The document you’ve uploaded is a mathematics test for a subject on Boundary Value Problems, Transforms, and Probability, specifically for B.E. (Civil, EEE, Mech). Let me guide you through the questions:

PART A (5 * 2 = 10 marks)

1. Find the discrete time Fourier transform of $X[n]$ where $X[n] = \{1, 0 \leq n \leq 5; 0, \text{otherwise}\}$
   This question asks for the DTFT of the signal $X[n]$, which is non-zero for $n = 0$ to $n = 5$.

2. State Parseval’s identity in Fourier transform.
   Parseval’s identity relates the total energy of a signal in the time domain to its energy in the frequency domain.

3. Classify the PDE: $x u_{xx} + u_{yy} = 0$
   You are asked to classify the given partial differential equation based on its characteristics.

4. The solution which has an equal number of arbitrary constants and independent variables is known as ----
   The options are:

   • (A) General integral
   • (B) Complete integral
   • (C) Particular integral
   • (D) Singular integral

5. Which of the following equations represents Clairaut’s partial differential equation?
   The options involve equations of the form $z = p x + q y + f(p,q)$, and you are asked to select the correct one.

PART B (10+15 = 25 marks)

6. (a) Let $X[n]$ be a discrete-time sequence given by $X[n] = \{1, 2, 3, 0, -3, -2, -1\}$, where $X[0]$ is 1.

   i. The value of $X(e^{j\pi})$ is:

   • A) 1
   • B) 0
   • C) 6
   • D) 1/7
     This involves evaluating the discrete-time Fourier transform at a specific frequency.

   ii. The singular solution of $p = q = s$:
   Choose from the options which include equations involving $p$ and $q$.

   iii. Solve $p \tan x + q \tan y = \tan z$
   This asks you to solve a differential equation involving trigonometric functions.

Would you like detailed solutions for any of these questions, or do you want to focus on specific ones?

5 Related Questions:

1. How is the discrete Fourier transform (DFT) different from the discrete-time Fourier transform (DTFT)?
2. Can you explain how Parseval’s identity is used in signal processing?
3. What are the methods to classify second-order partial differential equations?
4. What is the significance of a singular integral in solving PDEs?
5. How do you derive Clairaut’s differential equation from general differential forms?

Tip:

When dealing with Fourier transforms, always ensure to differentiate between continuous and discrete domains to apply the right formulas.