List of practice Questions

The pair of physical quantities not having the same dimensions is:

The expression given below shows the variation of velocity \( v \) with time \( t \): \[ v = \frac{At^2 + Bt}{C + t} \] The dimension of \( A \), \( B \), and \( C \) is:

The coordinates of a particle with respect to origin in a given reference frame is \( (1, 1, 1) \) meters. If a force of \( \mathbf{F} = \hat{i} - \hat{j} + \hat{k} \) acts on the particle, then the magnitude of torque (with respect to origin) in the z-direction is:

In a hydraulic lift, the surface area of the input piston is 6 cm² and that of the output piston is 1500 cm². If 100 N force is applied to the input piston to raise the output piston by 20 cm, then the work done is kJ.

Match List - I with List - II:

Choose the correct answer from the given below options

Given below are some nitrogen containing compounds:
Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume ...... mg of HCl.
(Given Molar mass in g mol\(^{-1}\): C = 12, H = 1, O = 16, Cl = 35.5.)

The sum of sigma (σ) and pi (π) bonds in Hex-1,3-dien-5-yne is:

The molar mass of the water insoluble product formed from the fusion of chromite ore \(FeCr_2\text{O}_4\) with \(Na_2\text{CO}_3\) in presence of \(O_2\) is ....... g mol\(^{-1}\):

If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \( (\alpha, \beta) \), and \( X = \{ x \in \mathbb{Z} : \alpha<x<\beta \} \), then \( \sum_{x \in X} x^2 \) is equal to:

Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta\pi + \gamma \); \( \beta, \gamma \) are integers, then the value of \( |\beta - \gamma| \) equals:

If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \[ \log_{(x-1)} \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \] is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

Let a straight line \( L \) pass through the point \(P(2, -1, 3)\) and be perpendicular to the lines \[ \frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{-2} \quad \text{and} \quad \frac{x - 3}{1} = \frac{y - 2}{3} = \frac{z + 2}{4}. \] If the line \(L\) intersects the yz-plane at the point Q, then the distance between the points P and Q is:

Let \( S = \mathbb{N} \cup \{0\} \). Define a relation \( R \) from \( S \) to \( \mathbb{R} \) by: \[ R = \left\{ (x, y) : \log_e y = x \log_e \left(\frac{2}{5}\right), x \in S, y \in \mathbb{R} \right\}. \] Then, the sum of all the elements in the range of \( R \) is equal to:

If \(\alpha x + \beta y = 109\) is the equation of the chord of the ellipse \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \] whose midpoint is \(\left(\frac{5}{2}, \frac{1}{2}\right)\), then \(\alpha + \beta\) is equal to:

Let \( \alpha, \beta (\alpha \neq \beta) \) be the values of m, for which the equations \(x + y + z = 1\), \(x + 2y + 4z = m\), and \(x + 4y + 10z = m^2\) have infinitely many solutions. Then the value of \(\sum_{n=1}^{10} (n^4 + n^8)\) is equal to:

Let \( A = [a_{ij}] \) be a matrix of order 3 \(\times\) 3, with \(a_{ij} = (\sqrt{2})^{i+j}\). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), where \(\alpha, \beta \in \mathbb{Z}\), then \(\alpha + \beta\) is equal to:

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on \(3x + 2y + 2 = 0\). Then the length of the chord of the circle C, whose midpoint is (1, 2), is:

Let \( A = [a_{ij}] \) be a 2 \(\times\) 2 matrix such that \(a_{ij} \in \{0, 1\}\) for all \(i\) and \(j\). Let the random variable X denote the possible values of the determinant of the matrix A. Then, the variance of X is:

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains n white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability that the ball drawn is white is \(\frac{29}{45}\), then n is equal to:

The remainder, when \(7^{98}\) is divided by 23, is equal to:

Let \(f(x) = \int_{1}^{x} (t^2 - 9t + 20) \, dt\), \(1 \leq x \leq 5\). If the range of \(f(x)\) is \( [\alpha, \beta] \), then \(4(\alpha + \beta)\) equals:

If \[ 24 \left( \int_0^\frac{\pi}{4} \left[ \sin \left( 4x - \frac{\pi}{12} \right) + [2 \sin x] \right] dx \right) = 2n + \alpha, \] where [.] denotes the greatest integer function, then \( \alpha \) is equal to:

If \( \lim_{t \to \infty} \left( \int_0^{1} \left( 3x + 5 \right)^t dx \right) = \frac{\alpha}{5e} \left( \frac{8}{5} \right)^{\frac{3}{2}}, \) then \( \alpha \) is equal to ____ :

Let \( a_1, a_2, \dots, a_{2024} \) be an Arithmetic Progression such that \[ a_1 + (a_1 + a_0 + a_1 + a_2 + \cdots + a_{2020} + a_{2024}) = 2233. \quad \text{Then} \quad a_1 + a_2 + a_3 + \dots + a_{2022} \] is equal to ____ :

Let \( y^2 = 12x \) be the parabola and \( S \) its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ) = \frac{147}{4} \). Let \( C \) be the circle described by taking \( PQ \) as a diameter. If the equation of the circle \( C \) is: \[ 64x^2 + 64y^2 - \alpha x - 64\sqrt{3}y = \beta, \] then \( \beta - \alpha \) is equal to: