List of practice Questions

If \( u = x^y, v = y^x \) and quantity \( y \) remains 3 times the quantity \( x \) then find that amongst quantities \( u \) and \( v \), which changes more rapidly with respect to quantity \( x \) when \( x = 1 \). (Take \( \log_e 3 = 1.09 \))
If \( x = \frac{1-t^2}{1+t^2} , y = \frac{2t}{1+t^2} \) then prove that \( \frac{dy}{dx} + \frac{x}{y} = 0 \).
Correct Answer: Proved
If \( P(A) = \frac{1}{2}, P(B) = \frac{3}{8} \) and \( P(A \cap B) = \frac{1}{5} \), then \( P(A|B) \) is equal to:
Ram and Rahim are contesting for two vacancies in a company. Probability of selection of Ram is \( \frac{7}{9} \) and that of Rahim is \( \frac{4}{7} \). What is the probability that both will be selected?
Find \( \frac{d}{dx}(x^x) \) and evaluate whether this result is true \( \forall x \in \mathbb{R} \).
Evaluate \( \int \frac{2x-3}{x^2+1} dx \).
Using integration, find the area bounded by the circle whose centre is at origin and radius is 4 units.
The volume of spherical balloon is increasing at the rate of 25 c.c./s. Find the rate of change of its surface area at the instant when its radius is 5cm.
Order of differential equation \( \frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 3y = 0 \) is:
If \( \vec{a} \) is a non-zero vector then \( |\vec{a} \times \vec{a}| \) is equal to:
Name of the inequality \( |\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}| \) is:
Direction ratios of the straight line \( \vec{r} = 2\hat{i} - 3\hat{j} + \hat{k} + m(9\hat{i} - 2\hat{j} + 5\hat{k}) \) are:
Vector equation of the line \( \frac{x-5}{-4} = \frac{y-3}{5} = \frac{z+3}{-8} \) is:
Objective function of a linear programming problem is:
If \( y = e^{\log x} \), then \( \frac{dy}{dx} \) is:
If \( y = \tan x \), then at \( x = \frac{\pi}{4} \), \( \frac{dy}{dx} \) is equal to:
Radius of a circle is increasing at the rate of \( 2 \text{ m/s} \). Rate of change of its circumference is:
\( \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx \) is equal to:
\( \int_0^1 \frac{dx}{1+x^2} \) is equal to:
Range of function \( \cos^{-1} x \) is:
Principal value of \( \sin^{-1}(1) \) is:
If \( A \) is a square matrix of order \( 2 \times 2 \) and \( |A| = 5 \), then \( |Adj(A)| \) is:
If the order of the matrix \( A \) is \( 3 \times 2 \), then the order of the matrix \( (A')' \) is:
State Gauss’s theorem. Using this theorem, derive electric field intensity due to thin uniformly charged infinite plane sheet.
Define capacitance. Derive capacitance of a parallel plate capacitor having dielectric slab between its plates.