List of practice Questions

In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of:

Given below are two statements:

Statement-I: Figure shows the variation of stopping potential with frequency (v) for the two photosensitive materials M₁ and M₂. The slope gives value of \(\frac{h}{e}\) , where h is Planck's constant, e is the charge of electron.
Statement-II: M2 will emit photoelectrons of greater kinetic energy for the incident radiation having same frequency. In the light of the above statements 
Choose the most appropriate answer from the options given below.

If G be the gravitational constant and u be thee nergy density then which of the following quantity have the dimension as that the \(\sqrt{UG}\)

Following gates section is connected in a complete suitable circuit.

For which of the following combination, bulb will glow (ON):

Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is:

Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.

The number of distinct real roots of the equation \[ |x| \, |x + 2| - 5|x + 1| - 1 = 0 \] is _________.

Let \( a_1, a_2, a_3, \dots \) be in an arithmetic progression of positive terms.
Let \( A_k = a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2k-1}^2 - a_{2k}^2 \).  
If \( A_3 = -153 \), \( A_5 = -435 \), and \( a_1^2 + a_2^2 + a_3^2 = 66 \), then \( a_{17} - A_7 \) is equal to _________.

Let \( f \) be a differentiable function in the interval \( (0, \infty) \) such that \( f(1) = 1 \) and \(\lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1\) for each \( x > 0 \).
Then \( 2f(2) + 3f(3) \) is equal to _________.

If \( S = \{ a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\} \} \), where \([t]\) denotes the greatest integer less than or equal to \(t\) and \(\{t\}\) represents the fractional part of \(t\), then \( 72 \sum_{a \in S} a \) is equal to _________.

The number of ways of getting a sum 16 on throwing a dice four times is __________.

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable \( X \) denote the number of defective items in the sample. If the variance of \( X \) is \( \sigma^2 \), then \( 96\sigma^2 \) is equal to _________.

If the line \(\frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z\) makes a right angle with the line  \(\frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7},\) then \( 4\lambda + 9\mu \) is equal to:

The value of \(\int_{-\pi}^{\pi} \frac{2y(1 + \sin y)}{1 + \cos^2 y} \, dy\)

If A(l, –1, 2), B(5, 7, –6), C(3, 4, –10) and D(–l, –4, –2) are the vertices of a quadrilateral ABCD, then its area is :

Let \( f(x) = x^5 + 2x^3 + 3x + 1 \), \( x \in \mathbb{R} \), and \( g(x) \) be a function such that \( g(f(x)) = x \) for all \( x \in \mathbb{R} \). Then \( \frac{g(7)}{g'(7)} \) is equal to:

If \(\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \dots + \frac{1}{\sqrt{99} + \sqrt{100}} = m\) and  \(\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{99 \cdot 100} = n,\) then the point \( (m, n) \) lies on the line

Suppose \( \theta \in \left[ 0, \frac{\pi}{4} \right] \) is a solution of \( 4 \cos \theta - 3 \sin \theta = 1 \). Then \( \cos \theta \) is equal to:

Consider the following two statements:
Statement I: For any two non-zero complex numbers \( z_1, z_2 \),
\((|z_1| + |z_2|) \left| \frac{z_1}{|z_1|} + \frac{z_2}{|z_2|} \right| \leq 2 (|z_1| + |z_2|)\)
Statement II: If \( x, y, z \) are three distinct complex numbers and \( a, b, c \) are three positive real numbers such that  
\(\frac{a}{|y - z|} = \frac{b}{|z - x|} = \frac{c}{|x - y|},\)
then  
\(\frac{a^2}{y - z} + \frac{b^2}{z - x} + \frac{c^2}{x - y} = 1.\)
Between the above two statements,

Let the line 2x + 3y – k = 0, k > 0, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is x² + y² – 3x – 2y = 0 and the length of the latus rectum of the ellipse x² + 9y² = k² is m n , where m and n are coprime, then 2m + n is equal to

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

The coefficients a, b, c in the quadratic equation ax² + bx + c = 0 are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}. The probability of this equation having repeated roots is :

The integral \(\int_{0}^{\pi/4} \frac{136 \sin x}{3 \sin x + 5 \cos x} \, dx\) is equal to

If the function \(f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3},\) \(x \in \mathbb{R} \), is continuous at \( x = 0 \), then \( f(0) \) is equal to: