List of practice Questions

A line \( L \) passes through \( (1,2,-3) \) and \( (3,3,-1) \), and a plane \( \pi \) passes through \( (2,1,-2), (-2,-3,6), (0,2,-1) \). If \( \theta \) is the angle between \( L \) and \( \pi \), then \( 27 \cos^2 \theta = \) ?
\( \lim_{x \to 3} \frac{x^3 - 27}{x^2 - 9}. \)

If \( f(x) \) is given as: \( f(x) = \begin{cases} 3ax - 2b, & x<1 ax + b + 1, & x<1 \end{cases} \) and \( \lim_{x \to 1} f(x) \) exists, then the relation between \( a \) and \( b \) is:

.The function \( f(x) \) is given by: \[ f(x) = \begin{cases} \frac{2}{5 - x}, & x<3 \\ 5 - x, & x \geq 3 \end{cases} \] Which of the following is true

If \[ f(x) = \begin{cases} x^\alpha \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \] Which of the following is true?

 If the equation of the tangent at (2, 3) on y2 = ax3 + b is y = 4x - 5, then the value of a2 + b2 is:

If Rolle's theorem is applicable for the function \(f(x) = x(x+3)e^{-x/2}\) on \([-3, 0]\), then the value of \(c\) is:

 For all x ∈ [0, 2024] assume that f (x) is differentiable. f (0) = −2 and f ′(x) ≥ 5. Then the least possible value of f (2024) is:

Let f(x) = \(\int \frac{x}{(x^2+1)(x^2+3)} dx\). If f(3) = \(\frac{1}{4} \log(\frac{5}{6})\), then f(0) =

\(\int\frac{2\cos 2x}{(1+\sin 2x)(1+\cos 2x)}dx=\)

\(\int\left(\frac{x}{x\cos x - \sin x}\right)^2 dx = \)

If \(\lim_{n\rightarrow\infty}[(1+\frac{1}{n^{2}})(1+\frac{4}{n^{2}})(1+\frac{9}{n^{2}})\cdots(1+\frac{n^{2}}{n^{2}})]^{\frac{1}{n}}=ae^{b}\), then a+b=

If \(I_{n}=\int_{0}^{\pi/4}\tan^{n}x~dx,\) then \(I_{13}+I_{11}=\)
The area (in square units) of the smaller region lying above the X-axis and bounded between the circle \[ x^2 + y^2 = 2ax \] and the parabola \[ y^2 = ax \]
The difference of the order and degree of the differential equation \[ \left(\frac{d^2y}{dx^2} \right)^{-7/2} - \left(\frac{d^3y}{dx^3} \right)^2 - \left(\frac{d^2y}{dx^2} \right)^{-5/2} - \left(\frac{d^4y}{dx^4} \right) = 0 \]

A force of \( (6x^2 - 4x + 3) \, \text{N} \) acts on a body of mass 0.75 kg and displaces it from \( x = 5 \, \text{m} \) to \( x = 2 \, \text{m} \). The work done by the force is

The displacement of a particle executing simple harmonic motion is \( y = A \sin(2\pi t + \phi) \, \text{m} \), where \( t \) is time in seconds and \( \phi \) is the phase angle. At time \( t = 0 \), the displacement and velocity of the particle are 2 m and 4 ms-1. The phase angle, \( \phi \) =

The displacement of a damped oscillator is \( x(t) = \exp(-0.2t) \cos(3.2t + \phi) \), where \( t \) is time in seconds. The time required for the amplitude of the oscillator to become \( \frac{1}{e^{1.2}} \) times its initial amplitude is
If the work done in stretching a wire by 1 mm is 2 J, the work necessary for stretching another wire of same material but with double radius of cross section and half the length by 1 mm is
An ideal heat engine operates in Carnot cycle between 127\(^\circ\)C and 27\(^\circ\)C. It absorbs \( 5 \times 10^4 \) cal of heat at higher temperature. Amount of heat converted to work is

Two particles of equal mass \( m \) and equal charge \( q \) are separated by a distance of 16 cm. They do not experience any force. The value of \( \frac{q}{m} \) is                (if \( G \) is the universal gravitational constant and \( g \) is the acceleration due to gravity).

In the following diagram, the work done in moving a point charge from point P to point A, B and C are \( W_A, W_B, W_C \) respectively. Then (A, B, C are points on semicircle and point charge \( q \) is at the centre of semicircle)
 

Four condensers each of capacitance 8 muF are joined as shown in the figure. The equivalent capacitance between the points A and B will be
 

The resistance between points A and C in the given network is 

A wire shaped in a regular hexagon of side 2 cm carries a current of 4 A. The magnetic field at the centre of the hexagon is.