AMIETE – ET/CS/IT (OLD SCHEME)

Code: AE01/AC01/AT01 Subject: 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: (2 x 10)

a. The value of limit is

(A) 0 (B) 1

(C) -1 (D) limit does not exist

b. If then

(A) (B)

(C) (D)

c. The value of the integral is

(A) (B)

(C) (D) 0

d. The value of integral where R is the region given by R: 2y≤x≤2, 0≤y≤1 equal to

(A) (B)

(C) (D)

e. The solution of the differential equation y dx – x dy +e^1/x dx = 0, is given by

(A) y + x e^1/x = cy (B) y + x e^2/x = cy

(C) y + x e^1/x = cx (D) x + x e^1/x = cy

f. The particular integral of the differential equation is

(A) (B)

(C) (D)

g. The product of the eigen values of is equal to

(A) 6 (B) -8

(C) 4 (D) -6

h. Let T be a linear transformation from R³ into R² defined by the relation Tx=Ax, A=. The value of Tx when x is given by [3 4 5]^T

(A) (B)

(C) (D)

i. The value of as a polynomial in x is equal to

(A) 3x²-4x-4 (B) 3x²+4x-4

(C) 3x²-4x+4 (D) 3x²+4x+4

j. The value of the is

(A) (B)

(C) (D)

Answer any FIVE Questions out of EIGHT Questions.

Each Question carries 16 marks.

Q.2 a. Show that the function is continuous at (0, 0) but its partial derivatives f_x and f_y does not exist at (0, 0). (8)

b. If where x^-1+y^-1+z^-1=1, show that the stationary value of u is given by . (8)

Q.3 a. Expand f(x, y) = tan^-1(y/x), in powers of (x-1) and (y-1) upto third degree terms. Hence compute f (1.1, 0.9) approximately. (8)

b. Evaluate where D is the region bounded by x = 0,

y = 0, x + y =1, using the transformation x + y = u, y = uv. (8)

Q.4 a. Solve the differential equation . (8)

b. Solve by the method of undetermined coefficients, . (8)

Q.5 a. Find the general solution of the equation . (8)

b. Reduce the matrix to the diagonal form. (8)

Q.6 a. If the following system has non-trivial solution, prove that a + b + c = 0 or a = b = c; ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0. (8)

b. Prove that the matrix is unitary and find A^-1 (8)

Q.7 a. Test for consistency the following system of equations, and if consistent, solve them: (8)

b. If u = log (x³ + y³ + z³ - 3xyz) show that . (8)

Q.8 a. Find the power series solution about the origin of the equation . (11)

b. Prove that (5)

Q.9 a. Show that . (8)

b. Solve . (8)