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

Q. Find $\int \left(\frac{1+x\log x}{x}\right)e^{x}dx.$

Q. If $-1$, $2$, $\alpha$ are the roots of the equation $2x^3+x^2-7x-6=0$, then find $\alpha$.

Q. Write Simpson's rule to find an approximate value of $\stackrel{b}{\underset{a}{\int }}f\left(x\right)dx.$

Q. If $X$ is a Poisson variable with $P(X=0)=P(X=1)=k$, then show that $k=e^{-1}$.

Q. Solve the equation $x^5-5x^4+9x^3-9x^2+5x-1=0$.

Q. Evaluate $\stackrel{\pi }{\underset{0}{\int }}\frac{x\sin x}{1+\sin x}dx$.

1. $0$
2. $\pi$
3. $\frac{{\pi }^2}{2}$
4. $\frac{{\pi }^2}{2}-\pi$

Q. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$; $B=\left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$ and $2X+A=B$, then find $X$.

Q. Find the length of the chord intercepted by the circle $x^2+y^2-x+3y-22=0$ on the line $y=x-3$.

Q. Solve $(x+y+1)\frac{dy}{dx}=1$.

Q. Find the number of ways of arranging the letters of the word $ENGINEERING$.

Q. If the equation $x^2-15-m(2x-8)=0$ has equal roots, find the value of $m$.

Q. Find the coordinates of the points on the parabola $y^2=2x$ whose focal distance is $\frac{5}{2}$.

Q. If $A$, $B$ are two events with $P(A\cup B)=0.65$, $P(A\cap B)=0.15$, then find the value of $P\left(A^c\right)+P\left(B^c\right)$.

Q. If $n$ is a positive integer, prove that $\sum _{r=1}^n{r}^3{\left(\frac{{}^n{C}_r}{{}^n{C}_{r-1}}\right)}^2=\frac{n{(n+1)}^2(n+2)}{12}$.

Q. Find $\int \frac{1+{\cos }^{2}x}{1-\cos 2x}dx$.

Q. Solve the equations $x+y+4z=6$, $3x+2y-2z=9$, $5x+y+2z=13$ by using Cramer's Rule.

Q. Solve: $xdy=(y+x{\cos }^2\left(\frac{y}{x}\right))dx$.

Q. Resolve $\frac{x^4}{(x-1)(x-2)}$ into partial fractions.

Q. Find the area enclosed between the curves $y=x$ and $y=x^2$ from $x=0$ and $x=1$

Q. Find the equation of the sphere passing through $(4,3,-1)$ and having its centre at $(3,8,1)$.

Q. Show that the circles $x^2+y^2-4x-6y-12=0$ and $x^2+y^2+6x+18y+26=0$ touch each other. Also find the point of contact and common tangent at this point of contact.

Q. Find the sum of all $4$ digit numbers that can be formed using the digits $1,3,5,7$ and $9$ (without repetition).

Q. If $f\left(x\right)=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+...\infty$, then show that $f^{\prime }\left(x\right)=\sinh x$.

Q. Evaluate ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}{\sin }^{2}x{\cos }^{4}xdx.$

Q. If $A=\left[\begin{array}{ccc}1& 0& 0\\ 2& 3& 4\\ 5& -6& x\end{array}\right]$ and $detA=45$; then find $x$.

Q. If the eccentricity of a hyperbola is $\frac{5}{4}$, then find the eccentricity of its conjugate.

Q. Find the order and degree of ${\left[\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^3\right]}^{\frac{6}{5}}=6y$.

Q. Find $\int \frac{1}{3+2\sin x+\cos x}dx$.

Q. Find the number of terms in the expansion of ${(2x+3y+z)}^7$.