WRITE YOUR ANSWERS IN THE ANSWER BOOKLET.
SECTION (A)
(Answer ALL questions.)
(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 $b$.
(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 $5$.
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.
(Answer Any FOUR questions.)
(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 $x + 2$.
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$.
(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 $n$.
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.
(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 $u_{mn} = 1$.
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.
(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 $.
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)$.
(b). Draw a tree diagram to list all possible outcomes when four fair coins are tossed simultaneously. Hence determine the probability of getting:
(i) all heads,
(ii) two heads and two tails,
(iii) more tails than heads,
(iv) at least one tail,
(v) exactly one head.
(Answer Any THREE questions.)
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.
(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 $\alpha(ΔABC)$?
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.
(b). Solve the equation $\displaystyle \sqrt{3}\cos \theta +\sin \theta =\sqrt{2}$ for $ \displaystyle 0{}^\circ \le \theta \le 360{}^\circ $.
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.
(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.
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}}.$
(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.