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Posted Date: 26 Jul 2008 Posted By: Nitin Member Level: Platinum
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2007 The Institution of Engineers,India A.M.I.E.T.E Electronics & Tele Communication Engineering Code: AE01/AC01/AT01 Subject: MATHEMATICS-I University Question paper
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 best alternative in the following: (2 x 10)
a. The value of the limit is
(A) limit does not exist (B) 0
(C) 1 (D) -1
b. If then the value of is equal to
(A) 0 (B)
(C) (D)
c. If then the value of is
(A) u (B) 2u
(C) 3u (D) 0
d. The value of integral is equal to
(A) 22 (B) 26
(C) 5 (D) 25
e. The solution of the differential equation is given by
(A) (B)
(C) (D)
f. The solution of the differential equation is
(A) (B)
(C) (D)
g. If 3x+2y+z= 0, x+4y+z=0, 2x+y+4z=0, be a system of equations then
(A) system is inconsistent
(B) it has only trivial solution
(C) it can be reduced to a single equation thus solution does not exist
(D) Determinant of the coefficient matrix is zero.
h. If ? is an eigen value of a non-singular matrix A then the eigen value of A-1 is
(A) 1/ ? (B) ?
(C) –? (D) –1/ ?
i. The product of the eigen values of the matrix
is
(A) 3 (B) 8
(C) 1 (D) –1
j. The value of the integral is
(A) (B)
(C) (D)
Answer any FIVE Questions out of EIGHT Questions.
Each Question carries 16 marks.
Q.2 a. Find the extreme value of the function f(x,y,z) = 2x + 3y + z such that x2+y2=5 and x + z =1 (8)
b. Show that the function is continuous at (0,0) but its partial derivatives of first order do not exist at (0,0). (8)
Q.3 a. Evaluate the integral where T is region bounded by the cone
and the planes z=0 to z=h in the first octant. (8)
b. Show that the approximate change in the angle A of a triangle ABC due
to small changes in the sides a, b, c respectively, is given by where is the area of the triangle.
Verify that (8)
Q.4 a. If Show that
(8)
b. Using the method of variation of parameter method, find the general
solution of the differential equation (8)
Q.5 a. Find the general solution of the equation . (8)
b. Find the general solution of the equation . (8)
Q.6 a. Solve (8)
b. The set of vectors {x1, x2}, where x1 = (1,3)T, x2 = (4,6)T is a basis in R2. Find a linear transformations T such that Tx1 = (-2,2,-7)T and Tx2 = (-2,-4,-10)T (8)
Q.7 a. Show that the matrix A is diagonalizable. . Hence,
obtain the matrix P such that is a diagonal matrix. (8)
b. Investigate the values of ? for which the equations
are consistent, and hence find the ratios of x:y:z when ? has the smallest of
these values. (8)
Q.8 a. Find the first five non-vanishing terms in the power series solution of the
initial value problem
(11)
b. Show that (5)
Q.9 a. Show that (8)
b. Show that . (8)
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