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JNTU B.Tech I MATHEMATICS-II Previous Question Paper (July/August - 2021 - R16)

B.Tech | I | II | R16 | July/August - 2021

Answer PaperPaper Code: 132ABMaximum Marks: 75Time: 3 Hours

Answer any five questions. All questions carry equal marks.

Main

115 marks
a

Find

L-1[1/(s2+s+1)]
.

7 marks

Answer

To find the inverse Laplace transform of 1/(s2 + s + 1), we complete the square in the denominator:

s2 + s + 1 = (s + 1/2)2 + 3/4 = (s + 1/2)2 + (√3/2)2

Using the shifting property L-1{F(s+a)} = e-atf(t), where F(s) = 1/(s2 + (√3/2)2), we know that L-1{1/(s2 + a2)} = (1/a)sin(at). Here a = √3/2.

L-1[1/((s+1/2)2 + (√3/2)2)] = e-t/2 * (1/(√3/2)) * sin(√3t/2)

Simplifying, the result is:

(2/√3) e-t/2 sin(√3t/2)
b

Find the Laplace transform of

f(t) = t2(cos t + sin t)
.

8 marks

Answer

Using the property L{tnf(t)} = (-1)n (dn/dsn) F(s), for n=2 we have:

L{t2(cos t + sin t)} = d2/ds2 [L{cos t} + L{sin t}] = d2/ds2 [s/(s2+1) + 1/(s2+1)] = d2/ds2 [(s+1)/(s2+1)]

First derivative:

d/ds [(s+1)/(s2+1)] = [(s2+1)(1) - (s+1)(2s)] / (s2+1)2 = (s2+1-2s2-2s) / (s2+1)2 = (-s2-2s+1) / (s2+1)2

Second derivative:

d/ds [(-s2-2s+1) / (s2+1)2] = [(-2s-2)(s2+1)2 - (-s2-2s+1)(2)(s2+1)(2s)] / (s2+1)4

Simplifying by dividing by (s2+1):

[(-2s-2)(s2+1) - 4s(-s2-2s+1)] / (s2+1)3 = (-2s3-2s-2s2-2 + 4s3+8s2-4s) / (s2+1)3 = (2s3+6s2-6s-2) / (s2+1)3
2

State the convolution theorem on Laplace transforms. Using it find the inverse Laplace transform of

1/(s3(s+1))
.

15 marks

Answer

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315 marks
a

Evaluate

∫0a x3(a-x)3 dx
using Beta and Gamma functions.

7 marks

Answer

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b

Prove that

β(m,n) = [Γ(m)Γ(n)] / Γ(m+n)
.

8 marks

Answer

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415 marks
a

Evaluate

∫0∞ e-ax bx dx
using Gamma function.

7 marks

Answer

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b

Express

∫01 xm-1 (1-x)n-1 dx
in terms of Beta function and hence evaluate
∫01 x1/2(1-x)3/2 dx
.

8 marks

Answer

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515 marks
a

Evaluate

∫01 ∫2y2 ex^2 dx dy
by changing the order of integration.

7 marks

Answer

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b

Evaluate

∬R xydx dy
, where R is the first quadrant of the circle
x2+y2=a2
.

8 marks

Answer

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615 marks
a

Find the angle between the two surfaces

x2+y2+z2=9
and
z=x2+y2-3
at (-2, 1, 2).

7 marks

Answer

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b

If

v = xi + yj - 12z k
, find a scalar function
f(x,y,z)
such that
v = ∇f
.

8 marks

Answer

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715 marks
a

Evaluate the line integral of

v = xyi + y2j + zk
over the circular path
x2+y2=4, z=0
from (2, 0, 0) to (0, 2, 0).

7 marks

Answer

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b

Evaluate

∮c (3x2-8y2)dx + (4y-6xy)dy
, where c is the boundary of the region defined by
x=0, y=0
and
x+y=1
, by Green’s theorem.

8 marks

Answer

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8

Verify Gauss’s divergence theorem for

v = xzi + yj + yz2k
on the surface of the cube bounded by the planes x=0, x=1, y=0, y=1, z=0 and z=1.

15 marks

Answer

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About this paper

This page contains the question paper of MATHEMATICS-II from JNTU for B.Tech MME-MMT I II, July/August - 2021, R16.

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