List of top Questions asked in KEAM- 2024

If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to
The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \). If \( | \vec{a} \cdot \vec{b} |^2 = 15 \), then \( | \vec{a} \times \vec{b} |^2 \) is equal to
The coordination number of Pt and Fe in the complexes [PtCl\(_6\)]\(^{2-}\) and [Fe(C\(_2\)O\(_4\))\(_3\)]\(^{3-}\) are respectively
If \( E \) is the electric field intensity between the plates of a charged parallel plate capacitor, energy stored per unit volume in it is (permittivity of free space = \( \epsilon_0 \))
The elastic energy stored per unit volume in a stretched wire is: \[ \left( Y = \text{Young's modulus of the material of the wire}; \quad S = \text{stress acting on the wire} \right) \]
Let \( A = \{ 1, 3, 5, 7, \dots, 21 \} \). The number of ways 4 numbers, containing always 11, can be selected from the set A is equal to
If \[ f(x) = \sqrt[3]{x^2} + \sqrt{x}, \] then the value of \( f'(64) \) is equal to
The general solution of the differential equation \[ 2y \tan x + \frac{dy}{dx} = 5 \sin x \] is:
An alternative form of Biot-Savart's law is
Find the integrating factor of the differential equation: \[ (3 \sin x \cos x) \, dy = (1 + 3y \sin^2 x) \, dx, \quad {where} \quad 0<x<\frac{\pi}{2} \] is

Ice is coated uniformly around a sphere of radius 15 cm. If ice is melting at the rate of \( 80 { cm}^3/{min} \) when the thickness is 5 cm, then the rate of change of thickness of ice is

Let \( z = x + iy \), where \( y>0 \). If \( z + \overline{z} = 6 \) and \( |z| + | \overline{z} | = 10 \), then \( z = \)
Let \( A \) be a \( 3 \times 3 \) matrix with \( |A| = 7 \). If \( B = 3A \), then the value of \[ \left| \frac{{adj } A}{B} \right| \] is equal to
If \[ A = \begin{bmatrix} -7 & 3 \\ 3 & -1 \end{bmatrix} \] then \( \det(A^5) \) is equal to:
The means of two samples of size 30 and 40 are 35 and 42 respectively. Then the mean of the combined sample of size 70 is
For a smoothly running analog clock, the ratio of the angular velocity of the minute hand to the angular velocity of the hour hand is

The area bounded by the parabola \( y = x^2 + 4 \) and the straight line passing through the points \( (-1,2) \quad {and} \quad (1,6) \) is (in square units)

The value of \( \int_{-2}^{2} x |x| \,dx \) is:
A particle of 100 g mass is projected vertically up with a kinetic energy of 20 J. The maximum height reached by the particle is (\( g = 10 \) m/s\( ^2 \)) (neglecting air resistance).
Evaluate the integral: \[ \int_{0}^{3} |x - 2| \,dx \] is equal to
The function \( f(x) = 6x^4 - 3x^2 - 5 \) is increasing in the set:
Let \( \alpha \) and \( \beta \) be real numbers such that \( f(x) \) is defined as: \[ f(x) = \begin{cases} 2x^2 + 4x + \alpha, & \text{if } x < 1 \\ \beta x^2 + 5, & \text{if } x \geq 1 \end{cases} \] and is differentiable at \( x = 1 \). Then \( \alpha + \beta \) is equal to:
The range of the function \( f(x) = 8 + \sqrt{x - 5} \) is:

The centre of the hyperbola \(16x^2 - 4y^2 + 64x - 24y - 36 = 0\) is at the point:

The foci of the ellipse \(\frac{x^2}{49} + \frac{y^2}{24} = 1\) are: