List of top Geophysics Questions asked in Common University Entrance Test

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

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

The general value of \( \log(1 + i) + \log(1 - i) \) is:

Statement (I): The determinant of a matrix A and its transpose A^T are equal.
Statement (II): The determinant of the product of two matrices A and B is the product of their determinants.
In light of the above statements, choose the most appropriate answer from the options given below:

If \( (\sqrt{3} + 1)^n + (\sqrt{3} - 1)^n = 4 \), then the value of \( n \) is:

In the system of linear equations AX = B, if A is a singular matrix and B is a null matrix, then which of the following is correct?

The surface integral \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \] where \( \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( z = 0 \) and \( z = 4 \), equals:

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:

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

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:

The asymptote of the spiral \( r = \frac{\theta}{\sin \theta} \) is:

For Fresnel half period zones:
(A) The amplitude of disturbance is directly proportional to the area of the zone.
(B) The radii of half period zones are directly proportional to the square root of the natural number.
(C) The area of each zone is directly proportional to the wavelength of the light.
(D) The radii of half period zones are directly proportional to the square root of the wavelength of the light.
Choose the correct answer from the options given below:

	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:

If A is a skew-symmetric matrix of odd order, then the determinant of A is:

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

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:

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

In Michelson Interferometer, the distance traversed by the mirror between two successive disappearances is $0.289 \, \text{mm}$. The difference between the wavelengths of two lines is (Assume the wavelength of one line is $5890 \, \text{\AA}$):

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:

Match List I with List II:

List I	List II
(A) The linear momentum of the system remains constant	(IV) The net external force acting on a system of particles is zero
(B) The angular momentum of the system remains constant	(III) The external torque acting on a system of particles is zero
(C) Inertial frame	(I) The frames relative to which an unaccelerated body appears unaccelerated
(D) Non-inertial frame	(II) The frames relative to which an unaccelerated body appears accelerated

Choose the correct answer from the options given 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 shape of air film formed between the plano-convex lens and the glass slab in Newton’s ring experiment is