JNTU
JNTU B.Tech II MATHEMATICS-II Previous Question Paper (March-2021 - R15)
Answer any five questions. All questions carry equal marks.
Main
Find the unit normal vector to the surface x3+y3+z3=3 at the point (1, 2, -1).
Answer
Let the surface be defined by the function f(x, y, z) = x3 + y3 + z3 - 3 = 0. The normal vector to the surface at any point is given by the gradient vector ∇f.
Evaluating the gradient at the point (1, 2, -1):
The magnitude of the gradient vector is:
The unit normal vector n is given by ∇f / |∇f|:
Evaluate by stoke’s theorem ∫(ex dx + 2y dy - dz) where c is the curve x2+y2=9 and z=2.
Answer
Stokes' theorem states that ∫C F · dr = ∫∫S (∇ × F) · n dS. Here, F = exi + 2yj - k.
First, calculate the curl of F:
| ∂/∂x ∂/∂y ∂/∂z |
| ex 2y -1 |
= i(0 - 0) - j(0 - 0) + k(0 - 0) = 0
Since the curl of the vector field F is the zero vector, the surface integral over any surface S bounded by the curve C is equal to zero.
Therefore, the line integral ∫C (ex dx + 2y dy - dz) = 0.
If f = 3x2yz2i + x2z2j + 2x3yz k. Show that ∫f∙dr over C is independent of the path of integration. Hence evaluate the integral when C is any path joining (0, 0, 0) to (1, 2, 3).
Answer
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Obtain the half range cosine and sine series for the function f(x) = x when 0 < x < π.
Answer
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Find the missing term in the following data:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 1 | 3 | 9 | - | 81 |
Answer
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Fit a straight line to the following data by the method of least square:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 1 | 1.8 | 3.3 | 4.5 | 6.5 |
Answer
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Solve the system of equations:
using LU decomposition method.
Answer
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Determine the largest eigen value and the corresponding eigenvector of the matrix:
to 3 correct decimal places using the power method.
Answer
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Fit a curve of the form y = a/x + bx to the data given below:
| x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 |
|---|---|---|---|---|---|
| y | 0 | 0.1280 | 0.5440 | 1.2960 | 2.4320 |
Answer
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Evaluate ∫(1/(1+x2)) dx from 0 to 1 by taking h = 1/6 using:
Simpson’s 1/3rd rule
Answer
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Simpson’s 3/8th rule
Answer
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