AMIETE – ET/CS/IT (NEW SCHEME) – Code: AE51/AC51/AT51

Subject: ENGINEERING MATHEMATICS - I

Time: 3 Hours Max. Marks: 100

NOTE: There are 9 Questions in all.

· Question 1 is compulsory and carries 20 marks. Answer to Q.1 must be written in the space provided for it in the answer book supplied and nowhere else.

· Out of the remaining EIGHT Questions, answer any FIVE Questions. Each question carries 16 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or the best alternative in the following: (210)

a. The sum and product of the eigen value of the matrix A = are respectively

(A) 7 and 7 (B) 7 and 5

(C) 7 and 6 (D) 7 and 8

b. If where , then A is

(A) (B)

(C) (D)

c. If z = then is

(A) z (B) 2z

(C) tan z (D) sin z

d. The order of convergence of Newton-Raphson method is

(A) 0 (B) 1

(C) 2 (D) 3

e. The Wronskian of x and is

(A) (B)

(C) (D)

f. To transform into a linear differential equation with constant coefficients, the required substitution is

(A) x = sin t (B)

(C) x = log t (D)

g. The particular integral (PI) of differential equation is

(A) (B)

(C) (D)

h. Which of the following is ‘Rodrigue formula’

(A) (B)

(C) (D)

i. The value of is

(A) (B)

(C) (D)

j. The value of is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each question carries 16 marks.

Q.2 a. Find the maximum and minimum distances of the point (3, 4 , 12) from the sphere . (8)

b. Using differentiation under integral sign, evaluate . (8)

Q.3 a. Change the order of integration in and hence evaluate the same. (8)

b. Calculate, by double integration, the volume generated by the revolution of the cardioids about its axis. (8)

Q.4 a. Find the values of k for which the system of equations

has a non-trivial solution. (8)

b. Find the eigen values and eigen vectors of the matrix . (8)

Q.5 a. (i) Develop the Newton-Raphson method to find the square root a number N. (where N > 0)

(ii) Give the geometrical interpretation of Newton-Raphson method. (4+4=8)

b. Determine the largest eigen value and the corresponding eigen vector of the matrix A, using power method.

(8)

Q.6 Solve the following differential equations

(a) (8)

(b) (8)

Q.7 a. Solve by the method of variation of parameters . (8)

b. Solve . (8)

Q.8 a. Prove that (8)

b. Solve in series the equation . (8)

Q.9 a. Prove the following

(i)

(ii) (8)

b. Express in terms of Legendre polynomials. (8)