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Question:

Using the method of Laplace transforms, solve the following initial value problem:

$$y'' - 3y' + 2y = 6e^{-t}; \quad y(0) = 3, \quad y'(0) = 3$

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Question:

Determine the Laplace transform of $f(t) = t^2 e^{2t}$ .

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Question:

Determine the Fourier transform of the function :

$$f(t) = \begin{cases} \sin 3t & -\pi \le t \le \pi \\ 0 & \text{otherwise} \end{cases}$

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Question:

Using the method of residues, show that:

$$\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2}$

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Question:

Obtain the Laurent series expansion of the function $\frac{e^z}{(z-2)^2}$ about the singularity $z = 2$ .

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Question:

Evaluate the integral

$$\oint_C \frac{5z^2 - 3z + 2}{(z - 1)^3} dz$
where C is any simple closed curve enclosing $z = 1$ .

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Question:

Show that the function $\omega = \sin z$ satisfies the Cauchy-Riemann and Laplace equations.

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Question:

Show that the roots of the equation $z^4 - 1 = 0$ form a cyclic group of order 4.

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Question:

Define covariant and contravariant tensors of rank two. Show that the velocity and acceleration are contravariant vectors.

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Question:

Identify the conic section whose equation is

$$5x^2 - 4xy + 5y^2 = 4$

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Question:

Show that every eigenvalue of a unitary matrix is of unit modulus.

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Question:

Determine the eigenvalues and eigenvectors of the following matrix A:

$$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 3 & 2 & 3 \end{bmatrix}$

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Question:

Define skew-hermitian matrix, orthogonal matrix and unitary matrix with an example of each.

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Question:

रोड्रिगेज़ सूत्र का प्रयोग करके हर्मिट बहुपद H₄(x) का व्यंजक प्राप्त करें और सिद्ध करें कि $H_3'(x) = 6 H_2(x)$ ।

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Question:

फलन $f(x) = x^4 - 1$ का प्रसार $\sum_{k=0}^{\infty} A_k P_k(x)$ के रूप की श्रेणी में करें।

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Question:

सिद्ध करें कि :

$$J_0(x) - J_2(x) = 2 \frac{d}{dx} [J_1(x)]$

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Question:

सिद्ध करें कि P_l(x), अंतराल (-1, 1) में P_k(x) के लांबिक है।

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Question:

लाप्लास रूपांतरण विधि का प्रयोग करके निम्नलिखित आदि मान समस्या का हल प्राप्त करें :

$$y'' - 3y' + 2y = 6e^{-t}; \quad y(0) = 3, \quad y'(0) = 3$

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Question:

फलन $f(t) = t^2 e^{2t}$ का लाप्लास रूपांतर निर्धारित करें।

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Question:

निम्नलिखित फलन का फूरिये रूपांतर निर्धारित करें :

$<div style=$ $f(t)=\begin{cases}\sin&space;3t,&-\pi\le&space;t\le\pi,\\0,&\text{otherwise}.\end{cases}$ ">

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