Evaluate $\textstyle\int$ sec² (7 – 4x) dx.

Evaluate $\textstyle\int$ sin³ x dx.

Find the solution of the differential equation ${\frac{d y}{d x}}=x^{3}e^{-2y}$.

Find the value of x + y from the following equation : 2 $\left(\begin{array}{l l }x & 5 \\ 7 & y-3\end{array}\right)$ + $\left(\begin{array}{l l }3 & -4 \\ 1 & 2\end{array}\right)$ = $\left(\begin{array}{l l }7 & 6 \\ 15 & 14\end{array}\right)$ .

For what values of $\stackrel{\rightarrow}{a}\$, the vectors $2\stackrel{\wedge}{i}-3\stackrel{\wedge}{j} +4\stackrel{\wedge}{k}$ and $a\,{\hat{\boldsymbol{i}}}+6\,{\hat{\boldsymbol{j}}}-8\,{\hat{\boldsymbol{k}}}$ are collinear?

Give an example of vectors $\stackrel{\rightarrow}{a}\$ and $\stackrel{\rightarrow}{b}\$ such that $\vert{\vec{b}}\vert=$ $\vert{\vec{b}}\vert$ but $\stackrel{\rightarrow}{a}\neq \stackrel{\rightarrow}{b}\;.$  

If $\left(\begin{array}{l l l }1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{l }x \\ -1 \\ z\end{array}\right)$ = $\left(\begin{array}{l }1 \\ 2 \\ 1\end{array}\right)$ , find x + y + z.

If $\left(\begin{array}{l l }1 & 0 \\ y & 5\end{array}\right)$ + 2 $\left(\begin{array}{l l }x & 0 \\ 1 & -2\end{array}\right)$ = , where I is a 2×2 unit matrix, find (x – y).

If A = $[a_{i j}]$ is is a matrix of order 2 × 2, such that |A| = –15, and $C_{i j}$ represents the cofactor of $a_{i j}$ , then find $a_{21}c_{21}+a_{22}c_{22}$

If A' = $\left(\begin{array}{l l }3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right)$ and B = $\left(\begin{array}{l l l }-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right)$ then find A' – B'.

If A, B and C are the vertices of a $\Delta A B C$, then what is the value of ${\vec{AB}}+{\vec{BC}}+{\vec{C A}}$ ?

If for any 2 × 2 square matrix A, A(adj A) = $\left(\begin{array}{l l }8 & 0 \\ 0 & 8\end{array}\right)$ , then write the value of |A|.

If matrix A = $[a_{i j}]_{2\times2}$, where $a_{i j}$ = $\left\{\begin{array}{l}{{2,\ i\neq j}}\\ {{0,\ i=j^{\prime}}}\end{array}\right.$ , Write the matrix A.

If x $\left(\begin{array}{l }2 \\ 3\end{array}\right)$ + y $\left(\begin{array}{l }-y \\ y\end{array}\right)$ = $\left(\begin{array}{l }10 \\ 5\end{array}\right)$ , write the value of x.

If Δ = $\begin{vmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{vmatrix}$ , write the minor of the element $a_{23}.$