Question

Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\;{\cot{x}}=$ 4x cosec x, (x ≠ 0), given, that y = 0,when $x={\frac{\pi}{2}}$

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$

Evaluate : $\large\textstyle\int{\frac{3x+1}{(x^{3}-x^{2}-x+1)}}d x\,.$

Sand is pouring from the pipe at the rate of 12 cm³/s. The falling sand forms a cone on a ground in such a way that the height of cone is always one-sixth of radius of the base. How fast is the height of sand cone increasing when the height is 4 cm ?

If A = $\left(\begin{array}{l l }2 & 3 \\ 1 & -4\end{array}\right)$ , B = $\left(\begin{array}{l l }1 & -2 \\ -1 & 3\end{array}\right)$ , verify that $(A B)^{-1}=B^{-1}A^{-1}$

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.