2010_midsem_m201

National Institute of Technology-Durgapur Department of Mathematics (M-201) Class Test CxCy Date 24.03.2010 Full Marks: 20 Time: 50 min Answer any FOUR 4 × 5 1. If I n = R sin nθ sinθ dθ, prove that (n - 1)(I n - I n-2 )=2 sin(n - 1)θ. 2. Prove that R ∞ 0 e -x 2 dx = √ π 2 using Gamma function. Hence ﬁnd the value of R ∞ -∞ e -x 2 dx. 3. Solve (D 2 + 1)y =2 sin 3 x. 4. Find the rank of the matrix A = 1 1 3 1 3 -3 -2 -4 -4 . 5. Determine the values of λ for which the following equations have non-zero solutions: x +2y +3z = λx, 3x + y +2z = λy and 2x +3y + z = λz . 1

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midsem questions nit durgapur 2010

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National Institute of Technology-Durgapur

Department of Mathematics (M-201)Class Test CxCy Date 24.03.2010

Full Marks: 20 Time: 50 min

Answer any FOUR 4 5

1. If In = sin n

sind, prove that (n 1)(In In2) = 2 sin(n 1).

2. Prove that0 e

x2dx =pi2

using Gamma function. Hence find the value of e

x2dx.

3. Solve (D2 + 1)y = 2 sin3x.

4. Find the rank of the matrix A =

1 1 31 3 32 4 4

.5. Determine the values of for which the following equations have non-zero solutions: x+2y+3z = x,

3x+ y + 2z = y and 2x+ 3y + z = z.

1