Using the property L{tnf(t)} = (-1)n (dn/dsn) F(s) , where F(s) = L{cos 2t} = s / (s2 + 4) , we h...

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-II Previous Question Paper (May - 2018 - R16)

B.Tech | I | I | R16 | May - 2018

Answer PaperPaper Code: 131ABMaximum Marks: 75Time: 3 hours

Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.

Part A

125 marks

Find

L{t³ cos 2t}

2 marks

Answer

Using the property

L{tⁿf(t)} = (-1)ⁿ (dⁿ/dsⁿ) F(s)

, where

F(s) = L{cos 2t} = s / (s² + 4)

, we have

L{t³ cos 2t} = (-1)³ (d³/ds³) [s / (s² + 4)] = - (6s(s² - 4) / (s² + 4)⁴) = (6s³ - 24s) / (s² + 4)⁴

Find

L^-1{s / (s+1)(s+2)}

3 marks

Answer

Using partial fractions:

s / ((s+1)(s+2)) = A/(s+1) + B/(s+2)

. Solving gives

A = -1

and

B = 2

. Thus,

L^-1{-1/(s+1) + 2/(s+2)} = -e^-t + 2e^-2t

Evaluate

∫₀¹ x⁵(1-x)⁷ dx

2 marks

Answer

Evaluate

∫₀^∞ x²e^-4x dx

3 marks

Answer

Evaluate

∫₀¹ ∫₀^x xy dy dx

2 marks

Answer

Evaluate

∫₁² ∫₁³ ∫₁³ dx dy dz

3 marks

Answer

r = xi + yj + zk

then find

div r

2 marks

Answer

State Green’s theorem on a plane.

3 marks

Answer

Evaluate

∇(x² + y²z + z)

2 marks

Answer

is a constant vector then find

curl(r x a)

3 marks

Answer

Part B

210 marks

Find

L{te^t sin 3t}

5 marks

Answer

Find

L^-1{s / (s²+4)(s²+25)}

5 marks

Answer

Solve the differential equation

(d²x/dt²) + 9x = sin t

using Laplace transform, given that

x(0)=1, x(π)=1

10 marks

Answer

Prove that

β(m, n) = (Γ(m) Γ(n)) / Γ(m+n)

10 marks

Answer

Show that

β(m, n+1) = β(m+1, n) = (n/m)β(m+1, n)

[Note: content adjusted per curriculum standards].

10 marks

Answer

Change the order of integration and solve

∫₀^a ∫_x^√(a²-x²) xy dy dx

10 marks

Answer

Find the area of the loop of the curve

r = a(1 + cos θ)

10 marks

Answer

810 marks

Prove that

∇ x (A x B) = A(∇.B) - B(∇.A) + (B.∇)A - (A.∇)B

5 marks

Answer

Find the directional derivative of

x² + y² + z²

at (1, -1, 3) in the direction of

i + j + 2k

5 marks

Answer

Show that

∇² f(r) = f''(r) + (2/r)f'(r)

where

r = |r|

10 marks

Answer

Verify Green’s theorem for

∫_C (xy + y²)dx + x² dy

where ‘C’ is bounded by y = x and y = x².

10 marks

Answer

Verify the Stoke’s theorem for

F = yi + zj + xk

and surface is the part of the plane

x + y + z = 1

above the xy - plane.

10 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-II Previous Question Paper (May - 2018 - R16)

Part A

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Part B

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