Engineering Maths 2 - Jan 2013

First Year Engineering (C 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 correct answers for the following:

1 (a) (i) The general solution of the equation p²-5p+6=0 is:
(a) (y-2x-c)(y-3x-c)=0
(b) (y+2x-c)(y+3x-c)=0
(c) (y-2x-c)(y+3x-c)=0
(d) (y-x-c)(y+x-c)=0(1 marks) 1 (a) (ii) If a differentiable equation is solvable for y then it is of the form:
(a) x=f(y,p)
(b) y=f(x,p)
(c) y=f(x²,py)
(d) x=f(y²,p)(1 marks) 1 (a) (iii) The differentiable equation of the form y=px+f(p) whose general solution is y=cx+f(c) is known as?
(a) Glairauts equation
(b) Cauchys equation
(c) Lagranges equation
(d) None of these(1 marks) 1 (a) (iv) The singular solution of the equation y=px-logp is:
(a) y=1-logx
(b) y=1-log(1/x)
(c) y=logx-2x
(d) None of these(1 marks) 1 (b) Solve the equation p² + p(x+y) + xy = 0(5 marks) 1 (c) Solve the equation xp²- 2yp + ax = 0(5 marks) 1 (d) Obtain the general solution and singular solution of the equation sinpx�cosy = cospx�siny+p(6 marks)

Choose correct answers for the following:

2 (a) (i) The homogeneous linear differential equation whose auxiliary equation has roots 1,1,-2 is:
(a) D³ + 3D² + D + 1 = 0
(b) D³ - 3D + 2 = 0
(c) (D+1)²(D+2) = 0
(d) D³ + 3D + 2 = 0(1 marks) 2 (a) (ii) The complementary function for the differential equation (D²+2D+1)y = 2x+x² is:
(a) c₁e^-x + x²c₂e^-x
(b) c₁e^x +c₂e^-x
(c) (c₁+c₂)e^x
(d) (c₁+c₂)e^-x(1 marks) 2 (a) (iii) The particular integral of (D²+a²)y = cosax is:
(a) (-x/2a)sinax
(b) (x/2a)cosax
(c) (-x/2a)cosax
(d) (x/2a)sinax(1 marks) 2 (a) (iv) The general solution of an n^th order linear differential equation contains:
(a) at most n constants
(b) exactly n independent constants
(c) atleast n independent constants
(d) more than n constants(1 marks) 2 (b) Solve: y'' - 2y' + y = xe^xsinx(5 marks) 2 (c) Solve: d²y/dx² - 4dy/dx + 4y = e^2x + cosx + 4(6 marks) 2 (d) Solve: dx/dt = 2x-3y, dy/dt = y-2x given x(0) =8 and y(0) = 3(6 marks)

Choose correct answers for the following:

3 (a) (i) By the method of variation of parameters,the value of W is called:
(a) Demorgans function
(b) Eulers function
(c) Wronskian function
(d) None of these(1 marks) 3 (a) (ii) The differential equation of the form a₀(ax+b)²y''+a₁(ax+b)y'+a₂y = φ(x) is called:
(a) Simultaneous equation
(b) Legendres equation
(c) Cauchys equation
(d) Eulers equation(1 marks) 3 (a) (iii) The equation x³ d³y/dx³ + 3x² d²y/dx² + x dy/dx = x³logx by putting x=e^t with D=d/dt reduces to?
(a) (D³+D²+D)y = 0
(b) D³y = 0
(c) D³y = te^3t
(d) none of these(1 marks) 3 (a) (iv) To find the series solution for the equation 4xd²y/dx² + 2dy/dx + y = 0,we assume the solution as:
(1 marks) 3 (b) Using the variation of parameters method,solve the equation y'' - 2y' + y = e^x/x(4 marks) 3 (c) Solve the equation : x²y'' - xy' + 2y = xsin(logx)(6 marks) 3 (d) Obtain the Frobenius type series solution of the equation xd²y/dx² + y = 0(6 marks)

Choose correct answers for the following:

4 (a) (i) The partial differential equation obtained by eliminating arbitrary constants from the relation Z = (x-a²) + (y-b²) is:
(a) p² + q² = 4z
(b) p² - q² = 4z
(c) p + q = z
(d) p - q = 2z(1 marks) 4 (a) (ii) The auxiliary equations of Lagranges linear equation Pp + Qq = R = are:
(a) dx/p = dy/q = dz/R
(b) dx/P = dy/Q = dz/R
(c) dx/x = dy/y = dz/z
(d) dx/x + dy/y - dz/z = 0(1 marks) 4 (a) (iii) General solution of the equation ∂²z/(∂x∂y) = x²y is:
(a) (1⁄6) x³y² + f(y) + g(x)
(b) (1⁄6) x²y² + f(y)
(c) (1⁄6) x³y³
(d) None of these(1 marks) 4 (a) (iv) By the method of separation of variables,we seek a solution in the form:
(a) X = X(x)Y(y)
(b) Z = X+Y
(c) Z = X²Y²
(d) Z =X/Y(1 marks) 4 (b) Form a partial differential equation from the relation Z = f(y) + φ(x+y) (5 marks) 4 (c) Solve the equation (x² - y² - z²)p + 2xyq = 2xz(5 marks) 4 (d) Use the method of separation of variables to solve ∂u/∂x = 2∂u/∂t + u; given that u(x,0) = 6e^-3x (6 marks)

Choose correct answers for the following:

5 (a) (i) ∫₀¹∫₀^x2 e^y/x dydx is equal to:
(a) 1⁄2
(b) -1⁄2
(c) 1⁄4
(d) 2⁄5(1 marks) 5 (a) (ii) The integral ∫₀^∞∫₀^∞e^-(x2+y2) dxdy by changing to polar form becomes:
(1 marks) 5 (a) (iii) β (3,1⁄2)  is equal to:
(a)  16⁄11
(b)  16⁄15
(c)  15⁄16
(d) 2π/3 (1 marks) 5 (a) (iv) The integral 2∫₀^∞e^-x2 dx  is:
(a) Γ(3⁄2)
(b) Γ(n+1)
(c) Γ(-1⁄2)
(d) Γ(1⁄2)(1 marks) 5 (b) Evaluate by changing the order of integration ∫₀^a∫₀^2√(xa)x² dydx; a>0 (4 marks) 5 (c) Evaluate the integral ∫₀¹∫₀^√(1-x2)∫₀^√(1-x2-y2) xyz dzdydx (6 marks) 5 (d) Prove that ∫₀^∞xe^-x8dx × ∫₀^∞x²e^-x4 dx = π/(16√2) (6 marks)

Choose correct answers for the following:

6 (a) (i) If f = (5xy-6x²)i + (2y-4x)j then ∫_Cf·dr where C is the curve y=x³ from the points (1,1) to (2,8) is:
(a) 35
(b) -35
(c) 3x+4y
(d) None of these(1 marks) 6 (a) (ii) In Green's theorem in the plane ∫_C(Mdx+Ndy) = ?
(1 marks) 6 (a) (iii) If ∫_Cf·dr^→ = 0 then f is called:
(a) Rational
(b) Irrotational
(c) Solenoidal
(d) Rotational (1 marks) 6 (a) (iv) If all the surfaces are closed in a region containing volume V then the following theorem is applicable:
(a) Stokes theorem
(b) Greens theorem
(c) Gauss divergence theorem
(d) none of these(1 marks) 6 (b) If f = (2x²-3z)i - 2xyj - 4xk. Evaluate∫_v curl f·dv where v is the volume of the region bounded by the planes x=0, y=0, z=0 and 2x+2y+z = 4 (4 marks) 6 (c) Verify Green's theorem for ∫_c (3x² - 8y²)dx + (4y-6xy)dy  where c is the triangle formed by x=0, y=0 and x+y=1 (6 marks) 6 (d) Verify the Stokes theorem for f = -y³ i^{^}+ x³ j^{^} where s is the circular disc x² + y² ≤ 1, z=0(6 marks)

Choose correct answers for the following:

7 (a) (i) The Laplace transform of f(t)/t when L[ f(t) ] = F(s) is:
(a) ∫₀^∞F(s) ds
(b) ∫_s^∞F(s) ds
(c) ∫₀^∞F(s-a) ds
(d) ∫₀^∞F(s+a) ds(1 marks) 7 (a) (ii) L[t³e^2t] = ?
(a) 3!/(s-2)⁴
(b) 3!/(s+2)⁴
(c) 3/(s-2)⁴
(d) 3/(s-2)(1 marks) 7 (a) (iii) L [f(t-a)H(t-a)] is equal to:
(a) e^-asL[f(t)]
(b) e^asL[f(t)]
(c) e^-as/s
(d) L[f(t)]/se^-as(1 marks) 7 (a) (iv) L[δ(t)] is equal to:
(a) 0
(b) -1
(c) e^-as
(d) L(1 marks) 7 (b) Evaluate L[sint sin2t sin3t](4 marks) 7 (c) A periodic function of 2π/ω is defined by:
f(t) = Esinωt for 0 ≤ t ≤ π/ω
f(t) = 0  for  π/ω ≤ 1 ≤ 2π/ω
Find  L[f(t)].(6 marks) 7 (d) Express in terms of unit step function and hence find L[f(t)]
f(t) = 2t , 0 < t ≤ π
f(t) = 1,    t > π(6 marks)

Choose correct answers for the following:

8 (a) (i) L^-1[F(s)/s] is equal to:
(a) ∫₀¹ f(t)·dt
(b) ∫₀^∞ f(t)·dt
(c) ∫₀^∞ f(t-a)·dt
(d) ∫₀^t f(t-a)·d(1 marks) 8 (a) (ii) L^-1[1/(s²+2s+5)] is equal to:
(a) e^tsin2t
(b)  1⁄2e^-tsin2t
(c)  1⁄2e^tcos2t
(d) e²tcos2t(1 marks) 8 (a) (iii) f(t)×g(t) is defined by:
(a) ∫₀^t f(t-u)g(u)du
(b) ∫₀^∞ f(t)dt
(c) ∫₀^t f(t)g(t)du
(d) ∫₀^t f(u)g(u)du(1 marks) 8 (a) (iv) L^-1[1/(s²+a²)] is:
(a) cosat
(b) secat
(c) sinat
(d) (1/a)sinat(1 marks) 8 (b) Find L^-1{(2s-1)/(s²+2s+17)}(4 marks) 8 (c) By employing the convolution theorem evaluate L^-1{s/(s²+a²)²}(6 marks) 8 (d) Solve the initial value problem y'' - 3y' + 2y = 4t + e^3t; y(0)=1, y'(0) = -1 using Laplace transforms(6 marks)