Engineering Maths 2 - Jun 2013

First Year Engineering (P Cycle) (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

Choose the correct answers for the following:

1 (a) (i) A differential equation of the first order but of higher degree, solvable for y, has the solution is:
(a) f(x,y,c)=0
(b) f(x,c₁,c₂)=0
(c) f(x,p,c)=0
(d) f₁(x,y,c)·f₂(x,y,c)=0(1 marks) 1 (a) (ii) If c²x²+1=2cy is the general solution of a differential equation then its singular solution is:
(a) y = x
(b) y = -x
(c) both (a) and (b)
(d) None of these(1 marks) 1 (a) (iii) The general solution of the differential equation p=log(px-y) is:
(a) y = px+e^p
(b) y = px-e^p
(c) y = px-e^c
(d) y = cx-e^c(1 marks) 1 (a) (iv) The differential equation xp² + x = 2yp can be solvable for:
(a) p
(b) y
(c) x
(d) All of these(1 marks) 1 (b) Solve xyp² + p(3x²-2y²) - 6xy = 0(5 marks) 1 (c) Solve y = psinp + cosp(5 marks) 1 (d) Solve y²logy = xyp + p²(6 marks)

Choose the correct answers for the following:

2 (a) (i) (1 marks) 2 (a) (ii) The roots of auxiliary equation of (D⁴ + 2D³ - 5D² - 6D)y = 0 are:
(a) -1, -1, 2, -3
(b) 0, -1, 2, -3
(c) 0, 1, -2, 3
(d) 0, -1, 2, 3(1 marks) 2 (a) (iii) The particular integral of (-D+2)³y = 3e^2x is:
(a) (x³e^2x)/3
(b) (x³e^2x)/2
(c) -(x³e^2x)/2
(d) -(x³e^2x)/6(1 marks) 2 (a) (iv) If dx/dt - 2y = 0, dy/dt - 2x = 0 then y is a function of:
(a) e^2t and e^-2t
(b) e^2it and e^-2it
(c) e^t and e^-2t
(d) none of these(1 marks) 2 (b) Solve (D³ - 6D² + 11D - 6)y = 2^x + cos2x(5 marks) 2 (c) Solve: (D² - 4D + 4)y = 8x²e^2xsin(2x)(5 marks) 2 (d) Solve dx/dt + dy/dt + 2x + y = 0; dy/dt + 5x + 3y = 0.(6 marks)

Choose the correct answers for the following:

3 (a) (i) The complementary function of x²y" + 4xy' + 2y = e^x is:
(a) c₁e^-x + c₂e^-2x
(b) c₁(-x) + c₂(-2x)
(c) c₁e^-2 + c₂e^2z
(d) c₁/x + c₂/x²(1 marks) 3 (a) (ii) If y = u(x)·1 + v(x)·e^2x is a particular integral of y" + y = cosecx in the method of variation of parameters then v(x) = ?
(a) e^-x
(b) e^-2x
(c) e^2x
(d) -e^-x(1 marks) 3 (a) (iii) The roots of the auxiliary equation of the transformed equation of: (2x+1)²y" - 2(2x+1)y' - 12y = 6x+5 are:
(a) 3, -1
(b) -3, 1
(c) 12, -4
(d) None of these(1 marks) 3 (a) (iv) Indicial equation is related to:
(a) Singular point
(b) Regular singular point
(c) Ordinary point
(d) None of these(1 marks) 3 (b) Solve (D²+1)y = tanx by method of variation of parameters.(5 marks) 3 (c) Solve x²y" - xy' + 2y = xsin(logx)(5 marks) 3 (d) Solve (1+x²)y" + xy' - y = 0 in series solution.(6 marks)

Choose the correct answers for the following:

4 (a) (i) z = (x-a)² + (y-b)², a and b are arbitrary constants, is a solution of:
(a) z = 2p² + 2q²
(b) 4z = p² + q²
(c) p = 2(x-a)
(d) q = 2(y-b)(1 marks) 4 (a) (ii) For z = (x+a)(x+b), z=0 is a:
(a) Singular solution
(b) General solution
(c) Particular solution
(d) Complete solution(1 marks) 4 (a) (iii) Suitable set of multipliers to solve (y² + z²)p + xyq = zx.
(a) 0, 1, 1
(b) x, -y, -z
(c) 1, -y/x, -z/x
(d) All of these(1 marks) 4 (a) (iv) Taking Z=X(x)·Y(y) is a solution of a partial differential equation then this procedure is called:
(a) Separation of derivatives
(b) Lagranges method
(c) Separation of variables
(d) Partial separation of variables(1 marks) 4 (b) Form a partial differential equation by eliminating arbitrary function from the relation z = f(xy/z).(5 marks) 4 (c) Solve xp - yq = y² - x²(5 marks) 4 (d) Solve ∂²z/∂x² - 2∂z/∂x + ∂z/∂y = 0 by the method of separation of variables. (6 marks)

Choose the correct answers for the following:

5 (a) (i) ∫₀¹ ∫₀^1-y (x² - y²)dxdy = ...
(a) 0
(b) 1/12
(c) 1/6
(d) None of these(1 marks) 5 (a) (ii) ∫₀¹ ∫₀² ∫₀^(2-x-y) dzdydx = ...
(a) 3
(b) 2
(c) 1
(d) None of these(1 marks) 5 (a) (iii) ∫₀¹[log(1/x)]^1/2dx = ...
(a) Γ(1/2)
(b) Γ(3/2)
(c) Γ(5/2)
(d) None of these (1 marks) 5 (a) (iv) ∫₀^π⁄2 cos^mdx = ...
(a) 1/2 β((m-1)/2,1/2)
(b) β((m+1)/2,1/2)
(c) 1/2 β((m+1)/2,1/2)
(d) 2β((m+1)/2,1/2)(1 marks) 5 (b) Change into polar coordinates and evaluate ∫₀^∞∫₀^∞e^-(x2+y2)dydx (5 marks) 5 (c) Evaluate ∫_-c^c ∫_-b^b ∫_-a^a (x²+y²+z²)dzdydx (5 marks) 5 (d) Prove that β(m,n)=Γ(m)Γ(n)/Γ(m+n) (6 marks)

Choose the correct answers for the following:

6 (a) (i) Which theorem gives a relation between surface integral and volume integral?
(a) Greens
(b) Stokes
(c) Divergence
(d) None of these(1 marks) 6 (a) (ii) If c is x + y = 1 from (0,1) to (1,1) then∫_C(y²dx + x²dy) = ?
(a) 0
(b) 1
(c) 2
(d) 3(1 marks) 6 (a) (iii) The work done by the force F(bar) = yI+xJ+zK moves a particle from (0,0,0) to (2,1,1) along the curve x = t², y = t, z = 0 is:
(a) 3t²
(b) 0
(c) 1
(d) None of these(1 marks) 6 (a) (iv) If S is any closed surface enclosing the volume V then by divergence theorem, the value of ∫_SR^–.dS^– is:
(a) V
(b) 2V
(c) 3V
(d) None of these(1 marks) 6 (b) Use Green's theorem to evaluate ∫_C[(y - sinx)dx + cosxdy] where C is enclosed by y = 0, x = π/2, y = 2x/π (5 marks) 6 (c) Use Stoke's theorem to evaluate ∫_S curl F^–.d(S^–) where F^–= yI + (x-2xz)J - xyK and S is the surface of the sphere x²y² + z² = a² above the x-y plane. (5 marks) 6 (d) By transforming to a triple integral, evaluate: ∫_S{x³ dydz + x²y dzdx + x²z dxdy} where S is the closed surface bounded by the planes z=0, z=b and the cylinder x² + y² = a² (6 marks)

Choose the correct answers for the following:

7 (a) (i) L(2cosh2t) = ?
(a) 4/(s² - 4)
(b) 4s/(s² - 4)
(c) 2s/(s² - a²)
(d) None of these(1 marks) 7 (a) (ii) L{sint/t} = ?
(a) cot^-1s
(b) 1/(s²+1)
(c) tan^-1s
(d) cot^-1(s-1)(1 marks) 7 (a) (iii) L(f'(t)) = ?
(a) sf(t) - f(0)
(b) sf'(s) - f(0)
(c) f(s)··f(0)
(d) None of these (1 marks) 7 (a) (iv) L(sin2t.δ(t-2)) = ?
(a) e^2ssin4
(b) e^-2ssin2
(c) e^-4ssin2
(d) e^-2ssin4(1 marks) 7 (b) Prove that L(tⁿ) = n! / sⁿ⁺¹ if n is a positive integer.(5 marks) 7 (c) Find L((e^-tsint)/t) and hence find ∫₀^∞(e^-t)sint/t dt (5 marks) 7 (d) Express in terms of unit step function and hence find L{f(t)}
f(t) = t-1; 1 < t < 2
f(t) = -t-3; 2 < t < 3
0,otherwise(6 marks)

Choose the correct answers for the following:

8 (a) (i) L^-1(s^-5/2) = ?
(a) 2t^3⁄2/√π
(b) 4t^3⁄2/(3√π)
(c) 8t^3⁄2/(15√π)
(d) None of these(1 marks) 8 (a) (ii) L^-1(f^-(s).g^-(s)) = ?
(a) f(t).g(t)
(b) ∫₀^t f(u)g(t-u)du
(c) ∫₀^t f(t-u)g(u)du
(d) either (b) or (c) (1 marks) 8 (a) (iii) L^-1{1/(s²+5)} = ?
(a) 1/5 sin√t
(b) 1/√5 sin√5t
(c) 1/√5 sin√5t
(d) sin√5t(1 marks) 8 (a) (iv) L^-1(∫_s^∞F(s)ds) = ?
(a) t·f(t)
(b) f(t)/t
(c) f(s)/s
(d) None of these(1 marks) 8 (b) Find L^-1{log (s+1)/(s-1)}(5 marks) 8 (c) Find L^-1[1/(4s²-9)] by using convolution theorem.(5 marks) 8 (d) Solve by using Laplace transformation y''' + 2y'' - y' - 2y = 0 where y=1, dy/dt = 2 = d²y/dt² at t=0(6 marks)