b$.
(3 marks)
2.  (a) If the first four terms in the expansion of $\displaystyle (x^2−2)^5$ in descending powers of $\displaystyle x$ are $\displaystyle x^{10}−10x^8+40x^6+Ax^4+...$, find the value of $\displaystyle A$.
(3 marks)
(b) In a geometric progression, $\displaystyle {{u}_{1}}=\frac{1}{{81}}$ and $\displaystyle {{u}_{4}}=\frac{1}{{3}}$. Find the common ratio.
(3 marks)
3  (a) Find the two matrices of the form $\displaystyle P=\left( {\begin{array}{*{20}{c}} 4 \\ {x-2} \end{array}\ \ \ \begin{array}{*{20}{c}} {{{x}^{2}}-2x} \\ {-1} \end{array}} \right)$ such that $\displaystyle P=P'$.
(3 marks)
(b) A fair coin is tossed 5 times. What is the probability of getting at least one head?
(3 marks)
4.  (a) In the figure, $\displaystyle ∠ABC=30°$, $\displaystyle AB=BC$ and $\displaystyle AD$ is a tangent. Find $\displaystyle ∠BDA$.
(3 marks)
(b)  $\displaystyle A,B$, and $\displaystyle C$ are with position vectors $\displaystyle \hat{\text{i}}+3\hat{\text{j}}$, $\displaystyle 2\hat{\text{i}}+5\hat{\text{j}}$ and $\displaystyle k\hat{\text{i}}-4\hat{\text{j}}$  respectively. Find the value of $\displaystyle k$ if $\displaystyle A,B$, and $\displaystyle C$ are collinear.
(3 marks)
5.  (a) Prove that $\displaystyle {{(1-\tan x)}^{2}}+{{(1-\cot x)}^{2}}={{(\sec x-\operatorname{cosec}x)}^{2}}$.
(3 marks)
(b) Evaluate (i) $\displaystyle \underset{{x\to 1}}{\mathop{{\lim }}}\,\frac{{\sqrt[3]{x}-1}}{{x-1}}$   (ii) $\displaystyle \underset{{x\to \pi }}{\mathop{{\lim }}}\,\frac{{\cos \frac{x}{2}}}{{\pi -x}}$.
(3 marks)
Section (B)
Answer any FOUR Questions.

6.    (a) A function $\displaystyle f$ is defined by $\displaystyle f:x\mapsto \frac{x}{p}+q,p\ne 0$. If $\displaystyle f(8)=1$ and$\displaystyle {{f}^{{-1}}}(-2)=2$, show that $\displaystyle \frac{p}{2}+{{q}^{2}}=10$.
(5 marks)
(b)  Given that the $\displaystyle (p + 1)^{\text{th}}$ term of an A.P. is twice the $\displaystyle (q + 1)^{\text{th}}$ term. Prove that$\displaystyle (3p + 1)^{\text{th}}$ term is twice the $\displaystyle (p +q+ 1)^{\text{th}}$ term.
(5 marks)
7.    (a) Find the term in $\displaystyle x^2$ and $\displaystyle x^3$ in the expansion of  $\displaystyle(2x+1)^5$. Hence find the term in $\displaystyle x^3$ in the expansion of  $\displaystyle (x+3)(2x+1)^5$.
(5 marks)
(b) Let $\displaystyle {{\text{J}}^{+}}$ be the set of all positive integers. Is the operation $\displaystyle \odot$ defined by $\displaystyle x\odot y=x^2+3y$ a binary operation on  $\displaystyle {{\text{J}}^{+}}$? If it is a binary operation, solve the equation $\displaystyle \left( {k\odot 5} \right)-\left( {3\odot k} \right)=3k+1$.
(5 marks)
8.    (a) If $\displaystyle k+4, k$ and $\displaystyle 2k-15$ where $\displaystyle k>0$ are the first three terms of a geometric progression, find the value of $\displaystyle k$. Hence find the first term and the common ratio and determine whether the sum to infinity exists or not. Find the sum to infinity of the progression if exists.
(5 marks)
(b) Find the solution set in $\displaystyle \text{R}$ of the in equations $\displaystyle (x+2)^2>2x+7$ and illustrate it on the number line.
(5 marks)
9.   (a) Given that when $\displaystyle f(x)=6x^3+3x^2+ax+b$, where $\displaystyle a$ and $\displaystyle b$ are constants, is divided by $\displaystyle (x+1)$ the remainder is $\displaystyle 45$, show that $\displaystyle b – a=48$. Given also that $\displaystyle (2x+1)$ is a factor of $\displaystyle f(x)$, find the value of $\displaystyle a$ and of $\displaystyle b$. Hence factorise $\displaystyle f(x)$ completely.
(5 marks)
(b) If $\displaystyle A=\left( {\begin{array}{*{20}{c}} {-2} & 3 \\ {-3} & 4 \end{array}} \right)$ show that $\displaystyle A+{{A}^{{-1}}}-2I=O$ where $\displaystyle I$ is a unit matrix of order 2.
(5 marks)
10.  (a) The matrices $\displaystyle P,Q$ and $\displaystyle R$ such that$\displaystyle P=\left( {\begin{array}{*{20}{c}} 2 & 1 \\ 3 & 2 \end{array}} \right)$, $\displaystyle Q=\left( {\begin{array}{*{20}{c}} {-1} & 0 \\ 2 & 1 \end{array}} \right)$ and $\displaystyle R=PQ$. Verify that $\displaystyle {{Q}^{{-1}}}{{P}^{{-1}}}={{R}^{{-1}}}$.
(5 marks)
(b) The probability that a student will receive an $\displaystyle A, B, C$ or $\displaystyle D$ grade are 0.3, 0.38, 0.22 and 0.1 respectively. What is the probability that student will receive
(i) at least $\displaystyle B$ grade?
(ii) at most $\displaystyle C$ grade?
(iii) not an $\displaystyle A$ grade?
(iv) $\displaystyle B$ or $\displaystyle C$ grade?
(5 marks)
Section (C)
Answer any THREE Questions.

11.   (a) Two unequal circles are tangent externally at $\displaystyle O$. $\displaystyle AB$ the chord of the first circle is tangent to the second circle at $\displaystyle C$, and $\displaystyle AO$ meets this circle at $\displaystyle E$. Prove that $\displaystyle ∠BOC=∠COE$.
(5 marks)
(b) In $\displaystyle ΔABC$, $\displaystyle D$ is a point of $\displaystyle AC$ such that $\displaystyle AD=2CD$. $\displaystyle E$ is on $\displaystyle BC$ such that $\displaystyle DE \parallel AB$. Compare the areas of $\displaystyle ΔCDE$ and $\displaystyle ΔABC$. If $\displaystyle α (ABED)=40$, what is $\displaystyle α (ΔABC)$?
(5 marks)
12.    (a) The position vectors, relative to an origin $\displaystyle O$, of three nonlinear points $\displaystyle A, B$ and $\displaystyle C$ are $-2\hat{\text{i}}+3\hat{\text{j}}$, $3\hat{\text{i}}+2\hat{\text{j}}$ and $-\hat{\text{i}}-5\hat{\text{j}}$. Show that $\displaystyle ΔABC$ is isosceles.
(5 marks)
(b) The tangent at the point $\displaystyle C$ on the circle meets the diameter $\displaystyle AB$ produced at $\displaystyle T$. If $\displaystyle ∠BCT=27°$, calculate $\displaystyle ∠CTA$. If $\displaystyle CT=t$ and $\displaystyle BT=x$, prove that the radius of the circle is $\displaystyle {\frac{{{{t}^{2}}-{{x}^{2}}}}{{2x}}}$.
(5 marks)
13.   (a) Find the angles of a triangle, given that $\displaystyle \angle A$ is obtuse and $\sec (B+C)=\operatorname{cosec}(B-C)=2$.
(5 marks)
(b) Differentiate $\displaystyle \frac{1}{\sqrt{x}}$ with respect to $\displaystyle x$ from the first principles.
(5 marks)
14.   (a) A cruise ship travels at a bearing of $\displaystyle 45°$ at $\displaystyle 15$ mph for $\displaystyle 3$ hours, and changes course to a bearing of $\displaystyle 120°$. It then travels $\displaystyle 10$ mph for $\displaystyle 2$ hours. Find the distance of the ship from its original position and also its bearing from the original position.
(5 marks)
(b) Find the two positive numbers $\displaystyle x$ and $\displaystyle y$ such that their sum is $\displaystyle 60$ and $\displaystyle xy^3$ is maximum.
(5 marks)
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