JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-II Previous Question Paper (November/December - 2020 - R18)

B.Tech | I | I-II | R18 | November/December - 2020

Answer PaperPaper Code: 152AAMaximum Marks: 75Time: 2 hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

Solve

dy/dx + y/x = y^2 sin x

7 marks

Answer

The given equation is a Bernoulli differential equation of the form dy/dx + Py = Qy^n, where P = 1/x, Q = sin x, and n = 2. Dividing both sides by y^2 gives:

y^-2(dy/dx) + (1/x)y^-1 = sin x

Let v = y^-1. Then dv/dx = -y^-2(dy/dx), or -dv/dx = y^-2(dy/dx). Substituting this into the equation:

-dv/dx + (1/x)v = sin x

dv/dx - (1/x)v = -sin x

This is a linear differential equation. The integrating factor (IF) is:

IF = e^(∫ -1/x dx) = e^(-ln x) = 1/x

Multiplying the linear equation by the IF:

(1/x)(dv/dx) - (1/x^2)v = -(sin x)/x

d/dx(v/x) = -(sin x)/x

Integrating both sides:

v/x = -∫ (sin x)/x dx + C

v = -x ∫ (sin x)/x dx + Cx

Substituting back v = 1/y, the general solution is:

1/y = -x ∫ (sin x)/x dx + Cx

If the air is maintained at 30°C and the temperature of the body cools from 80°C to 60°C in 12 minutes, find the temperature of the body after 24 minutes.

8 marks

Answer

According to Newton's Law of Cooling, dT/dt = -k(T - T_s), where T_s = 30°C. Separating variables and integrating gives:

ln(T - 30) = -kt + C

T - 30 = Ce^(-kt)

At t = 0, T = 80, so 80 - 30 = Ce^0, which gives C = 50. Thus, T = 30 + 50e^(-kt).

At t = 12, T = 60:

60 = 30 + 50e^(-12k)

30 = 50e^(-12k) ⇒ e^(-12k) = 0.6

We need to find T at t = 24:

T = 30 + 50e^(-24k)

T = 30 + 50(e^(-12k))^2

T = 30 + 50(0.6)^2

T = 30 + 50(0.36) = 30 + 18 = 48°C

The temperature of the body after 24 minutes is 48°C.

215 marks

Solve

dy/dx = e^(x-y)(e^x - e^y)

7 marks

Answer

Radium decomposes at a rate proportional to the amount present. If p% of the original amount disappears in 1 year, how much will remain at the end of 21 years.

8 marks

Answer

315 marks

Solve

(d^2y/dx^2) - (a+b)(dy/dx) + aby = e^ax + e^bx

8 marks

Answer

Solve

(D^2 - 1)y = e^x cos x

7 marks

Answer

415 marks

Solve

(D^2 - 4D + 3)y = sin 3x cos 2x

7 marks

Answer

Solve

x^2(d^2y/dx^2) - x(dy/dx) + y = 2 log x

8 marks

Answer

515 marks

Check the equality of the two double integrals

integral_0^1 integral_0^1 (x-y)/(x+y)^3 dy dx

and

integral_0^1 integral_0^1 (x-y)/(x+y)^3 dx dy

8 marks

Answer

Evaluate

integral_0^1 integral_0^(sqrt(1-x^2)) integral_0^(sqrt(1-x^2-y^2)) dz dy dx

7 marks

Answer

615 marks

Change the order of integration in the integral and evaluate

integral_0^(4a) integral_(x^2/4a)^(2 sqrt(ax)) dy dx

7 marks

Answer

Calculate the volume of the solid bounded by the planes x = 0, y = 0, x+y+z = a and z = 0.

8 marks

Answer

Find the directional derivative of the function

φ = xy^2 + yz^3

at the point (2, -1, 1) in the direction of the normal to the surface

x log z - y^2 + 4 = 0

at (-1, 2, 1).

15 marks

Answer

Verify Green’s theorem in the plane for

integral_C (x^2 - xy^3) dx + (y^2 - 2xy) dy

where C is a square with vertices (0, 0), (2, 0), (2, 2), (0, 2).

15 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-II Previous Question Paper (November/December - 2020 - R18)

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