JNTU
JNTU B.Tech I MATHEMATICS-II Previous Question Paper (December 2019/January 2020 - R16)
Part A is compulsory which carries 25 marks. Answer all questions. Part B consists of 5 Units. Answer any one full question from each unit. Each question in Part B carries 10 marks.
PART-A
Find L[e2t]
Answer
Using the standard Laplace transform formula for an exponential function, L[eat] = 1 / (s - a), where s > a. Substituting a = 2, we get:
This result is valid for s > 2.
Find inverse Laplace of 1/(s2+2s+2)
Answer
To find the inverse Laplace transform of the given function, first complete the square in the denominator:
The expression becomes:
Using the first shifting theorem, which states that L-1{F(s + a)} = e-atf(t), where f(t) = L-1{F(s)}, we identify F(s) = 1 / (s2 + 1). Since L-1{1 / (s2 + 1)} = sin(t), the inverse transform is:
Evaluate the integral from 0 to π/2 of sin3x cos4x dx using Beta-Gamma functions.
Answer
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Show that Γ(1/2) = √π, using the relation β(m,n) = (Γ(m)Γ(n))/Γ(m+n).
Answer
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Evaluate the triple integral of dxdydz over the limits 0 to 1 for x, y, and z.
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Evaluate the double integral of dydx over the region 0 to x and 0 to 1.
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Find the directional derivative of r = x2+y2+z2 at the point (0, r, 0) in the direction of j.
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Find Curl(grad f) where f = x2+y2+z2.
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State Green’s theorem in a plane.
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Find the work done by F = x2i + y2j + z3k in moving a particle from (-1, 2, 1) to (2, 3, 4) along the line joining them.
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PART-B
Find inverse Laplace transform of L-1[log((s+1)/(s-1))].
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Find L{t et sin 3t}.
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Using Laplace transform solve y'' + 3y' + 2y = 3, given y(0) = 0, y'(0) = 1.
Answer
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Prove the relation between Beta and Gamma functions.
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If β(m,n) = β(m+1,n) + β(m,n+1), find the relationship between variables.
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Prove that Γ(m)Γ(m + 1/2) = (sqrt(π)/22m-1)Γ(2m).
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Show that ∫ (xm-1(1-x)n-1)/(x+a)m+n dx = (1/an) β(m,n).
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Evaluate ∫∫ e-(x2+y2) dxdy by changing to polar coordinates.
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Evaluate ∫∫∫ (xy+yz+zx) dxdydz where V is the region bounded by x=0,1; y=0,2; z=0,3.
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Evaluate ∫∫ xy dydx with limits 0 to x and 1-x to 2.
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Evaluate ∫∫∫ (x+y+z) dxdydz.
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Prove that div(A x B) = B . curl(A) - A . curl(B).
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Evaluate the angle between the normals of the surface xy = z2 at the points (4,1,2) and (3,3,-3).
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Find the unit normal vector to the surface x2+y=xz2 at (2,-2,3).
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For any vector field V, prove that div(curl V) = 0.
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Verify Divergence Theorem for F = x2i + y2j + z2k over the surface S of the solid cut off by the plane x+y+z=a in the 1st octant.
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Find the work done by the force F = (-xy)i + (zj) - (x2k) along the curve x=t, y=t2, z=t3 from t=0 to 1.
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Apply Stokes Theorem to evaluate ∫ F . dR where C is the curve of intersection of the sphere x2+y2+z2=a2 and x+z=a.
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