🏫 National Institute of Technology Silchar

Click to view the Question Paper

Answer any 8 questions

Qn	Details	Marks	CO
1	Obtain the half-range Fourier sine series and half-range Fourier cosine series expansion of the function shown in the figure below: y = f(x) (triangular wave from 0 to L with peak at L/2)	10	CO1
2(a)	Find the Fourier sine integral of f(x) = e-kx, where k > 0 and x > 0	5	CO1
2(b)	Find the inverse Fourier sine transformation f(x), of (e-aλ)/λ , a ≥ 0. Hence find the value of f(x) when a = 0	5	CO1
3(a)	Eliminate the arbitrary functions f(x) and g(y) from z = yf(x) + xg(y) to form a partial differential equation	5	CO2
3(b)	Find the general solution of (y - z)∂z/∂x + (x - y)∂z/∂y = z - x	5	CO2
4(a)	Find a complete integral of (p² + q²)y = qz, where p = ∂z/∂x and q = ∂z/∂y	5	CO2
4(b)	Find a general solution of ∂²z/∂x² + ∂²z/∂x∂y - 6∂²z/∂y² = y cos x	5	CO2
5(a)	Find the general solution of the following PDE: ∂²z/∂x² + 2∂²z/∂x∂y - 8∂²z/∂y² = √(2x + 3y)	5	CO2
5(b)	A tightly stretched string with fixed end points x = 0 and x = 5 is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points a velocity λx(5 - x), find the displacement of the string at any distance x from one end at any time t (complete derivation is not necessary)	5	CO1
6	With complete derivation find the temperature in a bar of length l whose two ends are kept at 0°C, lateral surface is insulated and the initial temperature is sin(nπx/l)	10	CO1
7(a)(i)	A random variable X has the following probability function with values 1, 2, 3, 4, 5, 6, 7 and P(X) as k, 2k, 2k, 9k-1, k², 2k², 3k²+k. Determine the value of k	6	CO3
7(a)(ii)	Evaluate P(X < 4), P(X ≥ 4), and P(0 < X < 4)		CO3
7(a)(iii)	If P(X ≤ a) > 0.5, find the minimum value of a. Find the distribution function of X ?		CO3
7(b)	The probability that a man aged 60 will live to be 70 is 0.65. What is the probability that at least 7 men out of 10 men who are 60 now will live to be 70?	4	CO3
8(a)(i)	If the probability that an individual suffers a bad reaction from a certain injection is 0.001, using Poisson distribution determine the probability that out of 2000 individuals exactly 3 ?	5	CO3
8(a)(ii)	more than 2 individuals		CO3
8(a)(iii)	none		CO3
8(a)(iv)	more than one individual will suffer a bad reaction?		CO3
8(b)	In a normal distribution, 31% of items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution	5	CO3
9(a)	Two lines of regression are given by 5y - 8x + 17 = 0 and 2y - 5x + 14 = 0. If σy² = 16, find (i) the mean values of x and y (ii) σx² (iii) the coefficient of correlation between x and y	5	CO3
9(b)	Employ the method of least-squares to fit a parabola y = a + bx + cx² to the following data: x: -1, 0, 0, 1 and y: 2, 0, 1, 2	5	CO3

Click to view the Question Paper

Answer any 5 Questions

Q.N.	Questions	Marks
1. (a)	Find the general solution of (D² - 4DD' + 4D'²)z = ey + 2x, where D = ∂/∂x and D' = ∂/∂y	5
(b)	Find the general solution of (D² + 2DD' + D'² - 2D - 2D')z = sin(2y + x), where D = ∂/∂x and D' = ∂/∂y	5
2.	A thin rectangular homogeneous thermally conducting plate lies in the xy-plane defined by 0 ≤ x ≤ a, 0 ≤ y ≤ B. The edge y = 0 is held at the temperature T̄x (x - a), where T̄ is a constant, while the remaining edges are held at 0°. The other faces are insulated and no internal sources and sinks are present. By using separation of variable method derive all the possible solutions of this problem and find the steady state temperature in the plate.	10
3.(a)	A tightly stretched string with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating by giving to each of its points a velocity Ax(l - x), find the displacement of the string at any distance x from one end at any time t.	5
(b)	Find the temperature in a laterally insulated bar of length 2 cm whose two ends are kept at 0°C and if the initial temperature is given by (sin πx/2 + 3 sin 5πx/2)°C.	5
4. (a)	A random variable X has the following probability function: x 1 2 3 4 5 6 7 f(x) k 2k 2k 3k k² 2k² k + 7k² (i) Determine the value of k. (ii) Evaluate P(X < 6), P(X >= 6), and P(0 < X < 5). (iii) If P(X ≤ a) > 0.5, find the minimum value of a. (iv) Find the distribution function of X?	6
(b)	The probabilities of 1 and 2 successes are 0.4096 and 0.2048 respectively in a binomial distribution consisting of 5 independent trials. Find the parameter 'p' of the distribution, and the probability that there are 3 successes.	4
5. (a)	In a Poisson distribution, probability corresponding to 3 successes is 2/3 times the probability corresponding to 4 successes. Find the mean and standard deviation of the distribution	4
(b)	Suppose that the weight (X) in pounds, of 40-year-old men follows normal distribution with mean μ pounds and standard deviation σ pounds. If 5% of this population is heavier than 214 pounds and 95% of this population is heavier than 100 pounds. Find the values of the mean μ and standard deviation σ of this distribution. Given that 1/√(2π) ∫01.645 e-z²/2 dz = 0.95, where z denotes the standard normal variable.	6
6. (a)	Fit a parabola by using the method of least square to the following bivariate data: (2,4), (3,7), (5,15), (7,28), (9,45).	5
(b)	Consider the following bivariate data of individuals: Height(X)(in cm) 160 165 170 175 180 185 Weight (Y)(in Kg) 60 70 75 80 90 95 Find the appropriate equation of the regression line to predict the weight of an individual, whose height is 173 cm.	5

Click to view the Question Paper

Answer any 5 Questions

S No	Questions	Marks	CO
1. (a)	Obtain a half range Fourier cosine series for f(x) = {-kx, 0 ≤ x < l/2; k(l - x), l/2 ≤ x ≤ l} Hence deduce the sum of the series 1/1 + 1/3² + 1/5² + ... ∞	5	CO-2
(b)	Find Fourier sine integral of f(x) = e-kx, where k > 0 and x > 0.	5	CO-2
2. (a)	Find the inverse Fourier sine transformation, f(x), of e-aλ/λ, a ≥ 0. Hence find the value of f(x) when a = 0.	3+1	CO-2
(b)	Write the general form of Lagrange's linear PDE of first order and the corresponding auxiliary equations (AE) for a function of n variables, u = u(x₁, x₂, ..., xₙ). Hence, solve the following PDE (z² - 2yz - y²)p + (xy + zx)q = (xy - zx), where p = ∂z/∂x and q = ∂z/∂y	2+4	CO-1
3. (a)	Find a complete integral of p²x + q²y = z.	5	CO-1
(b)	Find the general solution of the following PDE: (D + D')(D² + D'² + 2DD' - 4D - 4D' + 3)z = ex+ysin(2x + y), where D = ∂/∂x, D' = ∂/∂y	5	CO-1
4.	Using method of separation of variables, derive the general solution of the partial differential equation given by: u_tt = c²u_xx; 0 ≤ x ≤ L, t > 0 subject to the boundary conditions u(0,t) = u(L,t) = 0; t ≥ 0, as well as initial conditions u(x,0) = f(x), u_t(x,0) = g(x), 0 ≤ x ≤ L.	10	CO-1
5.(a)	Out of 800 families with 5 children each, how many would expect to have (i) 3 boys, (ii) 5 girls and (iii) either 2 or 3 boys; assuming the births of boys and girls occur with equal probability.	6	CO-3
(b)	The probability mass function of a random variable X is given by P(i) = (cαi) / (i!)   for i = 0, 1, 2, ..., and a being some positive value. Find the value of c. hence find, (i) P(X = 0) and (ii) P(X > 2).	4	CO-3
6.(a)	Suppose that the height (X) in inches, of 25-year-old men is a normal random variable with mean μ = 70 inches. If P(X > 79) = 0.025, what is the standard deviation of this normal variable? Given that 1/√(2π) ∫01.96 e-z²/2 dz = 0.975, where z denotes the standard normal variable.	4	CO-3
(b)	Fit a straight line to the following data and estimate the value of y corresponding to x = 6. x 0 5 10 15 20 25 y 12 15 17 22 24 30	6	CO-3