List of top Geophysics Questions asked in CUET (PG)

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:

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.
Choose the most appropriate answer from the options given below:

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

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:

The value of the integral $$ \oint_C \left[ (x^3 + xy) \, dx + (x^2 - y^3) \, dy \right] $$ where $C$ is the square formed by the lines $x = \pm 1$, $y = \pm 1$, is: 

	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)

Choose the correct answer from the options given below:

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

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:

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:

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:

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

	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:

	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:

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

The shortest distance between the lines
$$ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} $$ and
$$ \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5} $$ 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.
Choose the most appropriate answer:

The Hamiltonian’s equation of motion is

The equation of motion for a compound pendulum is:

Find the amplitude of the electric field in a parallel beam of light of intensity $2.0 \, \text{W/m}^2$:

	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:

Displacement current is not a conventional current but it is:

Which of the following statements are true for ideal PN junction current voltage relationship?
(A) The abrupt depletion layer approximation applies.
(B) The Maxwell-Boltzmann approximation applies to carrier statistics.
(C) The concept of low injection applies.
(D) The total current is not constant throughout the entire pn structure.
Choose the correct answer from the options given below:

Consider phosphorus doping in silicon, for $T = 300 \, \text{K}$, at concentration $N_d = 10^{16} \, \text{cm}^{-3}$. The fraction of total electrons still in the donor states is: