Q. Find the number of ways of forming a committee of $5$ members from $6$ men and $3$ women.

Q.

Find the adjoint matrix and inverse matrix of the matrix
$\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$

Q. If the abscissae of points $A,B$ are the roots of the equation $x^2+2ax-b^2=0$ and ordinates of $A,B$ are roots of $y^2+2py-q^2=0$, then find the equation of a circle for which $\overline{AB}$ is a diameter

Q. Find the extreme value of Quadratic expression $2x-7-5x^2$. Also state whether it is maximum or minimum with reason.

Q. Find the order and degree of:
${\left(\frac{d^{3}y}{d{x}^3}\right)}^2-3{\left(\frac{dy}{dx}\right)}^3-e^x=4$

Q. Find the sum of the infinite series $1+\frac{1}{2!}+\frac{1}{4!}+\frac{1}{6!}+....$

Q. Find the value of the integral
${\int }_0^{2\pi }{\sin }^{2}x{\cos }^{4}x\cdot dx$

Q. When two dice are thrown, the sum on the two dice happened to be $7$. Find the probability that none of the dice shows a number $2$.

Q. Find the range of the expression $\frac{x^2+x+1}{x^2-x+1}$.

Q. Evaluate :
$\int \frac{dx}{(x+1)(x+2)}$

Q. Find the number of ways permuting the letters of the word $PICTURE$ so that all the vowels come together.

Q. Find the length of major axis, minor axis, latus rectum and eccentricity of the ellipse $9x^2+16y^2=144$

Q. Obtain the parametric equation of the circle represented by $x^2+y^2=4$

Q. Evaluate :
${\int }_0^{\frac{\pi }{2}}\frac{{\sin }^{5}x}{{\sin }^{5}x+{\cos }^{5}x}\cdot dx$

Q. Find the co-ordinates of the points on the parabola $y^2=8x$, whose focal distance is $10$

Q. Define Rectangular Hyperbola and find its eccentricity

1. $1$
2. $\sqrt{2}$
3. $2$
4. $\infty$

Q. If the equation $x^4+4x^3-2x^2-12x+9=0$ has two pairs of equal roots, find the roots of the equation.

Q. Show that the angle between the circles
$x^2+y^2=a^2,x^2+y^2=ax+ay$ is $\frac{3\pi }{4}$

Q. The mean and variance of a Binomial distribution is $4$ and $3$ respectively. Find its parameters.

Q. Resolve $\frac{1}{{(x-1)}^2(x-2)}$ into partial fractions.

Q. If ${{}^{22}C}_r$ is the largest binomial coefficient in the expansion of ${(1+x)}^{22}$, find the value of ${{}^{13}C}_r$.

1. $24$
2. $65$
3. $26$
4. $78$

Q. $3x+4y+5z=18$, $2x-y+8z=13$, $5x-2y+7z=20$. Solve the above system of equations by Cramer's method.

Q. If $A=\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$, then find $A^{-1}$.

Q. Evaluate :
$\int \frac{1}{(x+3)\sqrt{x+2}}dx$, on $x\in I\subset (-2,\infty )$

Q. If $1\le r\le n$, then prove that ${{}^{n}C}_r+{{}^{n}C}_{r-1}={{}^{n+1}C}_r$

Q. If $A=\left[\begin{array}{cc}2& 4\\ -1& k\end{array}\right]$ and $A^2=O$ (null matrix), then find the value of $k$.

Q. Prove that $C_0+\frac{C_1}{2}+\frac{C_2}{3}+\frac{C_3}{4}+....+\frac{C_n}{n+1}=\frac{2^{n+1}-1}{(n+1)}$

Q. Prove that
$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|={(a+b+c)}^3$

Q. If $1,\alpha ,\alpha$ are the roots of the equation $x^3-6x^2+9x-4=0$, then find the value of $\alpha$.

Q. Find the equation of the Circle, which is concentric with $x^2+y^2-6x-4y-12=0$ and passing through $(-2,14)$