Find the second difference of the polynomial x4 - x3 + x2 - 12x + 30 with interval of difference h=2.

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JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (September/October - 2021 - R15)

B.Tech | I Year | I-I | R15 | September/October - 2021

Answer PaperPaper Code: 121ALMaximum Marks: 75Time: 3 Hours

Answer any five questions. All questions carry equal marks.

Main

115 marks

Find the second difference of the polynomial

x⁴ - x³ + x² - 12x + 30

with interval of difference h=2.

7 marks

Answer

Let f(x) = x⁴ - x³ + x² - 12x + 30 and the interval of difference be h = 2. The forward difference operator is defined as Δf(x) = f(x + h) - f(x). The second difference is Δ²f(x) = f(x + 2h) - 2f(x + h) + f(x).

Substituting h = 2:

Δ²f(x) = f(x + 4) - 2f(x + 2) + f(x)

Calculating the terms:

f(x) = x⁴ - x³ + x² - 12x + 30

f(x + 2) = (x + 2)⁴ - (x + 2)³ + (x + 2)² - 12(x + 2) + 30 = x⁴ + 7x³ + 17x² + 4x + 18

f(x + 4) = (x + 4)⁴ - (x + 4)³ + (x + 4)² - 12(x + 4) + 30 = x⁴ + 13x³ + 65x² + 100x + 126

Substituting these into the expression for Δ²f(x):

Δ²f(x) = (x⁴ + 13x³ + 65x² + 100x + 126) - 2(x⁴ + 7x³ + 17x² + 4x + 18) + (x⁴ - x³ + x² - 12x + 30)

Δ²f(x) = 0x⁴ - 2x³ + 32x² + 80x + 120

Fit a curve

y = a + bx

by the method of least squares for the data:

x	1	2	3	4	5	6	7	8	9
y	9	8	10	12	11	13	14	16	15

8 marks

Answer

To fit the line y = a + bx, we use the normal equations:

Σy = na + bΣx

Σxy = aΣx + bΣx²

Given n = 9, we calculate the sums:

x	y	xy	x²
1	9	9	1
2	8	16	4
3	10	30	9
4	12	48	16
5	11	55	25
6	13	78	36
7	14	98	49
8	16	128	64
9	15	135	81
Total	108	597	285

Here, Σx = 45, Σy = 108, Σxy = 597, Σx² = 285. Substituting into the normal equations:

108 = 9a + 45b

597 = 45a + 285b

Solving the system: Dividing the first by 9 gives a + 5b = 12, so a = 12 - 5b. Substituting into the second:

45(12 - 5b) + 285b = 597

540 - 225b + 285b = 597

60b = 57 => b = 0.95

Then a = 12 - 5(0.95) = 7.25. The fitted curve is y = 7.25 + 0.95x.

215 marks

Find y(3.4) from the following table using Newton’s forward interpolation formula.

x	3	4	5	6
y	31	69	131	223

7 marks

Answer

Fit a curve using Lagrange’s interpolation formula for the data:

x	0	5	10	15	20	25
y	12	15	17	22	24	30

8 marks

Answer

315 marks

Find a root of the equation

f(x) = x + log₁₀x - 2

using Newton Raphson method.

7 marks

Answer

Given that

dy/dx = xy + 1

and y(0)=1. Compute y(0.1) using Picards method upto three decimals.

8 marks

Answer

Find y(0.1) and y(0.2) using Runge Kutta method given that

y' = x - y², y(0) = 1.

15 marks

Answer

Find the Fourier sine transform of

1 / (x(x² + a²))

15 marks

Answer

Obtain a Fourier series for

f(x) = sin(x)

(0 ≤ x ≤ π)

15 marks

Answer

715 marks

Solve the partial differential equation

p² + q² = z²(p+q)

8 marks

Answer

Solve the partial differential equation

(y-z)p + (z-x)q = (x-y)

7 marks

Answer

Verify Green’s theorem for

∫_c ((3x - 8y²)dx + (4y - 6xy)dy)

where C is the region bounded by x=0, y=0 and x+y=1.

15 marks

JNTU B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (September/October - 2021 - R15)

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