Question

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

Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?

Find the equation of tangent to the curve $y={\frac{x-7}{x^{2}-5x+6}}$ at the point, where it cuts the x-axis.

Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..

Using properties of determinants, prove that : $\begin{vmatrix}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}$ = $a b c+b c+c a+a b.$

A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

A problem in mathematics is given to 4 students A, B, C, D. Their chances of solving the problem, respectively, are 1/3, 1/4, 1/5 and 2/3. What is the probability that (i) the problem will be solved ? (ii) at most one of them solve the problem ?