List of top Geophysics Questions

Given below are two statements:
Statement (I): The nth derivative of the function \( e^x \cos x \cos 2x \) is:
\[\frac{e^x}{2} \big[ (10^n) \cos(3x + n \tan^{-1} 3) + (2^n) \cos(x + \frac{n \pi}{4}) \big]\]
Statement (II): The nth derivative of the function \( x \cos 2x \cos 3x \) is:
\[\frac{1}{2} \big[ (2^n) \cos(2x + \frac{n \pi}{2}) + (4^n) \cos(4x + \frac{n \pi}{2}) + (6^n) \cos(6x + \frac{n \pi}{2}) \big]\]
In light of the above statements, choose the most appropriate answer from the options given below:

	LIST I (Type of the Matrix)		LIST II (Property)
A.	Symmetric Matrix		I. aij = aji, for values of i and j
B.	Hermitian Matrix		II. aij = āji, for values of i and j
C.	Skew-Hermitian matrix		III. aij = -āji, for values of i and j
D.	Skew-Symmetric matrix		IV. aij = -aji, for values of i and j

Choose the correct answer from the options given below:

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.
Choose the most appropriate answer:

In order to introduce a phase difference of $\pi/2$ in a Quarter wave plate, the thickness of the crystal should have a value of... (Given: wavelength = $5893 \times 10^{-10} \, \text{m}$, refractive index for O-ray is $1.65836$ and refractive index for E-ray is $1.48641$)

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 four satellites are lying close to the earth at distances $h_1, h_2, h_3$, and $h_4$ meters respectively, away from the center of the earth. If the values of $h_i$'s are given in terms of $R$ (radius of the earth), write the time period of the satellite in increasing order:
$h_1 = R/3$
$h_2 = R/4$
$h_3 = R/5$
$h_4 = R/2$
Choose the correct answer from the options given below:

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:

If the radius of Earth becomes half of its present value, with its mass remaining the same, the duration of one day will become:

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
Choose the correct answer from the options given below:

The Hamiltonian’s equation of motion is

The value of curl (\(\text{grad } f\)), where \( f = x^2 - 4y^2 + 5z^2 \), is:

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

The image of the point (1 3 4) in the plane 2x − y + z + 3 = 0 is

If \[ \theta = r e^{-x^2}, \] then for what value of \( n \), the following holds: \[ \frac{1}{2} \left( \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial r^2} \right) = \frac{\partial \theta}{\partial x}. \]

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:

	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

Choose the correct answer from the options given below:

	LIST I (Name of the surface)		LIST II
A.	Ellipsoid		I. x²/a² + y²/b² + z²/c² = 1
B.	Hyperboloid of one sheet		II. x²/a² - y²/b² - z²/c² = 1
C.	Hyperboloid of two sheets		III. -x²/a² + y²/b² + z²/c² = 1
D.	Central conicoid		IV. x²/a² + y²/b² + z = 1

Choose the correct answer from the options given below:

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 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
Choose the most appropriate answer from the options given below

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 $y''(0) = 6$.
Choose the correct answer from the options below:

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
Choose the correct answer from the options given below

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 \( y = \frac{1}{x} \).
Choose the most appropriate answer from the options given below: