List of practice Questions

The Laplace transform of \( \cos\sqrt{t} \) is:

The value of \( \oint_S \vec{F} \cdot d\vec{s} \) where \( \vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k} \) taken over the cylinder \( x^2+y^2=4, z=0 \) and \( z=3 \) is:

The directional derivative of \( \nabla \cdot (\nabla f) \) at the point (1, -2, 1) in the direction of the normal to the surface \( xy^2z = 3x + z^2 \) where \( f = 2x^3y^2z^4 \) and \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) is

Let \( \vec{F} \) be the vector valued function and f be a scalar function. Let \( \nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z} \) then,
(A) div (grad f) = \( \nabla^2 f \)
(B) curl curl \( \vec{F} \) = grad curl \( \vec{F} \) - \( \nabla^2 \vec{F} \)
(C) div curl \( \vec{F} \) = \( \vec{0} \)
(D) curl grad f = \( \vec{0} \)
(E) div (\(f\vec{F}\)) = f div \( \vec{F} \) + (grad f) \( \times \vec{F} \)
Choose the correct answer from the options given below:

If two stones are thrown vertically upwards with their velocities in the ratio 2:5, then the ratio of the maximum heights attained by the stones is

If three forces of magnitudes 8 newtons, 5 newtons and 4 newtons acting a point are in equilibrium, then the angle between the two smaller forces is

Two forces acting at a point of a body are equilibrium if and only if they
(A) are equal in magnitude
(B) have same direction
(C) have opposite direction
(D) act along the same straight line
(E) are not equal in magnitude but have same direction
Choose the correct answer from the options given below:

The value of \( \int_2^3 \vec{A} \cdot \frac{d\vec{A}}{dt} dt \) if \( \vec{A}(2) = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \vec{A}(3) = 4\hat{i} - 2\hat{j} + 3\hat{k} \) is

If R is a closed region in the xy-plane bounded by a simple closed curve C and if M(x, y) and N(x, y) are continuous functions of x and y having continuous derivative in R, then

The surface area of the plane \(x + 2y + 2z = 12\) cut off by \(x=0, y=0\) and \(x^2+y^2=16\) is

A necessary and sufficient condition that the general equation of second degree \(ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\) may represent a pair of straight lines is

The plane \(x + y + z = \sqrt{3}\lambda\) touches the sphere \(x^2 + y^2 + z^2 - 2x - 2y - 2z - 6 = 0\) if:

The equation of cone with vertex at (0, 0, 0) and passing through the circle given by
\(x^2 + y^2 + z^2 + x - 2z + 3y - 4 = 0, x - y + z = 2\), is

Which of the following statements are true?
(A) The equations of the plane passing through the point (1, -1, 2) having 2, 3, 2 as direction ratios of normal to the plane is 2x + 3y + 2z = 3
(B) Angle between the normal to the plane 2x - y + z = 6 and x + y + 2z = 7 is \(\frac{\pi}{3}\)
(C) The angle at which the normal vectors to the plane 4x + 8y + z = 5 is inclined to the z-axis is \( \sin^{-1}(\frac{1}{9}) \)
(D) The equation of the plane passing through the point (3, -3, 1) and normal to the line joining the points (3, 4, -1) and (2, -1, 5) is x + 5y + 6z = -18
(E) A normal vector to the plane 2x - y + 2z = 5 is \( \frac{1}{3}(2\vec{i} - \vec{j} + 2\vec{k}) \)
Choose the correct answer from the options given below:

The surface area of the solid generated by revolving the curve \(x = e^t \cos t, y = e^t \sin t\) about y-axis for \(0 \le t \le \pi/2\) is

If the radius of curvature (\(\rho\)) at (0, 1) of \(y = e^x\) is \(\alpha\sqrt{\beta}\), then \(\alpha^2+\beta\) is:

The function \(y = \tan^{-1}x\) satisfies differential equation

Which of the following statements are true?
(A) For f(x) = |x|, for all x in [-1, 2]; Lagrange's mean value theorem is satisfied
(B) For f(x) = cosx, for all x in [0, \(\pi\)/2]; Lagrange's mean value theorem is satisfied
(C) For f(x) = \( \frac{1}{x} \), for all x in [-1, 2]; Lagrange's mean value theorem is satisfied
(D) For f(x) = x(x-1)(x-2), for all x in [0, 1/2]; Lagrange's mean value theorem is satisfied
(E) For f(x) = \(x^{1/3}\), for all x in [-1, 1]; Lagrange's mean value theorem is satisfied
Choose the correct answer from the options given below:

Which of the following statements are true?
(A) The curve r = a(1 + cos\(\theta\)) is symmetrical about the initial line
(B) The curve r = 2(1 - 2 sin\(\theta\)) is symmetrical about the initial line
(C) The curve r = a(1 + sin\(\theta\)) is symmetrical about the line \(\theta\) = \(\pi\)/2
(D) The curve r = a sin(3\(\theta\)) is symmetrical about the initial line
(E) The curve r² = a²cos(2\(\theta\)) is symmetrical about pole
Choose the correct answer from the options given below:

The system of equations
x + 2y + z = 6
x + 4y + 3z = 10
x + 4y + \(\lambda\)z = \(\mu\)
is inconsistent if

The complex number \(z_1, z_2\) and origin, form an equilateral triangle only if:

Let z, \(z_1, z_2\) be complex numbers. Then which of the following statements are True?
(A) \(e^z\) is never zero
(B) \(|e^{ix}|=1\) if x is real
(C) \(e^z = 1\) if z is an integral multiple of \(2\pi i\)
(D) \(e^{z_1} = e^{z_2}\) if and only if \(z_1 - z_2 = \frac{2\pi i n}{\sqrt{3}}\), where n is an integer
(E) \(|e^z|>e^z\) for \(z \neq 0\)
Choose the correct answer from the options given below:

For what values of n, \( \tan^{-1}3 + \tan^{-1}n = \tan^{-1}\left(\frac{3+n}{1-3n}\right) \) is valid

The asymptotes of the curve \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \) are

In Photo- electric effect
(A) There is no time interval (very small ~ 10⁻⁹s) between incidence of light and emissions of photo electrons.
(B) Higher the frequency of light, more is the kinetic energy of photo electrons emitted.
(C) A bright light yields more photo-electrons than dim light.
(D) Blue light emits slower electrons than red light.
Choose the correct answer from the options given below: