List of top Geophysics Questions

A Fraunhofer diffraction pattern is produced by a circular aperture of radius $0.05 \, \text{cm}$ at the focal plane of a convex lens of focal length $20 \, \text{cm}$. If the wavelength $\lambda = 5 \times 10^{-5} \, \text{cm}$, the radius of the first dark ring is:

Solve the following differential equation by the method of Laplace transform: \[ y''' + 2y'' - y' - 2y = 0 \] given that $y(0) = 0$, $y'(0) = 0$, and $y''(0) = 6$.
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The position vector of a point in the frame S moving with constant velocity 10 cm/s along the X-axis is given by (11,9.8) cm. The position with respect to S if the two frames were coincident only 1/2 second earlier.

In the matrix equation: \[ \begin{bmatrix} 3 & -1 \\ 2 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}, \] the values of \( x \) and \( y \) are:

The equation of motion for a compound pendulum is:

If the radius of curvature of the Folium \( x^3 + y^3 - 3xy = 0 \) at the point \( (3/2, 3/2) \) is 5, then the value of \( b^2 + 2a + 1 \) is:

Statement (I): If a given component of the total applied force vanishes, the corresponding component of the linear momentum is not conserved.
Statement (II): If the component of applied torque along the axis of rotation vanishes, then the component of total angular momentum along the axis of rotation is conserved.
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Let P and Q be two matrices such that P Q = 0 and P is non-singular, then
(a) Q is also non-singular
(b) Q = 0
(c) Q is singular
(d) P = Q
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The value of curl (\(\text{grad } f\)), where \( f = x^2 - 4y^2 + 5z^2 \), is:

	LIST I		LIST II
A.	Intrinsic semiconductor		I. Used as a rectifier circuit
B.	N-Type Semiconductor		II. Pure form of Semiconductor
C.	P-Type Semiconductor		III. Doping of pentavalent impurity in semiconductor
D.	P-N Junction diode		IV. Doping of trivalent impurity in semiconductor

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The shape of air film formed between the plano-convex lens and the glass slab in Newton’s ring experiment is

Given below are two statements
Statement I Every homogeneous equation of second degree in x y and z represents a cone whose vertex is at the origin
Statement II If two equations representing the guiding curve are such that the one equation is of the first degree then the required cone with vertex at the origin is obtained by making the other equation homogeneous with the help of the first equation
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If \[ \theta = r e^{-x^2}, \] then for what value of \( n \), the following holds: \[ \frac{1}{2} \left( \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial r^2} \right) = \frac{\partial \theta}{\partial x}. \]

In a Fraunhofer N-slit diffraction experiment, a grating has $5000$ lines/cm, and monochromatic light of wavelength $5 \times 10^{-5}$ cm is used. What is the highest order spectrum that may be observed?

Given below are two statements:
{Statement (I):} Two families of curves such that every member of either family cuts each member of the other family at right angles are called orthogonal trajectories of each other.
{Statement (II):} The orthogonal trajectories of the curve \( xy = c \) is \( y = \frac{1}{x} \).
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Let f(x) be a differentiable function for all values of x with f′(x) ≤ 32 and f(3) = 21, then the maximum value of f(8) is:

The general solution of the differential equation \[ \frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = e^x \cos 2x \] is:

Let \( a \) be the magnitude of the directional derivative of the function: \[\phi(x, y) = \frac{x}{x^2 + y^2}\] along a line making an angle of \( 45^\circ \) with the positive x-axis at the point \( (0, 2) \). Then, the value of \( 1/a^2 \) is:

The principal value of \( i^i \) is:

The function f(x) is said to be piecewise continuous, if it satisfies the following conditions (Dirichlet conditions):
(A) The function must have a finite number of maxima and minima.
(B) The function must have a finite number of infinite discontinuities, in a period of one oscillation.
(C) The function must have an infinite number of maxima and minima.
(D) The function must have a finite number of finite discontinuities, in a period of one oscillation
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	LIST I		LIST II
A.	d²y/dx² + 13y = 0		I. ex(c1 + c2x)
B.	d²y/dx² + 4dy/dx + 5y = cosh 5x		II. e2x(c1 cos 3x + c2 sin 3x)
C.	d²y/dx² + dy/dx + y = cos²x		III. c1ex + c2e3x
D.	d²y/dx² - 4dy/dx + 3y = sin 3x cos 2x		IV. e-2x(c1 cos x + c2 sin x)

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Given below are two statements:
Statement (I): If F is an irrotational vector field, then the angular velocity of the vector field is always greater than zero.
Statement (II): For a solenoidal vector function, the divergence is always zero.
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Match List I with List II:

List I	List II
A. Fraunhofer Diffraction	I. Interaction of the light waves from two different wave fronts.
B. Fresnel Diffraction	II. The distance between the source and the screen are effectively at infinite distance.
C. Interference of Light	III. It’s a phenomenon in which the wave vibrations are restricted to a particular direction in a plane.
D. Polarization of Light	IV. The source and screen or both are at finite distances from the aperture or obstacle.

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In a submarine telegraph cable, the speed of signaling varies as \( x^2 \log(1/x) \), where \( x \) is the ratio of the radius of the core to that of the covering. To attain the greatest speed, the value of this ratio is:

Match List I with List II:

LIST I	LIST II
A. Maxwell's First Equation	I. Modified Ampere's Law
B. Maxwell's Second Equation	II. Faraday's Laws of Electromagnetic Induction
C. Maxwell's Third Equation	III. Gauss Law in Electrostatics
D. Maxwell's Fourth Equation	IV. Gauss Law in Magnetostatics

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