Question

Write the value of x + y + z if $\left(\begin{array}{l l l }1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$ $\left(\begin{array}{l }x \\ y \\ z\end{array}\right)$ = $\left(\begin{array}{l }1 \\ -1 \\ 0\end{array}\right)$.

If a unit vector $\stackrel{\rightarrow}{a}\$ makes an angle $\frac{\pi}{3}$ with $\stackrel{\wedge}{i},$ $\frac{\pi}{4}$ with $\stackrel{\wedge}{j}$ and an acute angle θ with $\stackrel{\wedge}{k}$, then find the value of θ.

Let R be the equivalence relation in the set A = {0, 1, 2, 3, 4, 5} given by R = {(a, b) : 2 divides (a –b)}. Write the equivalence class [0].

Write the sum of the order and degree of the differential equation $1+\left(\frac{d y}{d x}\right)^{4}=7\left(\frac{d^{2}y}{d x^{2}}\right)^{3}\,.$

If A is a matrix of order 3 x 4 and B is a matrix of order 4 x 3, then find order of matrix (AB.)

Evaluate $\textstyle\int$ sin³ x dx.

Write the position vector of the point which divides the join of points with position vectors $3\stackrel{\rightarrow}{a}-2\stackrel{\rightarrow}{b}$ and $2\stackrel{\rightarrow}{a}+3\stackrel{\rightarrow}{b}$ in the ratio 2:1.