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Q2 (a) If u = ( ) )x/y(x/yx ψ+φ , prove that 0y

uyyxuxy2

xux

2

22

2

2

22 =

∂

∂+

∂∂∂

+∂

∂

Answer

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Q2 (b) Show that the maximum and minimum of the radii vectors of the sections of

the surface 2

2

2

2

2

22222

cz

b

y

ax)zyx( ++=++ by the plane

0zyx =γ+µ+λ are given by the equation-

0rc1

crb1

bra1

a22

22

22

22

22

22=

−

γ+

−

µ+

−

λ

Answer

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Q3 (a) Evaluate dxdy)yx( 2∫∫ + over the area bounded by the ellipse 1by

ax

2

2

2

2

=+

Answer

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Q3 (b) Evaluate ∫∫∫ dxdydzz2 over the sphere x2+y2 +z2 = 1

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Answer

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Q4 (a) For what values of η , the equations x + y + z =1, x + 2y + 4z = η , x + 4y +

10z = 2η have a solution and solve them completely in each case.

Answer

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Q4 (b) Determine the eigenvalues and the corresponding eigenvectors of the matrix

A =

−−−

−

312132

226

Answer

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Q5 (a) Find by Newton – Raphson method, the real root of the equation:

3x = cosx + 1

Nearer to 1, correct to three decimal places.

Answer

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Q5 (b) Apply Runge-Kutta method of fourth order to find approximate value of y

for x = 0.2, in steps of 0.1, if 2y x dxdy

+= , given that y = 1 where x = 0

Answer

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Q6 (a) Solve the equation cos x dy = y (sin x – y) dx

Answer

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Q6 (b) Find the orthogonal trajectories of the family of curves:

1)b(

y)a(

x2

2

2

2=

λ++

λ+

Where λ is the parameter.

Answer

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Q7 (a) Find the solution of the equation y4dx

yd2

2+ = 8 cos 2x

given that y = 0 and dxdy = 2 when x = 0

Answer

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Q7 (b) Solve by the method of variation of parameters:

x2

2

e12y

dxyd

+=−

Answer

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Q8 (a) Show that )n,m(Ba)ba(

1dx)bxa(

)x1(xnm

1

0nm

1n1m

+=

+

−∫ +

−−

where B (m, n) is Beta function.

Answer

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Q8 (b) Solve following differential equation 2x2xxy4dxdyx2

dxyd 222

2++=+− in

power of x.

Answer

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Q9 (a) Prove that:

)x(xJ)x(J)xnn()x("Jx 1nn22

n2

++−−=

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Answer

Q9 (b) Prove that

0dx'P'P)x1( n

1

1m

2 =−∫+

−

Where m and n are distinct positive integers.

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Answer

Text Books

1. Higher Engineering Mathematics, Dr. B.S. Grewal, 40th edition 2007, Khanna Publishers, Delhi.

2. Text book of Engineering Mathematics, N.P. Bali & Manish Goyal, 7th edition 2007, Laxmi Publication (P) Ltd.