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JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (March/April - 2023 - R16)

B.Tech | I Year | I-II | R16 | March/April - 2023

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

Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.

Part A

(25 Marks)

125 marks

State the conditions for the existence of Laplace transform of a function.

2 marks

Answer

The Laplace transform of a function f(t) exists if: 1) f(t) is piecewise continuous on every finite interval in the range t ≥ 0, and 2) f(t) is of exponential order, i.e., |f(t)| ≤ Me^ct for some constants M, c > 0 as t → ∞.

State the convolution theorem for Laplace transforms.

3 marks

Answer

If L{f(t)} = F(s) and L{g(t)} = G(s), then the convolution theorem states that L{f(t) * g(t)} = F(s)G(s), where the convolution (f * g)(t) is defined as

∫₀^t f(u)g(t-u) du

Show that

β(m,n) = β(n,m)

2 marks

Answer

Show that

Γ(n+1) = nΓ(n)

3 marks

Answer

Evaluate

∫₀^∞ ∫₀^x x e^-x/y dy dx

2 marks

Answer

Evaluate

∫∫ xy dxdy

over the positive quadrant of the circle

x² + y² = a²

3 marks

Answer

Show that

∇(rⁿ) = nr^n-2 R

where

R = xi + yj + zk

2 marks

Answer

If F = ∇u, show that curl F = 0.

3 marks

Answer

F = xyi - 3j

, evaluate

∫ F.dR

, where C is the curve in the xy – plane

y = x²

from (0,0) to (1,2).

2 marks

Answer

Find the total work done by the force

F = xyi - yj + zxk

in moving a particle around the circle

x² + y² = 4

3 marks

Answer

Part B

(50 Marks)

210 marks

Find

L{cos t / t}

5 marks

Answer

Using convolution theorem find the inverse Laplace transform of

s / ((s²+a²)(s²+b²))

5 marks

Answer

310 marks

Find the inverse Laplace transform of

log((s²+a²)/(s²+b²))

5 marks

Answer

Using Laplace transforms, solve:

y'' - 3y' + 2y = e^3t

with

y(0)=1, y'(0)=0

5 marks

Answer

410 marks

Show that

β(m,n) = (Γ(m)Γ(n))/Γ(m+n)

5 marks

Answer

Evaluate

β(7/2, 1/2)

5 marks

Answer

510 marks

Evaluate

∫₀^∞ e^-ax x^m-1 sin bx dx

in terms of Gamma functions.

5 marks

Answer

Evaluate the expression provided in question paper context.

5 marks

Answer

610 marks

By double integration find the area enclosed by the curves:

y = x²

and

y = 4-x²

5 marks

Answer

Find the volume enclosed by the cylinders

x² + y² = ax

and

z² = ax

5 marks

Answer

710 marks

By double integration find the whole area of the curve

a²x² = y³(2a-y)

5 marks

Answer

Find the volume common to the cylinders

x² + y² = a²

and

x² + z² = a²

5 marks

Answer

810 marks

Show that

div(grad rⁿ) = n(n+1)r^n-2

5 marks

Answer

If r is the distance of a point (x,y,z) from the origin, prove that

curl(K x grad(1/r)) + grad(K . curl(1/r)) = 0

where K is the unit vector.

5 marks

Answer

910 marks

For a solenoidal vector F, show that

curl curl curl curl F = ∇⁴F

5 marks

Answer

f(x,y,z) = rⁿ

, determine n if

div(grad f) = 0

5 marks

Answer

1010 marks

Evaluate

∫_S F.n dS

where

F = 18zi - 12j + 3yk

and S is the portion of the plane

2x+3y+6z = -12

in the first octant.

5 marks

Answer

Verify the Greens Theorem for

∫_C [(3x-8y)dx + (4y-6xy)dy]

where C is the boundary of the region bounded by

x=0, y=0, x+y=1

5 marks

Answer

1110 marks

Verify the Stokes Theorem for

F = (x-z)i + (x^3+yz)j - 3xy^2k

(approximate re-statement) over the surface of the cube.

5 marks

Answer

Verify Divergence Theorem for

F = yi + xj + z²k

over the cylindrical region bounded by

x²+y²=9, z=0, z=2

5 marks

JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (March/April - 2023 - R16)

Part A

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Part B

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