Q. Evaluate: ${\int }_0^{\pi /2}\frac{\cos x}{(1+\sin x)(2+\sin x)}dx$

Q. If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, then find $A^2$.

Q. Solve the differential equation $2xydy=(x^2+y^2)dx$.

Q. The lines $\frac{1-x}{3}=\frac{7y-14}{2P}=\frac{z-3}{2}$ and $\frac{7-7x}{3P}=\frac{y-5}{1}=\frac{6-z}{5}$ are perpendicular to each other. Find the value of $P$.

Q. Find the general solution of differential equation $\frac{dy}{dx}+\sqrt{\frac{1-y^2}{1-x^2}}=0$; $x\ne 1$.

Q. If $A$ and $B$ are two independent events with $P\left(A\right)=\frac{1}{2},P(A\cup B)=\frac{3}{5}$ and $P\left(B\right)=x$, then find the value of $x$.

Q. Find a particular solution of the differential equation $(1+x^2)dy+2xydx=\cot xdx$, given that $y=0$ if $x=\frac{\pi }{2}$

Q. Find the projection of vector $\overrightarrow{a}=2\hat{i}+3\hat{j}+2\hat{k}$ on a vector $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$

Q. Solve the following linear programming problem graphically:
Maximize: $Z=60x+40y$
subject to the constraints:

Q. Evaluate: $\int a^{3{\log }_{a}x}dx$.

Q. Find the direction cosines of the unit vector perpendicular to the plane $\overrightarrow{r}.(6\hat{i}-3\hat{j}-2\hat{k})+1=0$

Q. Find the equation of the normals to the curve $2x^2-y^2=14$ which are parallel to the line $x+3y=6$.

Q. Find the value of $\cos [\frac{\pi }{2}-{\sin }^{-1}\left(\frac{1}{3}\right)]$

Q. Evaluate: $\int 5-4x-x^{2}dx$.

Q. A wire of length $28cm$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Q. Solve graphically $2x^2+x-6=0$

Q. Prove that $\frac{1}{{\sin }^2\theta }-\frac{1-{\sin }^2\theta }{1-{\cos }^2\theta }=1$

Q. If $A=\left[\begin{array}{cc}1& 5\\ 6& 7\end{array}\right]$, then find $A+A^{\prime }$.

Q. If the points $(a,1),(1,2)$ and $(0,b+1)$ are collinear, then show that $\frac{1}{a}+\frac{1}{b}=1$