List of top Geophysics Questions on Mathematics

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: 

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 curl (\(\text{grad } f\)), where \( f = x^2 - 4y^2 + 5z^2 \), 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 volume of the solid standing on the area common to the curves \( x^2 = y, y = x \) and cut off by the surface \( z = y - x^2 \) is:

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

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:

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

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?

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:

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

	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: