JNTU
JNTU B.Tech I Year MATHEMATICS-II Previous Question Paper (September - 2023 - R16)
Note: i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Find the Laplace transform of the function f(t) = t2.
Answer
The Laplace transform of a function f(t) = tn is given by the formula:
For the given function f(t) = t2, we have n = 2. Substituting this into the formula:
Find the Laplace transform of (sin 3t / t).
Answer
To find the Laplace transform of (sin 3t / t), we use the division by t property of the Laplace transform, which states that if L{f(t)} = F(s), then L{f(t)/t} = ∫s∞ F(u) du.
First, let f(t) = sin 3t. The Laplace transform of sin 3t is:
Applying the division property:
Evaluating the integral:
Using the identity π/2 - tan-1(x) = cot-1(x), the final result is:
Find the value of Γ(-1/2).
Answer
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Express ∫03 dx / √(9-x2) in terms of Beta functions.
Answer
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Evaluate ∫0π ∫0a sin θ r dr dθ. What is the region of integration?
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Find the area between two curves y = x; y = x2.
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Define Solenoidal and Irrotational of a vector.
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Find the angle between two surfaces x2 + y2 + z2 = 9 ; x2 + y2 - z - 3 = 0 at (2, -1, 2).
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Find the work done in moving a particle by the force field F̅ = 3x2i + j + zk̅ along the straight line from (0, 0, 0) to (2, 1, 3).
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If F̅ = (2x2 - 3z)i - 2xyj - 4xk̅ then evaluate ∫(∇ × F̅) dv over v, where v is the closed region bounded by x=0, y=0, z=0, 2x+2y+z=4.
Answer
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Part B
Use Laplace transform to prove that ∫0∞ (sin t / t) dt = π/4.
Answer
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Find L-1[s / ((s+1)2(s2+1))].
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Solve by using Laplace transforms y''(t) + 5y'(t) + 6y(t) = t with y(0) = 1 and y'(0) = 1.
Answer
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Evaluate ∫02a x3(2ax - x2)3/2 dx.
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Evaluate ∫0∞ sin(x2) dx.
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Show that ∫01 (xm-1 + xn-1) / (1 + x)m+n dx = β(m, n).
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Express ∫0π/2 sin5θ cos7/2θ dθ in terms of Beta and Gamma functions and hence evaluate.
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Change the order of integration and hence evaluate the double integral ∫04a ∫x2/4a2√(ax) dy dx.
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Find the volume of the tetrahedron bounded by the planes x=0, y=0, z=0 and x/a + y/b + z/c = 1 using triple integration.
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Prove that ∇(rn) = nrn-2 r̅.
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Find constants a, b, c so that the vector A̅ = (x + 2y + az)i + (bx - 3y - z)j + (4x + cy + 2z)k̅ is irrotational. Also find φ such that A̅ = ∇φ.
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Prove that ∇(∇ ⋅ (r̅/r)) = -2/r3 r̅.
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Prove that ∇2(rn) = nrn-2(n + 1).
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Verify Gauss Divergence theorem for F̅ = x3i + y3j + z3k̅ taken over the cube bounded by x=0, x=a, y=0, y=a, z=0, z=a.
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Verify Stokes theorem for F̅ = (y - z + 2)i + (yz + 4)j - xzk̅ where S is the surface of the cube x=0, y=0, z=0, x=2, y=2, z=2 above the xy-plane.
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