## Engineering Maths 2 - Jan 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 correct answers for the following:

**1 (a) (i)** The general solution of the equation p^{2}-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(x^{2},py)

(d) x=f(y^{2},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 p^{2} + p(x+y) + xy = 0(5 marks)
**1 (c) ** Solve the equation xp^{2}- 2yp + ax = 0(5 marks)
**1 (d)** Obtain the general solution and singular solution of the equation $$\sin px \cos y = \cos px \sin y+p$$(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) D^{3} + 3D^{2} + D + 1 = 0

(b) D^{3} - 3D + 2 = 0

(c) (D+1)^{2}(D+2) = 0

(d) D^{3} + 3D + 2 = 0(1 marks)
**2 (a) (ii)** The complementary function for the differential equation (D^{2}+2D+1)y = 2x+x^{2} is:

(a) c_{1}e^{-x }+ x^{2}c_{2}e^{-x}

(b) c_{1}e^{x }+c_{2}e^{-x }

(c) (c_{1}+c_{2})e^{x}

(d) (c_{1}+c_{2})e^{-x}(1 marks)
**2 (a) (iii)** The particular integral of (D^{2}+a^{2})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^{x}sinx(5 marks)
**2 (c) ** Solve: d^{2}y/dx^{2} - 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 a_{0}(ax+b)^{2}y''+a_{1}(ax+b)y'+a_{2}y = φ(x) is called:

(a) Simultaneous equation

(b) Legendres equation

(c) Cauchys equation

(d) Eulers equation(1 marks)
**3 (a) (iii)** The equation x^{3} d^{3}y/dx^{3} + 3x^{2} d^{2}y/dx^{2} + x dy/dx = x^{3}logx by putting x=e^{t} with D=d/dt reduces to?

(a) (D^{3}+D^{2}+D)y = 0

(b) D^{3}y = 0

(c) D^{3}y = te^{3t}

(d) none of these(1 marks)
**3 (a) (iv)** To find the series solution for the equation 4xd^{2}y/dx^{2} + 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 : x^{2}y'' - xy' + 2y = xsin(logx)(6 marks)
**3 (d)** Obtain the Frobenius type series solution of the equation xd^{2}y/dx^{2} + 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-a^{2}) + (y-b^{2}) is:

(a) p^{2} + q^{2} = 4z

(b) p^{2} - q^{2} = 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 ∂^{2}z/(∂x∂y) = x^{2}y is:

(a) (1⁄6) x^{3}y^{2} + f(y) + g(x)

(b) (1⁄6) x^{2}y^{2} + f(y)

(c) (1⁄6) x^{3}y^{3}

(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 = X^{2}Y^{2}

(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 (x^{2} - y^{2} - z^{2})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)** ∫_{0}^{1}∫_{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)}x^{2} dydx; a>0
(4 marks)
**5 (c) ** Evaluate the integral ∫_{0}^{1}∫_{0}^{√(1-x2)}∫_{0}^{√(1-x2-y2)} xyz dzdydx
(6 marks)
**5 (d)** Prove that ∫_{0}^{∞}xe^{-x8}dx × ∫_{0}^{∞}x^{2}e^{-x4} dx = π/(16√2)
(6 marks)

### Choose correct answers for the following:

**6 (a) (i)** If f = (5xy-6x^{2})i + (2y-4x)j then ∫_{C}f·dr where C is the curve y=x^{3} 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 ∫_{C}f·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 = (2x^{2}-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} (3x^{2} - 8y^{2})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 = -y^{3} i^{^}+ x^{3} j^{^ }where s is the circular disc x^{2} + y^{2} ≤ 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[t^{3}e^{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^{-as}L[f(t)]

(b) e^{as}L[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) ∫_{0}^{1} 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/(s^{2}+2s+5)] is equal to:

(a) e^{t}sin2t

(b) 1⁄2e^{-t}sin2t

(c) 1⁄2e^{t}cos2t

(d) e^{2}tcos2t(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/(s^{2}+a^{2})] is:

(a) cosat

(b) secat

(c) sinat

(d) (1/a)sinat(1 marks)
**8 (b)** Find L^{-1}{(2s-1)/(s^{2}+2s+17)}(4 marks)
**8 (c) ** By employing the convolution theorem evaluate L^{-1}{s/(s^{2}+a^{2})^{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)