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QUESTION: 1

For the function at the point (1, 2, -1), find its rate of change with distance in the direction

Solution:

Therefore,

Therefore, rate of change with distance in the direction of

QUESTION: 2

Evaluate the line integral (e^{x}y + cos x sin y) dx + (e^{x} + sin x cos y)dy, around the curve 'C'

Solution:

Note: . then vector function is conservative and hence line integral (work clone) along a closed path is zero)

QUESTION: 3

If a unitary matrix U is written as A + iB, where A and B are Hermitian matrix with non degenerate eigenval ues. then

Solution:

Since. U is unitary matrix.

or (A + iB) (A - iB) = I (since, A and B are Hermitian = B)

A^{2} + B^{2} + i(BA - AB) = I

Now. equating real and imaginary

Part. A^{2} + B^{2} = I and BA - AB = 0

QUESTION: 4

If (A)_{nxn} is a square matrix, and elements of matrix A(a_{ij}) are such that

then value of det (e^{A}) is equal to

Solution:

We know that, det (e^{A}) = e^{trace} A

*Answer can only contain numeric values

QUESTION: 5

If solution y(x), of the differential equation subjected to the boundary conditions y(0) = 2, y'(0) = 1 has the form y(x) = Ae^{-x} + Be^{x} + Ce^{2x}, then the value of A + B - C is

(upto two decimal places)

Solution:

D^{2} - 3D + 2 = 0

(D - 2) (D - 1) = 0

Therefore, complementary function Be^{x} + Ce^{2x} and

*Answer can only contain numeric values

QUESTION: 6

The range of values ofz, for which the following complex power series converges, is Izl < A, then A is .................. (answer should be an integer)

Solution:

QUESTION: 7

If the matrix can be diagonalised by a transformation of the form S* AS = A', where S has the normalized eigenvectors of A as its columns, then A' is

Solution:

A' will be a diagonal matrix, whose diagonal elements are the eigenvalues of A.

Therefore, to find eigenvalues of A.

Therefore, eigenvalues are λ = - 2, - 2, 4

QUESTION: 8

If is the fourier transformation of then the fourier trans form of f ' (t) is

Solution:

Fourier transfonn of f' (t)

QUESTION: 9

If a complex function f (z) lias a pole of order m at z = z_{0}. then f'(z) has a pole of order

Solution:

where g is analytic at z_{0} and g(z_{0}) ≠ 0

where h(z) = (z - z_{0}) g'(z) - mg(z) is analytic at z = z_{0} and h(z_{0}) = - mg (z_{0}) ≠ 0

∴ f' (z) has a pole of order (m + 1) at z = z_{0}.

QUESTION: 10

The inverse laplace transfonn of

Solution:

Using partial fraction.

*Answer can only contain numeric values

QUESTION: 11

If the fourier series for Isin θ| in the range -π __<__ θ __<__ π is given by then the value of A is .........

Solution:

*Answer can only contain numeric values

QUESTION: 12

Value of the integral

Solution:

Poles are at z = __+__ 2

Therefore, residue at 2 = 2

*Answer can only contain numeric values

QUESTION: 13

If the fourier expansion of then the value of ar if

Solution:

QUESTION: 14

If z = x^{y}, where y = tan^{-1} t and x = sin t, then the value of dz/dt is equal to

Solution:

QUESTION: 15

If y (x) is the solution of the differential equation with y(1) = -1, then

Solution:

Now, y (1) = -1, put x = 1, y = -1

∴ for y to be defined log x must be defined i.e.. x > 0

Also when log x - 1 = 0, log x = 1

So in the range 0 < x < 3 (option (b)). there will be a point x = 2.73 at whichy is not defined.

Also at x = e : y → ∞ (blow up).

QUESTION: 16

then the value of integral where ‘S’ is the surface given by z = 12, x^{2} - y^{2} __<__ 25 (taken anticlockwise), is

Solution:

∴

QUESTION: 17

The fourier complex transfoem of f(x), where

Solution:

Fourier transform of f (x)

(integrating by parts)

QUESTION: 18

then which of the following statements is true about A and B

Solution:

Since A is upper trinagular and B is lower triangular matrix. Hence, their eigenvalues are the principle diagonal elements.

Therefore, eigenvalues of A are 1, 0, 2 and eigenvalues of B are -1,1, 3

Since eigenvalues of A and B are all distinct, hence their corresponding eigenvectors will be linearly independent.

Hence, both A and B are digonalizable.

QUESTION: 19

The Laplace transform of f (t) = t^{2} cos at is

Solution:

*Answer can only contain numeric values

QUESTION: 20

The value of the integral where C is closecl contour defined by the equation 2 |z - 1| - 3 = 0, traversed in the clockwise direction, is _______________________ (answer should be an integers)

Solution:

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