Question

Find the points on the curve y = x³ – 3x² – 9x + 7 at which the tangent to the curve is parallel to the x-axis.

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Using properties of determinants, prove that : $\begin{vmatrix}x²+1 & xy & xz \\ xy & y²+1 & yz \\ xz & yz & z²+1\end{vmatrix}$

Find : $\textstyle\int$${\frac{x^{3}}{x^{4}+3x^{2}+2}}d x~.$

Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem, independently, then find the probability that (i) the problem is solved (ii) exactly one of them solves the problem

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.

Solve the following differential equation. $(x^{3}+x^{2}+x+1){\frac{d y}{d x}}=2x^{2}+x.$