List of practice Questions

Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
Two identical charges $ q $ are placed 1 m apart. The electrostatic force between them is $ 9 \times 10^{-9} \, \text{N} $. What is the magnitude of each charge? (Take $ k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 $)
The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:
There are two inclined surfaces of equal length inclined at an angle of \(45^\circ\) with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on the rough surface than on the smooth surface. The coefficient of kinetic friction (\(\mu_k\)) between the object and the rough surface is close to :
For the given graph of a Linear Programming
For the given graph of a Linear Programming Problem, write all the constraints satisfying the given feasible region.

Find:$\displaystyle \int \dfrac{dx}{\sin x + \sin 2x}$

Evaluate:
$\displaystyle \int_{0}^{3} x \cos(\pi x) \, dx$

Find $ k $ so that the function \[ f(x) = \begin{cases} \frac{x^2 - 2x - 3}{x + 1} & \text{if } x \neq -1 \\ k & \text{if } x = -1 \end{cases} \] is continuous at $ x = -1 $.
Find: \[ \int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}} \, dx \]
A cylindrical water container has developed a leak at the bottom. The water is leaking at the rate of 5 cm$^3$/s from the leak. If the radius of the container is 15 cm, find the rate at which the height of water is decreasing inside the container, when the height of water is 2 meters.
Check the differentiability of the function f(x) = |x| at x = 0.
If \[ A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \quad \text{and} \quad A^2 = kA, \quad \text{then find the value of k.} \]
The solution of the differential equation $ \frac{dy}{dx} = -\frac{x}{y} $ represents family of:
If $ \int e^{-3 \log x} \, dx = f(x) + C $, then $ f(x) $ is:
If A and B are invertible matrices of order $3 \times 3$ such that $\text{det} = 4$ and $\text{det}([AB]^{-1}) = \frac{1}{20}$, then $\text{det}$ is equal to:
The coordinates of the foot of the perpendicular drawn from the point $A(-2, 3, 5)$ on the y-axis are:
Domain of $ \sin^{-1}(2x^2 - 3) $ is:
Evaluate: \[ I = \int_0^{\frac{\pi}{4}} \frac{\sin x \cos x}{\cos^4 x + \sin^4 x} \, dx \]
Solve the following Linear Programming Problem graphically:
Minimise \( Z = 3x + 5y \)
subject to the constraints: \[ x + 2y \geq 10, \quad x + y \geq 6, \quad 3x + y \geq 8, \quad x, y \geq 0. \]
Find: \[ J = \int \frac{\sqrt{x^2 + 1} \left[ \log(x^2 + 1) - 2 \log x \right]}{x^2} \, dx \]
Consider the experiment of tossing a coin. If the coin shows head, toss it again; but if it shows a tail, then throw a die. Find the conditional probability of the event A: `the die shows a number greater than 3' given that B: `there is at least one tail'.

Probability Distribution of Random Variable X

X1232λ3λ4λ
P(X)\(\frac{11}{30}\)\(\frac{1}{15}\)\(\frac{10}{30}\)\(\frac{3\lambda}{10}\)\(\frac{1}{15}\)\(\frac{1}{10}\)

(i) Calculate \( \lambda \), if \( E(X) = 3.2 \)
(ii) Find P(X > 1).

Find the local maxima and local minima of the function \[ f(x) = \frac{8}{3} x^3 - 12x^2 + 18x + 5. \]
Find: \[ \int \frac{\cos x \, dx}{1 + \cos x + \sin x}. \]
If $\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right| = -57$, the product of the possible values of $x$ is: