List of top Geophysics Questions

Match List-I with List-II 

List-IBody / AxisList-IIMoment of Inertia
AM.I. of solid cone about its vertical axisIV\(\dfrac{3}{10}MR^2\)
BM.I. of solid cylinder about its own axisIII\(\dfrac{1}{2}MR^2\)
CM.I. of circular lamina about a diameterII\(\dfrac{1}{4}MR^2\)
DM.I. of annular ring about a diameterI\(\dfrac{1}{4}M(R^2+r^2)\)

Match List-I with List-II 

List-IConceptList-IIType / Relation
ATorqueIIRotational Motion
BForceIIILinear Motion
CRocketIVVariable Mass
DParallel LCR CircuitIRejector Circuit

Match List-I with List-II 

List-IConceptList-IICharacteristic / Application
AAvalanche BreakdownIVVery lightly doped regions
BZener BreakdownIIIDominant in heavily doped regions
CPoint Contact diodeIIMicrowave applications
DLED DiodeIGaAsP

Match List-I with List-II 

List-IConcept / DeviceList-IIDescription
AOptical PumpingIIPopulation Inversion
BHe-Ne LaserIFour Level Laser
CRuby LaserIIIThree Level Laser
DOptical ResonatorIVPair of optically plane parallel mirrors

Match List-I with List-II 

List-IThermodynamic Quantity / RelationList-IIExpression
AMaxwell Thermodynamic RelationIVMaxwell relation
BClausius-Clapeyron EquationIII\[ \frac{dP}{dT}=\frac{L}{T(V_2-V_1)} \]
CEnthalpyII\[ H=U+PV \]
DGibbs functionI\[ G=U-TS+PV \]

Match List-I with List-II 

List-IConcept / ExperimentList-IIDescription
AMichelson-Morley ExperimentIVTo detect ether and earth velocity relative to ether
BRaman effectIInelastic scattering of light by medium
CPlanck Radiation LawIIIResonators exchange energy in quanta
DEstimation of Temperature of SunIIStefan's or Wien's displacement law

Given Vector Field:

\[ \vec{A} = (bx+4y^2z)\hat{i} + (x^3\sin z-3y)\hat{j} - (e^x+4\cos(x^2y))\hat{k} \]

For a vector field to be solenoidal,

\[ \nabla \cdot \vec{A}=0 \]
 

Match List-I with List-II for the hyperbola 

\[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \]

List-IElementList-IICorresponding Value / Equation
ACoordinate of centreII\((0,0)\)
BOne coordinate of vertexI\((a,0)\)
CEquation of transverse axisIII\(y=0\)
DEquation of conjugate axisIV\(x=0\)

Match List-I with List-II 

List-IMatrix TypeList-IIProperty
AOrthogonal matrixIProduct of matrix and its transpose equals identity matrix
BSkew-Hermitian matrixIVDiagonal elements are either zero or pure imaginary
CReal skew symmetric matrixIIDiagonal elements are zero
DHermitian matrixIIIDiagonal elements are real

 

Assertion (A) and Reason (R)

Assertion (A):
\[ (\cos\theta+i\sin\theta)^{p/q} \] has \(q\) and only \(q\) different roots, where \(q\) is a positive integer.

Reason (R):
\[ (-1)^{1/3} \] has three different roots.

Assertion (A) and Reason (R)

Assertion (A):
If a polynomial \(f(x)\) is divisible by \((x-a)^m\), but not divisible by \((x-a)^{m+1}\), then \(a\) is called a root of multiplicity \(m\) of the equation \(f(x)=0\).

Reason (R):
\(x=-1\) is the root of multiplicity \(3\) of the equation \[ x^4+x^3-3x^2-5x-2=0. \]
 

Match List-I with List-II 

List-IFunctionList-II\(n^{\text{th}}\) Derivative Form
A\(e^{ax}\)IV\(a^n e^{ax}\)
B\(\log(ax+b)\)III\(\dfrac{(-1)^{n-1}(n-1)!a^n}{(ax+b)^n}\)
C\(\cos(ax+b)\)II\(a^n\cos\left(ax+b+\dfrac{n\pi}{2}\right)\)
D\(e^{ax}\cos(bx+c)\)I\(r^n e^{ax}\cos(bx+c+n\theta)\), 
where \(r=\sqrt{a^2+b^2}\), \(\theta=\tan^{-1}\left(\dfrac{b}{a}\right)\)

Principal Values of Logarithmic Functions

CaseComplex NumberPrincipal Value
A\(\log(-2)\)\(\log 2 + i\pi\)
B\(\log(5i)\)\(\log 5 + \dfrac{i\pi}{2}\)
C\(\log(\sqrt{3}-i)\)\(\log 2 - \dfrac{i\pi}{6}\)
D\(\log(2-3i)\)\(\dfrac{1}{2}\log 13 - i\tan^{-1}\left(\dfrac{3}{2}\right)\)

 

Match List-I with List-II 

List-I\(F(x)\)List-IIParticular Integral Rule
A\(F(x)=e^{ax}\)III\(\dfrac{e^{ax}}{f(a)}, \quad f(a)\neq 0\)
B\(F(x)=\sin ax\)II\(\dfrac{\sin ax}{f(-a^2)}\)
C\(F(x)=xe^{ax}\)I\(\left(\dfrac{x}{f(a)}-\dfrac{f'(a)}{[f(a)]^2}\right)e^{ax}, \quad f(a)\neq 0\)
D\(F(x)=\cos ax\)IV\(D^2\) is replaced by \(-a^2\) in \(f(D)\)
Consider the following statements related to a matrix:
A. Inverse of a matrix is unique if it exists.
B. A matrix \(A\) is non-singular if \(|A|=0\).
C. Matrix \[ A=\begin{pmatrix} \cos x & \sin x -\sin x & \cos x \end{pmatrix} \] is orthogonal.
D. Matrix \[ A=\begin{pmatrix} 0 & -2 & -8 2 & 0 & -4 8 & -4 & 0 \end{pmatrix} \] is skew symmetri
C.
The function of two variables is maximum if:
A. \(rt-s^2>0\),
B. \(r<0\),
C. \(rt-s^2<0\),
D. \(r>0\), E. \(rt-s^2=0,\ r<0\).
If the roots of the equation \[ x^3+3px^2+3qx+r=0 \] are in geometric progression, then:
\[ (1+i\sqrt{3})^{10}+(1-i\sqrt{3})^{10} \] equals to:
Expansion of following functions are given below:
A. \[ \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \]
B. \[ \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots \]
C. \[ \sin x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots \]
D. \[ \cos x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \] Choose the correct answer from the options given below:
Two forces of magnitude \(8\ \text{N}\) and \(6\ \text{N}\) act at a point making an angle of \(90^\circ\) between them. The magnitude of resultant force is:
Following are the parts of general solution of \[ \frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=x\cos x: \]
A. \((C_1+C_2x)e^{-x\),
B. \((C_1+C_2x)e^x\),
C. \(\frac{x\sin x}{2}+\frac{\cos x}{2}+\frac{\sin x}{2}\),
D. \(\frac{x\sin x}{2}+\frac{\cos x}{2}-\frac{\sin x}{2}\).}
Match List-I with List-II. List-I:
A. Node,
B. Center,
C. Saddle point,
D. Spiral when eigenvalues are complex with negative real parts. List-II: I. Stable, II. Unstable, III. Asymptotic stable, IV. Asymptotic stable or unstable.
The integral equation \[ y(x)=x+\int_0^3 \cos(x-t)y(t)\,dt \] is a:
The orthogonal trajectories of the cardioids \[ r=a(1-\cos\theta) \] is:
The linear differential equation \[ a_0\frac{d^ny}{dx^n}+a_1\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_ny=F(x) \] is non-homogeneous if: