Solve your IGNOU Doubts

Question:

Consider a triangle T in the z-plane with vertices at i, 1 - i, 1 + i. Determine the triangle T₀ into which T is mapped under the transformations

$$w = iz + 2 - i$

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Question:

Evaluate the integral

$$\int_0^{2\pi} \frac{d\theta}{1 + p \cos \theta}$

by the method of residues when -1 < p < 1.

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Question:

Evaluate the value of the contour integral $6 \oint_C \frac{z \, dz}{(z - 1)(z^2 + 9)}$ where C is a circle defined by $|z| = 4$ .

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Question:

Obtain the Laurent series expansion of $\frac{e^z}{(z - 1)^2}$ about $z = 1$ . Determine the type of singularity and the region of convergence.

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Question:

Show that the Series $\sum_{n=1}^{\infty} z^n (1 - z)$ converges for |z| < 1 and find its sum.

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Question:

Calculate the value of the integral $\oint_C \frac{\cos z}{z} dz$ when C is the circle $|z| = 2$ .

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Question:

Two objects of equal mass are connected by a three spring with spring constants as shown in figure. Obtain the normal mode frequencies.

Image ignouassignments-ignouacademy-com--p-solve-32526

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Question:

The orbit of a particle in a central force field is given by $r = k e^{\alpha \theta}$ , obtain the force law.

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Question:

Locate and name the singularity of the function:

$$\frac{z^2 - 2z}{z^2 + 2z + 2}$

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Question:

Obtain the analytic function whose real part is

$$u(x, y) = e^x \cos y$

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Question:

A central potential is given as V(r):

(i) Obtain the criterion for a stable circular orbit.

(ii) If the central potential is given as $V(r) = -k/r^n, n \geq 1$ . For what values of n are the circular orbits stable.

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Question:

Define covariant and cotravariant tensors of rank 2. Prove that $a_i = g_{ij}v^j$ transform covariantly, where g_ij are the components of the matrix tensor of rank 2 and vⁱ the components of a contravariant vector.

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Question:

Obtain the eigenvalues and eigenvectors of matrix A:

$$A = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}$

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Question:

The first Pauli matrix is

$$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
calculate $U_1(\theta) = \exp(i\theta \sigma_1) = 1 + i\theta \sigma_1 - \frac{\theta^2}{2} \sigma_1^2 - \dots$
For real $\theta$ , show that U₁ is unitary and has determinant

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Question:

Consider a simple pendulum of mass m whose length changes with time as $l = l_0 t^2, (x^2 + y^2 = l_0^2 t^4)$ . Obtain the Lagrange equation of motion, the generalised momentum and the energy function for the system.

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Question:

Show that the following vectors

$$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$

are linearly independent.

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Question:

Obtain orthogonality relation for Hermite polynomials using the generating function:

$$e^{2xt-t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}$

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Question:

Using the generating function for Bessel functions of the first kind and integral order

$$g(x, t) = \exp \left[ \frac{x}{2} \left( t - \frac{1}{t} \right) \right] = \sum_{n=-\infty}^{\infty} J_n(x) t^n$

Obtain the recurrence relation

$$J_{n-1}(x) + J_{n+1}(x) = 2 \frac{n}{x} J_n(x)$

Also using the generating function show that

$$J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \dots + 2J_{2k}(x) + \dots = 1$

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Question:

A pendulum of mass m is pivoted to a support of mass M sliding without friction along x-axis as shown in the figure. Obtain the Lagrange equation of motion for the given system.

Image ignouassignments-ignouacademy-com--p-your-68791

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Question:

Use the method of separation of variables to reduce the Laplace's equation $\nabla^2 f = 0$ into three ODEs.