For the function at the point (1, 2, -1), find its rate of change with distance in the direction
Therefore, rate of change with distance in the direction of
Evaluate the line integral (exy + cos x sin y) dx + (ex + sin x cos y)dy, around the curve 'C'
Note: . then vector function is conservative and hence line integral (work clone) along a closed path is zero)
If a unitary matrix U is written as A + iB, where A and B are Hermitian matrix with non degenerate eigenval ues. then
Since. U is unitary matrix.
or (A + iB) (A - iB) = I (since, A and B are Hermitian = B)
A2 + B2 + i(BA - AB) = I
Now. equating real and imaginary
Part. A2 + B2 = I and BA - AB = 0
If (A)nxn is a square matrix, and elements of matrix A(aij) are such that
then value of det (eA) is equal to
We know that, det (eA) = etrace A
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 + Bex + Ce2x, then the value of A + B - C is
(upto two decimal places)
D2 - 3D + 2 = 0
(D - 2) (D - 1) = 0
Therefore, complementary function Bex + Ce2x and
The range of values ofz, for which the following complex power series converges, is Izl < A, then A is .................. (answer should be an integer)
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
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
If is the fourier transformation of then the fourier trans form of f ' (t) is
Fourier transfonn of f' (t)
If a complex function f (z) lias a pole of order m at z = z0. then f'(z) has a pole of order
where g is analytic at z0 and g(z0) ≠ 0
where h(z) = (z - z0) g'(z) - mg(z) is analytic at z = z0 and h(z0) = - mg (z0) ≠ 0
∴ f' (z) has a pole of order (m + 1) at z = z0.
The inverse laplace transfonn of
Using partial fraction.
If the fourier series for Isin θ| in the range -π < θ < π is given by then the value of A is .........
Value of the integral
Poles are at z = + 2
Therefore, residue at 2 = 2
If the fourier expansion of then the value of ar if
If z = xy, where y = tan-1 t and x = sin t, then the value of dz/dt is equal to
If y (x) is the solution of the differential equation with y(1) = -1, then
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).
then the value of integral where ‘S’ is the surface given by z = 12, x2 - y2 < 25 (taken anticlockwise), is
The fourier complex transfoem of f(x), where
Fourier transform of f (x)
(integrating by parts)
then which of the following statements is true about A and B
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.
The Laplace transform of f (t) = t2 cos at is
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)