Maths 2 Model Question Paper

06MAT21 Eighth Semester B.E. Degree Examination, Dec.08/Jan.09 Second Semester B.E. Degree Examination Engineering Mathematics - II Time: 3 hrs.

Max. Marks:100 Note:1. Answer any FIVE full questions, choosing at least two from each part. 2. Answer all objectives type questions only in first and second writing pages. 3. Answer to the objective type questions should not be repeated.

1

Part A a. Select correct answer in each of the following: i)

The curvature at any point on the curve y= x 3 is, 2x 6x 5x A) B) C) 3 3 3 1+ 9 x 4 2 1+ 9 x 4 2 1+ 9 x 4 2 The radius of curvature of a curve in the pedal form is, 2 dr dr 2 d r C) p A) r B) r 2 dp dp dp

(

ii)

)

(

)

(

)

D) None of these.

D) r 2

dr dp

iii)

The value of ‘C’ of the Cauchy’s mean value theorem for f ( x ) = e x and g( x ) = e − x in [4, 5] is, A) 52 B) 32 C) 92 D) 21

iv)

Maclaurin’s series expansion of e x is, x2 x3 x3 x5 x3 x5 + +..... B) x − + ..... C) x + + +..... A) 1+ x + 3! 5! 3! 5! 2! 3!

x2 x4 D) 1− + ..... 2! 4! (04 Marks)

4

4

b. Find the radius of curvature of the curve x + y = 2 at the point (1, 1). c. State and prove Cauchy’s mean value theorem. d. Expand esin x using Maclaurin’s series upto the term containing x 4 . 2

(04 Marks) (06 Marks) (06 Marks)

a. Select correct answer in each of the following: a x −b x Lt i) = x →0 x a A) logab B) log C) a log b D) Does not exist b ii) The set of necessary conditions for f(x, y) to have a maximum or minimum is, ∂f ∂f ∂ 2f ∂ 2f A) =0 C) =0 D) None of these = 0, = 0 B) ∂x ∂y ∂y∂x ∂x 2 iii) The rectangular solid of maximum volume that can be inscribed in a sphere is, A) a cube B) a triangle C) a rectangle D) None of these iv) In a plane triangle ABC the maximum value of cosAcosBcosC is, A) 83 B) 81 C) 85 D) 25 (04 Marks) 8 b. Evaluate Lt x log tan x .

(04 Marks)

c. Expand cosxcosy in powers of x and y upto fourth degree terms. d. Find the extreme values of xy subject to the condition x 2 + xy + y 2 = a 2 .

(06 Marks)

x →0

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(06 Marks)

06MAT21 3

a. Select correct answer in each of the following: 45

i)

∫∫ x

2

ydydx =

13

A) 58

B) 168

C) 100

D) 125

a 2 b 2c 2 C) 27

a 2 bc 2 D) 27

C) 3

D) 4

abc

ii)

∫∫∫ x

2 2 2

y z dxdydz =

000

iii) iv)

a 3 b 3c 3 abc A) B) 27 3 The value of β(2, 1)+β(1, 2) is, A) 2 B) 1 The value of

( 12 ) is, B) π

π

A)

C)

π 2

D)

π

(04 Marks)

2

b. Find the area between the parabolas y 2 = 4ax and x 2 = 4ay . 1 1−x 1− x − y

c. Evaluate

∫ ∫ 0 ∞

d. Express

0

dx

∫ 1+ x 4

dzdydx

∫ (1+ x + y + z )3 0

(04 Marks)

.

(06 Marks)

in terms of Beta function and hence evaluate.

(06 Marks)

0

4

a. Select correct answer in each of the following: →

i)

If F is the force acting upon a particle moves from one end of a curve C to the other →

end then the total work done by F is, → →

→→

→

A) ∫ F × dr

B) ∫ F ⋅ dr

C) ∫ dr

D) None of these

→

ii)

The line integral of F = x 2i + xyj from O(0, 0) to P(1, 1) along the straight path is, A)

iii)

iv)

1 3

If M, N,

B) ∂N ∂x

,

∂M ∂y

5 3

C)

2 3

D)

4 3

are continuous functions, C is a simple closed curve enclosing the

region R in the xy plane then Green’s theorem states that,  ∂N ∂M   ∂N ∂M  A) ∫ Mdx + Ndy = ∫ ∫  − B) ∫ Mdx + Ndy = ∫ ∫  + dxdy dxdy ∂y  ∂y  C R  ∂x C R  ∂x  ∂M ∂N   ∂M ∂N  C) ∫ Mdx + Ndy = ∫ ∫  − D) ∫ Mdx − Ndy = ∫ ∫  − dxdy dxdy ∂x  ∂y  C R  ∂y C R  ∂x The cylindrical coordinate system is, A) Not orthogonal B)Orthogonal C)Coplanar D) Non-coplanar (04 Marks) →

b. Find the total work done by the force represented by F = 3xyi − yj+ 2 xzk in a moving particle round the circle x 2 + y 2 = 4 .

(04 Marks) →

c. Verify Stokes theorem for the vector field, F = (2 x − y)i − yz 2 j− y 2 zk over the upper half surface of x 2 + y 2 + z 2 =1 bounded by its projection on the xy-plane.

(06 Marks)

→

d. Express the vector F = zi − 2 xj+ yk in cylindrical coordinates. 2 of 4

(06 Marks)

06MAT21 5

a.

Part B Select the correct answer in each of the following : i) The solution of the different equation (D2 – a2)y = 0 is A) C1eax + C2e-ax B) (C1 + C2x)eax C) (C1 + C2x2)eax ii)

P.I. of the differential equation (D2 + 5D + 6)y = ex is

ex ex C) 12ex D) 12 6 x C.F. of y″ - 2y′ + y = x e sin x is A) C1ex + C2e-x B) (C1 + C2x) ex C) C1 + C2ex D) None of these P.I. of y″ - 3y′ + 2y = 4 is 1 1 3 A) 2 B) C) D) (04 Marks) 2 4 2

A) ex iii) iv)

+6

d2y

Solve

c.

Solve the equation

dx 2 a.

dx 3

dx 2

+ 11 d2y

dy + 6y = e2x . dx

(04 Marks)

dy + 5 y = e 2 x sin x . dx dx Using the method of undetermined coefficient solve the equation, d2y

6

d3y

B)

b.

d.

D) (C1x+C2x2)eax.

+2

2

−2

dy + 4 y = 2 x 2 + 3e − x . dx

(06 Marks)

(06 Marks)

Select the correct answer in each of the following : i) The Wronskian of cos x and sin x is A) 0 B) 1 C) 2

D) 4

2

dy + y = log x into a linear differential equation with dx dx 2 constant coefficients put x = A) et B) Log x C) ex D) t iii) The solution of the differential equation y″ + y = 0, satisfying the conditions y(0) = 1 π and y   =2 is 2 A) Cos x + 2 sin x B) 2Cos x + Sin x C) Cos x – Sin x D) None of these x iv) C1 Cos ax + C2 Sin ax Cos ax is the general solution of 2a A) (D2 + a2)y = Sin ax C) (D2 + a2)y = Cos ax B) (D2 – a2)y = Sin ax D) (D + a)y = Sin x. (04 Marks) ii)

b. c. d.

To transform x 2

2 2d y Solve x 2

d y

−x

dy + y = log x . dx dx Solve y″-2y′+y=ex log x by using method of variation of parameters. d2y dy dy d 2 y Solve +4 + 5 y = 0 , given that y = 2 and = when x = 0. dx dx dx 2 dx 2

−x

3 of 4

(04 Marks) (06 Marks) (06 Marks)

06MAT21 7

a.

Select the correct answer in each of the following : i) Laplace transform of f(t) : t ≥ 0 is defined as ∞

A)

∫e

− st

1

f ( t )dt

0

B) ∫ e

− st

∞

f ( t )dt

C)

0

∫e

− st

∞

f ( t )dt

D)

1

Laplace transform of sin at is 1 a B) A) S2 + a 2 S2 + a 2 3t iii) Laplace transform of e sint is 1 1 A) B) 2 2 S + S + 10 S − 6S + 10 iv) Laplace transform of f ′( t ) = A) s f(s) – f(0) B) s f′ (s) – f(0)

∫e

st

f ( t )dt

0

ii)

2t

b. c.

C)

C)

S S2 + a 2 1 2

D)

D)

S + 4S + 10 C) f(s) – f(0)

1 S2 − a 2 S 2

S − 6S+10

D) s f′ (s) – f(0).

2

Find the Laplace transform of e Cos t. Find the Laplace transform of the periodic function of period 4 defined by, 3t for 0 < t < 2 f(t) =   6 for 2 < t < 4  π − t for 0 < t < π d. Express the function f(t) =  t>π Sin t for in terms of unit step function and find its Laplace transform.

8

a.

(04 Marks) (04 Marks)

(06 Marks)

(06 Marks)

Select the correct answer in each of the following : s−a i) Inverse Laplace transform of is (s − a ) 2 + b 2 A) et cos bt B) eat Cos bt C) e-at Cos bt D) eat Sin bt. s 2 −3s + 4 is ii) Inverse Laplace transform of s3 A) 1 – 3t + 2t2 B) 10 – 3t + 2t2 C) 4 – 3t + 4t2 D) 5 – 6t + 2t2.  1  iii) L−1  n  is possible only when n is s  A) negative integer B) positive integer C) zero D) Real number.  1  iv) L−1  = 5  ( s + 3 )   A)

e −3 t t 4 24

B)

e5 t t 4 24

C)

e 2t t 4 24

D)

e −3 t t 3 24

3s + 2

b.

Find the inverse Laplace transform of 2 . s −s − 2

c.

Using the convolution theorem obtain the inverse Laplace transform of

(04 Marks) (04 Marks)

2 s 2 2 2 2 (s + a )(s + b )

.

(06 Marks)

d. Solve the differential equation

d2y dt

2

+4

dy + 4 y = e − t with y(0) = 0 = y′(0) , using Laplace dt

transform method.

(06 Marks)

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