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JNTU B.Tech II Year I Semester NUMERICAL METHODS AND COMPLEX VARIABLES Previous Question Paper (February - 2024 - R22)

B.Tech | II Year I Semester | II-I | R22 | February - 2024

Answer PaperPaper Code: 183BRMaximum Marks: 60Time: 3 Hours

Part-A is a compulsory question. Part-B consists of ten questions (2-11) carrying 10 marks each. Answer five questions from Part-B, choosing one from each unit.

Part A

Answer all sub-questions:

10 marks

Find the period of cos 3x.

1 marks

Answer

The period of a function cos(ax) is given by 2π/|a|. For cos(3x), a = 3, therefore the period is:

T = 2π/3

If f(x) = x^2 in -2 < x < 2, f(x + 4) = f(x), then find the coefficient of cos(nπx/2) in the Fourier series expansion of f(x).

1 marks

Answer

For a function defined on (-L, L) with period 2L, the Fourier coefficient a_n is given by (1/L) ∫_-L^L f(x) cos(nπx/L) dx. Here, L = 2 and f(x) = x². Since f(x) is an even function:

a_n = (2/2) ∫₀² x² cos(nπx/2) dx = ∫₀² x² cos(nπx/2) dx

Integrating by parts:

a_n = 16(-1)ⁿ / (n²π²)

What is the nth divided difference of a polynomial of the nth degree?

1 marks

Answer

Write the condition when the Newton-Raphson method fails while solving f(x)=0.

1 marks

Answer

While applying Simpson's 3/8th rule, to evaluate the integral, how many sub-intervals should the interval [a,b] be divided into?

1 marks

Answer

Write Euler's modified iterative formula.

1 marks

Answer

Give an example of a differentiable function but not analytic at a given point. Explain the reason.

1 marks

Answer

Describe the region Im(3/z) < 1/3. Is it bounded or unbounded?

1 marks

Answer

How to identify a singular point as a pole or isolated singularity from Laurent series expansion?

1 marks

Answer

Give an example of essential singularity.

1 marks

Answer

Part B

210 marks

Find the Fourier series expansion of the function f(x) = x + π, -π < x < π. Hence find the sum of the series 1 - 1/3 + 1/5 - 1/7 + ...

5 marks

Answer

Find the Fourier cosine integral representation of f(x) = e^-x + e^-3x, x > 0.

5 marks

Answer

310 marks

Use the integral

∫_-∞^∞ e^-y² dy = √π

to find the Fourier transform of f(x) = e^-x²/2.

5 marks

Answer

Find the complex form of the Fourier transform of f(x) = sin(x) if -2≤x≤2, and 0 elsewhere.

5 marks

Answer

410 marks

Find the smallest positive root of the equation that lies between 3 and 4, correct to 3 decimal places, using the method of false position.

5 marks

Answer

If y(10) = 35.3, y(15)=32.4, y(20)=29.2, y(25)=26.1, y(30)=23.2 and y(35)=20.5, find y(12) using (i) Newton's forward interpolation formula and (ii) Newton's backward interpolation formula.

5 marks

Answer

510 marks

Find the Newton-Raphson iterative formula to find the pth root of a positive number N and hence find the cube root of 17.

5 marks

Answer

Use Stirling's formula to compute tan 89° 26'.

5 marks

Answer

610 marks

Find the value of ∫₀^π/2 √(1 - 0.162sin²x) dx using Simpson's 1/3rd rule.

5 marks

Answer

Find the values of y at x = ±0.1, using the Taylor series method of third order with h = 0.1, given that dy/dx = 1/(x+y), y(0)=2.

5 marks

Answer

710 marks

Compute ∫₄^5.2 log_e x dx using 3/8th rule of integration, by dividing the interval of integration into 6 equal sub-intervals.

5 marks

Answer

Using Runge-Kutta 4th order method, find y(0.1) and y(0.2) for the initial value problem dy/dx = (y^2 - x^2)/(y^2 + x^2), y(0)=1 with h=0.1.

5 marks

Answer

810 marks

Find the image of the region |z + 1 + i| < 1 under the transformation w = (3-4i)z + 6+2i.

5 marks

Answer

Find the bilinear transformation that maps the points ∞, i, 0 in z-plane into points 0, i, ∞ in w-plane. Hence, find the fixed points of the map.

5 marks

Answer

910 marks

If w = f(z) = u + iv is an analytic function of z, and u - v = (x - y)(x^2 + 4xy + y^2). Find w in terms of z.

5 marks

Answer

If w = f(z) is an analytic function of z; prove that (∂²/∂x² + ∂²/∂y²) log |f'(z)| = 0.

5 marks

Answer

1010 marks

State the Cauchy’s Residue theorem. Use this theorem to evaluate ∫_C (z+1)/((z-1)(z²+5)) dz, where C is |z| = 3.

5 marks

Answer

If tan z is expanded about z = π/2 as Laurent series in 0 < |z - π/2| < π/2, find the principal part and classify the singularity at z = π/2.

5 marks

Answer

1110 marks

Apply calculus of residues to evaluate ∫₀^π 1/(a + b cos θ) dθ where 0 < b < a.

5 marks

Answer

Locate the zeros and classify the singularities of (i) tan z/z (ii) cot(πz)/z².

5 marks

Part A

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Part B

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