Question

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

If $|\stackrel{\rightarrow}{a}|$=2, $|\stackrel{\rightarrow}{b}|$=3 and $\stackrel{\rightarrow}{a}$.$\stackrel{\rightarrow}{b}$=3 then find the projection of $|\stackrel{\rightarrow}{b}|$ on $|\stackrel{\rightarrow}{a}|$.

Find : $\int{\frac{\sin^{2}x-\cos^{2}x}{\sin x\cos x}}\,d x$

Find a vector in the direction of $\stackrel{\rightarrow}{a}=\stackrel{\wedge}{i}-2\;\stackrel{\wedge}{j}$ that has magnitude 7 units.

Find the value of a if $\left(\begin{array}{l l }a-b & 2a+c \\ 2a-b & 3c+d\end{array}\right)$ = $\left(\begin{array}{l l }-1 & 5 \\ 0 & 13\end{array}\right)$ .

If A is a square matrix such that A² = A, then write the value of (I + A)² – 3A.

If A = $\left(\begin{array}{l l }4 & 6 \\ 7 & 5\end{array}\right)$ , then what is the value of $\ A{\cdot}(a d j A)$ ?