JNTU
JNTU B.Tech II MATHEMATICS - II Previous Question Paper (August/September - 2022 - R13)
Answer any five questions. All questions carry equal marks.
Main
Show that the vector πΉΜ = (π₯ + 3π¦)πΜ + (π¦ β 3π§)πΜ + (π₯ β 2π§)πΜ is solenoidal and also find πΉΜ β ππ’ππ πΉΜ .
Answer
A vector field F is solenoidal if its divergence is zero, denoted by div F = β Β· F = 0. Given F = (x + 3y)i + (y - 3z)j + (x - 2z)k, the divergence is calculated as:
Since the divergence is zero, the vector field is solenoidal. To find F Β· curl F, we first compute curl F = β Γ F:
Now, calculate the dot product F Β· curl F:
In what direction from (3, 1, β2) is the directional derivative of π = π₯Β³π¦Β² + π¦π§ maximum? Find also the magnitude of this maximum.
Answer
The directional derivative of a scalar field Ο is maximum in the direction of the gradient vector βΟ. Given Ο = xΒ³yΒ² + yz, the gradient is:
At the point (3, 1, -2):
The direction of maximum increase is the vector 27i + 52j + k. The magnitude of this maximum directional derivative is the magnitude of the gradient vector:
State Stokes theorem. Verify it for the vector field πΉΜ = (2π₯ β π¦)πΜ β π¦π§Β²πΜ β π¦Β²π§πΜ over the upper half surface of the sphere π₯Β² + π¦Β² + π§Β² = 1, bounded by its projection on the π₯π¦-plane.
Answer
Sign in to view the complete answer paper.
If π(π₯) = sin π₯ for 0 β€ π₯ β€ π/4 and cos π₯ for π/4 β€ π₯ β€ π/2, obtain the half-range sine series.
Answer
Sign in to view the complete answer paper.
Find the Fourier transform of π(π₯) = 1 for |π₯| β€ π and 0 for |π₯| > π. Hence prove that β«β^β (sinΒ² ππ₯)/π₯Β² ππ₯ = ππ/2.
Answer
Sign in to view the complete answer paper.
Determine the Fourier series for the function π-π₯Β² in βπ < π₯ < π.
Answer
Sign in to view the complete answer paper.
Find the Fourier sine transform of π(π₯) = πβ»|π₯|. Hence prove that β«β^β (π₯ sin ππ₯)/(1+π₯Β²) ππ₯ = (π/2)πβ»π, π > 0.
Answer
Sign in to view the complete answer paper.
Use Newtonβs formula of interpolation to find cos(0.17) given that cos(0) = 1, cos(0.05) = 0.9988, cos(0.10) = 0.9950, cos(0.15) = 0.9888, cos(0.20) = 0.9801, cos(0.25) = 0.9689 and cos(0.30) = 0.9553.
Answer
Sign in to view the complete answer paper.
Using Lagrangeβs interpolation, find the form of the function π(π₯) and hence evaluate π(0) from the data: (β1, 7), (1, 5), (2, 15) and (3, 19).
Answer
Sign in to view the complete answer paper.
Find π¦ββ given that π¦ββ = 24, π¦ββ = 32, π¦ββ = 35, π¦ββ = 40, using Gauss forward difference formula.
Answer
Sign in to view the complete answer paper.
Fit a parabolic curve to the following data and evaluate π¦(10):
π₯: 1, 2, 3, 4, 5, 6, 7
π¦(π₯): 14, 17, 22, 35, 67, 89, 102
Answer
Sign in to view the complete answer paper.
Find a real root of the equation π₯Β³ β 2 ln π₯ β 4 = 0 correct to three decimal places using the method of false position.
Answer
Sign in to view the complete answer paper.
Apply LU decomposition method to solve the system of equations: π₯ + π¦ + 2π§ = 4; 3π₯ + π¦ β 3π§ = β4; 2π₯ β 3π¦ β 5π§ = β5.
Answer
Sign in to view the complete answer paper.
Evaluate the integral β«βΒΉ ππ₯/(1+π₯Β²) using Gauss-Legendre one-point, 2-point and 3-point formulas. Also compare with the exact solution.
Answer
Sign in to view the complete answer paper.
Solve the Initial Value Problem: ππ¦/ππ₯ = π₯ + π¦, π¦(0) = 1 using Taylorβs series method and compute π¦(1).
Answer
Sign in to view the complete answer paper.