List of practice Questions

If \( P = (0,1,2) \), \( Q = (4,-2,-1) \) and \( O = (0,0,0) \), then \( \angle POQ \) is:
If the perpendicular distance from \( (1,2,4) \) to the plane \( 2x + 2y - z + k = 0 \) is 3, then \( k \) is:
Evaluate: \[ \lim_{x \to 0} \left[ \frac{1}{x} - \frac{1}{e^x - 1} \right] \]

Observe the following structure:
 
The formal charges on the atoms 1, 2, 3 respectively are:

Two statements are given below: Statement-I: The ratio of the molar volume of a gas to that of an ideal gas at constant temperature and pressure is called the compressibility factor. 
Statement-II: The RMS velocity of a gas is directly proportional to the square root of \( T(K) \). 

At 133.33 K, the RMS velocity of an ideal gas is \[ (M = 0.083 \text{ kg mol}^{-1}, R = 8.3 \text{ J mol}^{-1} \text{ K}^{-1}) \]

Given below are two statements: Statement-I: In the decomposition of potassium chlorate, Cl is reduced. 
Statement-II: Reaction of Na with \( O_2 \) to form \( Na_2O \) is a redox reaction. 

Let \( f(x) \) be defined as: \[ f(x) = \begin{cases} 0, & x = 0 \\ 2 - x, & 0 < x < 1 \\ 2, & x = 1 \\ 1 - x, & 1 < x < 2 \\ -\frac{3}{2}, & x \geq 2 \end{cases} \] Then which of the following is true?
If \( f(x) = \left(\frac{1+x}{1-x}\right)^{\frac{1}{x}} \) is continuous at \( x = 0 \), then \( f(0) \) is:
The function \( f(x) = |x - 24| \) is:
If \( y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots \infty}}} \), then the value of \( \frac{d^2y}{dx^2} \) at \( (\pi,1) \) is:
If \( f(0) = 0 \), \( f'(0) = 3 \), then the derivative of \( y = f(f(f(f(f(x))))) \) at \( x = 0 \) is:
The value \( c \) of Lagrange’s Mean Value Theorem for \( f(x) = e^x + 24 \) in \( [0,1] \) is:
Equation of the normal to the curve \( y = x^2 + x \) at the point \( (1,2) \) is:
Displacement \( s \) of a particle at time \( t \) is expressed as \( s = 2t^3 - 9t \). Find the acceleration at the time when the velocity vanishes.
If a running track of 500 ft. is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular portion is maximum is (in feet).
Evaluate the integral: \[ \int \frac{x^2 - 1}{x^3\sqrt{2x^4 - 2x^2 + 1}} \,dx. \]
Evaluate the integral: $$ \int \frac{x^3 \tan^{-1}(x^4)}{1 + x^8} \,dx. $$ 

Evaluate the integral: $$ I = \int \frac{2}{1 + x + x^2} \,dx. $$ 

Evaluate the integral: \[ I = \int \frac{1}{x^2\sqrt{1 + x^2}} \,dx. \]
Evaluate the integral: \[ I = \int_0^{\frac{\pi}{4}} \log(1 + \tan x) \,dx. \]
Evaluate the limit: \[ \lim_{n \to \infty} \left( \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2 - 1}} + \dots + \frac{1}{\sqrt{n^2 - (n-1)^2}} \right). \]
The area (in square units) bounded by the curves \( x = y^2 \) and \( x = 3 - 2y^2 \) is:
Evaluate the integral: \[ I = \int_{-\pi}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \,dx. \]
The general solution of the differential equation: \[ (1 + \tan y) (dx - dy) + 2x \, dy = 0. \]