CBSE Class 12 Mathematics — Short Answer Questions

Evaluate: integral from 0 to 1 of $x \tan^{-1} x \, dx$.

(a) Find: $\int \sqrt{\dfrac{x+2}{x-2}} \, dx$. OR (b) Find: $\int \dfrac{x^2}{(x^2+9)(x^2+16)} \, dx$.

If I_1 = integral from -pi/4 to pi/4 of dx/(1 + cos 2x) and I_2 = integral from -1/2 to 1/2 of |x| dx, then show that $I_1 - 4 I_2 = 0$.

(a) Find the general solution of the following differential equation: $x^2 \frac{dy}{dx} = x^2 + xy + y^2$ OR (b) Find the particular solution of the differential equation $xy \frac{dy}{dx} = (x + 2)(y + 2)$, given that $y(1) = -1$.

Solve the following linear programming problem graphically: Minimize $Z = 13x - 15y$ subject to constraints $x + y \le 7$, $2x - 3y + 6 \ge 0$, $x \ge 0$, $y \ge 0$.

(a) Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball. OR (b) The probability of simultaneous occurrence of at least one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that $P(X') + P(Y') = 2 - 2a + b$.