JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year MATHEMATICS – II Previous Question Paper (November/December - 2018 - R13)

B.Tech | II Year | I Semester | R13 | November/December - 2018

Answer PaperPaper Code: 113AAMaximum 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 and may have a, b, c as sub questions.

PART- A

125 marks

Find

∇× yz

2 marks

Answer

The curl of a scalar field is defined as the cross product of the del operator with the scalar function. Since the curl of a scalar function is not defined in standard vector calculus (curl is defined for vector fields), the expression ∇×(yz) is treated as the gradient of the scalar field or is considered zero if interpreting the scalar as a vector field with zero components. However, strictly speaking, ∇×(φ) is not defined; if the question implies the gradient, the result is z i + y k.

Evaluate by Green’s Theorem

∫_C (x²-y) dx + (x+y²) dy

where C is

x²+y²=1

3 marks

Answer

By Green's Theorem, ∫_C (M dx + N dy) = &iint;_R (∂N/∂x - ∂M/∂y) dA. Here, M = x²-y and N = x+y². Then ∂N/∂x = 1 and ∂M/∂y = -1. The integral becomes &iint;_R (1 - (-1)) dA = 2 &iint;_R dA. Since the region is a unit circle, the area is πr² = π(1)² = π. Thus, the integral equals 2π.

f(x) = sin x, in [-π, π], f(x+2π) = f(x)

. Find constant term in the fourier series of

f(x)

2 marks

Answer

F[f(t)] = F(s)

, then Prove that

F[e^iat f(t)] = F(s-a)

3 marks

Answer

Evaluate

Δ(1/x!)

, by taking step size equal to one.

2 marks

Answer

Write the three normal equations to fit the parabola

y = a + bx + cx²

, by method of least squares.

3 marks

Answer

Find any two values of x between which the root of

xe^x = cos x

lies.

2 marks

Answer

Find LU Decomposition of

[[2, 5], [2, 6]]

3 marks

Answer

Write the formula to evaluate

∫_a^b f(x) dx

by Trapezoidal rule.

2 marks

Answer

dy/dx = xy-1, y(0)=0

then find y(0.1) by Taylor’s series method taking upto 2 differentials.

3 marks

Answer

PART-B

210 marks

Find the unit normal vector to the surface

x³+y³+z³ = 3

at the point (1, 2, -1).

5 marks

Answer

Evaluate by stoke’s theorem

∫_c e^x dx + 2y dy - dz

where c is the curve

x²+y²=9

and

z=2

5 marks

Answer

Verify divergence theorem for

2x²yi - y²j + 4xz²k

taken over the region in first octant of the cylinder

x²+y²=9

and

z=2

10 marks

Answer

410 marks

Find the Fourier series to represent

f(x) = x², - π < x < π, f(x+2π) = f(x)

5 marks

Answer

Find the Fourier cosine transform of

e^-ax/x

5 marks

Answer

510 marks

Expand

f(x) = e^-x, -1 ≤ x ≤ 1; f(x+2) = f(x)

Fourier series.

5 marks

Answer

Find the finite fourier sine and cosine transform of

f(x) = x³, 0 < x < π

5 marks

Answer

610 marks

Find y(42) from the following data using newton’s forward formula:

x	20	25	30	35	40	45
y	352	332	291	260	231	204

5 marks

Answer

Fit a straight line

y = a + bx

by the method of least squares for the data:

x	1	2	3	4	5	6
y	14	33	40	63	76	85

5 marks

Answer

710 marks

Apply Lagrange’s formula to find f(5) given that f(1) = 2, f(2) = 4, f(3) = 8, f(4) = 16 and f(7) = 128.

5 marks

Answer

Fit a curve of the form

y = ab^x

by the method of least squares.

x	1	2	3	4	5	6	7	8
y	1.0	1.2	1.8	2.5	3.6	4.7	6.6	9.1

5 marks

Answer

Solve the system of equations by expressing the coefficient matrix as a product of lower triangular and upper triangular matrices:

x + 2y = 7, x - 3y - z = 4, 4y + 3z = 5

10 marks

Answer

Solve the system of equations using Gauss seidal method:
9x + 2y + 4z = 20
x + 10y + 4z = 6
2x - 4y + 10z = -15

10 marks

Answer

Find y(0.1) and y(0.2) using Runge Kutta fourth order formula given that

dy/dx = x² + y²

and y(0) = 1.

10 marks

Answer

Evaluate

∫₀¹ 1/(1+x²) dx

by taking h = 1/6 using: a) Simpson’s 1/3rd rule b) Simpson’s 3/8th rule.

10 marks

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II Year MATHEMATICS – II Previous Question Paper (November/December - 2018 - R13)

PART- A

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PART-B

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