CSIR NET-JRF Physical Sciences Quiz

A set of 100 data points yields an average x̄ = 9 and a standard deviation σx = 4. The error in the estimated mean is closest to

• 3.0

• 0.4

• 4.0

• 0.3

An integral is given by ∬ exp(-2xy + a) dx dy, where a is a real parameter. The full range of values of a for which the integral is finite, is

• a < -2

• -2 < a < 2

• -1 < a < 1

• -1 ≤ a ≤ 1

Vorticity of vector field B is defined as V = ∇ × B. Given B = kxyz, where k is a constant, which one of the following is correct?

• Vorticity is a null vector for all finite x, y, z

• Vorticity is parallel to the vector field everywhere

• The angle between vorticity and vector field depends on x, y, z

• Vorticity is perpendicular to the vector field everywhere

The matrix A is given by A = [[1, 2, 3], [0, 3, 2], [0, 0, 2]]. The eigenvalues of A^3 + 5A^2 + 6A + 2I, where I is the identity matrix, are

• 4, 9, 27

• 1, 9, 44

• 1, 110, 8

• 4, 110, 10

Probability density function of a variable x is given by P(x) = δ(x - a) + δ(x + a). The variance of x is

• 2a

• 0

• 2a^2

• 2a^2

An integral transform F of a function f(x) can be regarded as a result of applying an operator F to the function such that F(f(x)) = ∫ f(y) e^(-ixy) dy. If I is the identity operator, then the operator F is given by

• (4/2)I

• (2/2)I

• I

• (2/2)I

The integral I = ∫ (1/(x^2 + 1)) dx from 0 to 2 is estimated using Simpson's 1/3rd rule with a grid value of h = 0.5. The difference (I_estimated - I_exact) is closest to

• 0.007

• 0.001

• 0.0007

• -0.005

The following four matrices form a representation of a group. Which of the following represents the multiplication table for the same group?

• IA B C IIA B C A A IC B B B C A I C C BIA

• IA B C IIA B C A A B CI B B CIA C CIA B

• I A B C I I A B C A A C I B B B I C A C C B A I

• I A B C I I A B C A A I C B B B C I A C C B A I

The general solution for the second order differential equation dy/dx - 2sin(x) = 0 will be

• C1 e^(x) sin(x) + C2 e^(x) cos(x)

• C1 e^(x) sin(x) - C2 e^(x) cos(x)

• C1 e^(x) sin(x) + C2 e^(x) cos(x)

• C1 e^(x) sin(x) + C2 e^(x) cos(x)

A uniform plane square sheet of mass m is centered at the origin of an inertial frame. The sheet is rotating about an axis passing through the origin. At an instant when all its vertices lie on x and y axes, the angular momentum is L = I0 (i + j + k) ω, where I0 is the moment of inertia about the x axis. At this instant, the angular velocity of the sheet is

• (1) (0, 0, 2) (2) (0, 0, 2) (3) (0, 2, 0) (4) (0, 0, 1)

• (1) (0, 0, 2) (2) (0, 0, 2) (3) (0, 2, 0) (4) (0, 0, 1)

• (1) (0, 0, 2) (2) (0, 0, 2) (3) (0, 2, 0) (4) (0, 0, 1)

• (1) (0, 0, 2) (2) (0, 0, 2) (3) (0, 2, 0) (4) (0, 0, 1)