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JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (January/February - 2024 - R16)

B.Tech | I Year | I-II | R16 | January/February - 2024

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

State linearity property of Laplace transform.

2 marks

Answer

The linearity property of the Laplace transform states that for any two functions f(t) and g(t) and any two constants a and b, the Laplace transform of their linear combination is equal to the linear combination of their individual Laplace transforms. Mathematically, it is expressed as:

L{a f(t) + b g(t)} = a L{f(t)} + b L{g(t)} = a F(s) + b G(s)

where L{f(t)} = F(s) and L{g(t)} = G(s).

Find the inverse Laplace transform of

1 / ((s+a)(s+b))

3 marks

Answer

To find the inverse Laplace transform of the function 1 / ((s+a)(s+b)), we first apply partial fraction decomposition to the expression:

1 / ((s+a)(s+b)) = A / (s+a) + B / (s+b)

Solving for the constants, we find:

1 = A(s+b) + B(s+a)

Setting s = -a gives A = 1 / (b-a), and setting s = -b gives B = 1 / (a-b) = -1 / (b-a). Substituting these back, we get:

1 / ((s+a)(s+b)) = (1 / (b-a)) * [1 / (s+a) - 1 / (s+b)]

Using the linearity property and the standard inverse Laplace transform L^-1{1 / (s+k)} = e^-kt, the result is:

L^-1{1 / ((s+a)(s+b))} = (1 / (b-a)) * (e^-at - e^-bt)

If n is a positive integer, then prove that

Γ(n+1) = n!

2 marks

Answer

Prove that

β(m, n) = β(n, m)

3 marks

Answer

Evaluate

∬(x^2 y^3) dxdy

from 0 to 1 for both variables.

2 marks

Answer

Evaluate

∬ 1/(x^2 + y^2 + 1) dxdy

from 0 to 1 for both variables.

3 marks

Answer

Define a direction derivative of a scalar point function in the direction of a vector a.

2 marks

Answer

Define the divergence and curl of a vector point function F.

3 marks

Answer

State Stoke’s theorem.

2 marks

Answer

F = (5xy - 6x^2)i + (2y - 4x)j

, evaluate

∫ F·dr

along the curve C in xy-plane,

y = x^3

from the point (1, 1) to (2, 8).

3 marks

Answer

Part B

210 marks

Find the Laplace transform of the periodic function

f(t) = sin(ωt)

for

0 < t < π/ω

and

for

π/ω < t < 2π/ω

5 marks

Answer

Apply Convolution theorem to evaluate

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

5 marks

Answer

Solve by transform technique

d^2y/dt^2 - 2 dy/dt + 3y = sin t

, given

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

10 marks

Answer

410 marks

Show that

β(m, n+1) + β(m+1, n) = β(m, n)

5 marks

Answer

Prove that

Γ(1/2) = √π

5 marks

Answer

Define Beta and Gamma function and hence derive the relation between them.

10 marks

Answer

Evaluate

∬ e^(-x^2/y) dxdy

by changing the order of integration.

10 marks

Answer

Change the order of integration in

∫∫ xy dxdy

and then evaluate it.

10 marks

Answer

810 marks

Calculate the angle between the normal’s to the surface

xy^2 = z

at the points (4, 1, 2) and (3, 3, -3).

5 marks

Answer

Find the value of 'a' if the vector

(ax^2y + yz)i + (xy^2 - xz)j + (xyz - x^2y^2)k

has zero divergence.

5 marks

Answer

r = xi + yj + zk

, prove that

div(r^n) = (n+3)r^(n-2)

and

curl(r^n) = 0

10 marks

Answer

Verify Green's theorem for

∮ (x^2 + xy)dx + (x^2 + y^2)dy

, where C is the square formed by the lines

x = ±1, y = ±1

10 marks

Answer

Verify Stoke's theorem for a vector field defined by

F = (y^3)i + (x^3)j

, in the region

x^2 + y^2 ≤ 1, z = 0

10 marks

JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (January/February - 2024 - R16)

Part A

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Part B

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