JNTU

JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (December 2019/January 2020 - R16)

B.Tech | I Year | I-I | R16 | December 2019/January 2020

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.

PART A

Part A (25 Marks)

125 marks

Find

L{e^{2t}}

2 marks

Answer

Using the standard Laplace transform formula L{e^at} = 1/(s-a), for a = 2, we get L{e^2t} = 1/(s-2) for s > 2.

Find inverse Laplace of

1/(s^2+s+2)

3 marks

Answer

Complete the square in the denominator: s² + s + 2 = (s + 1/2)² + 7/4 = (s + 1/2)² + (sqrt(7)/2)². Using the shifting property L^-1{F(s+a)} = e^-atf(t) and L^-1{a/(s²+a²)} = sin(at), we get L^-1{1/((s+1/2)² + (sqrt(7)/2)²)} = (2/sqrt(7)) e^-t/2 sin((sqrt(7)/2)t).

Evaluate

Integral_0^{pi/2} sin^4 x cos^4 x dx

using Beta-Gamma functions.

2 marks

Answer

Show that

Gamma(1/2) = sqrt(pi)

, using the relation

Beta(m,n) = (Gamma(m)Gamma(n))/Gamma(m+n)

3 marks

Answer

Evaluate

Integral_0^1 Integral_0^1 Integral_0^1 dx dy dz

2 marks

Answer

Evaluate

Integral_0^2 Integral_0^x dy dx

3 marks

Answer

Find the directional derivative of

r = x^2+y^2+z^2

, at the point (0, r, 0) in the direction of j.

2 marks

Answer

Find Curl (grad f) where

f = x^2+y^2+z^2

3 marks

Answer

State Green’s theorem in a plane.

2 marks

Answer

Find the work done by

F = xi + yj + zk

in moving a particle from (-1, 2, 1) to (2, 3, 4) along the line joining them.

3 marks

Answer

PART B

Part B (50 Marks)

210 marks

Find inverse Laplace transform

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

5 marks

Answer

Find

L{ t e^{-t} sin 3t }

5 marks

Answer

Using Laplace transform solve

y'' + 3y' + 2y = 3, y(0)=0, y'(0)=1

10 marks

Answer

410 marks

Prove the relation between Beta and Gamma functions.

5 marks

Answer

Beta(m, n) = Beta(m+1, n) + Beta(m, n+1)

, find k+p in terms of m and n.

5 marks

Answer

510 marks

Prove that

Gamma(m)Gamma(m+1/2) = (sqrt(pi)/2^{2m-1})Gamma(2m)

5 marks

Answer

Show that

Integral_0^1 (x^{m-1} + x^{n-1}) / (1+x)^{m+n} dx = Beta(m,n)

5 marks

Answer

610 marks

Evaluate

Integral_0^inf Integral_0^inf e^{-(x^2+y^2)} dx dy

by changing to polar coordinates.

5 marks

Answer

Evaluate

IntegralIntegralIntegral_V (xy+yz+zx) dx dy dz

where V is the region bounded by x=0, 1; y=0, 2; z=0, 3.

5 marks

Answer

710 marks

Evaluate

Integral_0^1 Integral_0^{sqrt(1-x^2)} xy dy dx

5 marks

Answer

Evaluate

Integral_0^1 Integral_0^{x+z} Integral_{x-z}^{z+x} (x+y+z) dy dz dx

5 marks

Answer

810 marks

Prove the identity

Div(A cross B) = B . Curl A - A . Curl B

5 marks

Answer

Evaluate the angle between the normals of the surface

xy = z^2

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

5 marks

Answer

910 marks

Find the unit normal vector to surface

x^2+y+xz=4

at (2,-2,3). Also find the directional derivative of

x^2-z+ylog x = 1

at (-1, 2, 1).

5 marks

Answer

For any vector field V, prove that Div Curl V = 0.

5 marks

Answer

Verify Divergence Theorem for

F = x^2i + y^2j + z^2k

over the surface of the solid cut off by

x+y+z=a

in the 1st octant.

10 marks

Answer

1110 marks

Find the work done by

F = (-xy)i + (-z)j + (x)k

along the curve

x=t, y=t^2, z=t^3

from t=0 to t=1.

5 marks

Answer

Apply Stokes Theorem to evaluate

Integral_C F.dR

where C is intersection of

x^2+y^2+z^2=a^2

and

x+z=a

F = yi + zj + xk

5 marks

JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (December 2019/January 2020 - R16)

PART A

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

PART B

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

Answer

About this paper

Other papers in MATHEMATICS-II