iitm helping handiitmhelpinghand.webs.com/papers/mathematics-iii 2006-12.pdf... expand sin7 cos3 in...

END TERM EXAMINATION

THIRD SEMESTER [BCA]-DECEMBER 2006

Time: 3 Hours Maximum marks: 75

Q1 (a) Prove that sinh-1

x = log(x + ).

(b) If 2cos = x+1/x, then prove that 2cosn = xn + 1/x

n.

(c) The sequence < (-1)n> is bounded but not convergent. Give any other example of such a

sequence with justification.

(d) If = x and xn 0, for all n, then prove that x 0.

(e) Discuss the convergence of ).

(f) State Cauchy integral test. (g) For any vector in space, find grad .

(h) What are the Dirichlet’s conditions for Fourier series?

(i) For the vector = (x + y+1) + - (x + y) , evaluate . curl .

(j) Solve = (4x + y+1)2, if y (0) =1.

(10 2.5=25)

Q2 (a) Solve the equation x4 – x

3 + x

2 – x + 1 = 0. (6.5)

(b) Find , if xn = (5n

+ 7n + 9

n)1/n

. (6)

OR

(a) Examine the convergence of (x>0). (6.5)

(b) Expand sin7

cos3

in a series of sines of multiples of . (6)

Q3 (a) Prove that )=0, where is vector in space. (6)

(b) Find the directional derivative of the function (x, y, z) = x2 – y

2 +2z

2 at P (1, 2, 3) in the

direction of PQ, where Q has coordinate (5, 0, 4). (6.5)

OR

(a) Find , if = (y2 – 2xyz

3) + (3+2xy – x

2z

3) + (6z

3 – 3x

2yz

2) . (6.5)

(b) Find the angle between the surfaces x2 + y

2 + z

2 = 9 and z = x

2 + y

2 – 3 at the point (2, -1,

2). (6)

Q4 (a) Find Fourier expansion f(x) = x – x2, x (- , ) (6.5)

(b) Express f(x) = x as a half range sine series in (0, 2). (6)

OR

(a) Find Fourier expansion of f(x) = , x (- , ) (6.5)

(b) Express f(x) = as a half range sine series in (0, 1). (6)

Q5 Solve

(a) y= 2px + pn (3)

Paper code: BCA201 Subject: Mathematics – III Paper ID: 20201

Note: Q 1 is compulsory. Internal choice is indicated

IITM HELPING HAND

(b) + a2 y = secax. (4.5)

(c) - y = e3x

cos2x – e2x

sin3x. (5)

OR

(a) Solve (1 + xy2) = 1. (4)

(b) (D2 + 2D

2 + 1) y = x

2cosx. (5)

(c) + 2 +4y = 2x2 + 3 . (4)

IITM HELPING HAND




Q1 (a) Show the series 1 + r + r2 + r

3 +....................... is convergent if < 1.

(b)Find the characteristic equation of the matrix A = .

(c) A 3 3 real matrix has an Eigen value . Then its other two Eigen- values can be

(i) 0, 1 (ii) –i, i (iii) 2i, -2i (iv) 0, -i

(d) Solve y = (x-a) p-p2

(e) Find a unit vector normal to the surface x3 + 3y

2 + 2z

2 = 6 at the point (2, 0, 1).

(f) Find the P.I of (D2 + 4)y = sin2x.

(g) Consider a vector space V = R2(R). Let W = {(2a, a): a R}. Show that W is a subspace

of V.

(h) Define inner product space.

(i) Solve (x3 + 3xy

2)dx + (y

3 + 3x

2y)dy =0.

(j) State Sylvester’s inequality. (10 2.5=25)

Q2 (a) Show that convergent sequence of real no is bounded. (3)

(b) Examine the convergence or divergence of the following series: (4)

(i) ..................

(ii) .

(c) State the Cauchy integral test for convergence and examine the convergence of .

(3)

Q3 (a) Define the following with examples: (4)

(i) Vector space (ii) Linear dependence and independence s of vectors

(iii) Basis and dimension (iv) Eigen values and Eigen vectors.

(b) Let A =

Find the matrix P such that P-1

AP is a diagonal matrix. (3)

(c) Consider the vector space V = R3(R) with the usual Euclidean Inner Product. Transform

the basis V1 = (0, 1, 1), V2 = (0, 0, 1), V3 = (1, 1, 1) into an orthonormal basis by using Gram

Schmidt process.

Q4 Solve the following differential equation (any three) (3+3+4)

(a) = sec(x + y)

(b) D2 – 2D + 1)y = xe

xsinx

(c) P = log(px – y)



IITM HELPING HAND

(d) + (tanx + )y = secx/x.

Q5 (a) Solve +2

+ 9y = 6e

3x + 7e

-2x – log2. (3)

(b) Apply the method of variation of parameters to solve + y =

tanx . (3)

(c) Solve the following equations

(D + 2) x + (D + 1) y = 0

5x + (D + 3) y = 0. (4)

Q6 (a) Use the method of separation of variables and solve = 2 + u, u(x, 0) = 6e-3x

.

(3)

(b) Determine the solution of separation of one dimensional heat equation

= c2 subject to boundary conditions u (0, t) = 0, u (l, t) = 0, (l > 0) and the initial

condition u(x, 0) = x; l being the length of the bar. (7)

Q7 (a) Show that . (3)

(b) Find the directional derivative (x, y, z) = xyz at the point (1, -1, 2) in the direction of

vector (2i – 2j + 2k). (3)

(c) Find the divergence and curl of vector field (4)

V = (x2 – y

2)i + 2xyj + (y

2 – xy)k.

IITM HELPING HAND




Q1 (a) Prove that the arguments of the product of three complex numbers is equal to the sum of

their arguments.

(b) Find 3.

(c) Find the PI for the differential equation (D2- 3D + 2) y = 2e

xcos( ).

(d) Is it possible to have Fourier expansion of the function given by f(x) = sin (1/x) in the

interval (- , ).

(e) Prove that < (-1)n> is not a Cauchy sequence. Is it bounded?

(f) Evaluate

(g) Discuss the convergence of .

(h) If , then prove that = , where is a constant vector.

(i) Write the Fourier expansion of sin2x.

(j) State root test and ratio test for the series. Which of them is stronger?

(10 2.5=25)

Q2 (a) Solve:

(i) (ii) x4

+2x3

- x2

+xy = 1

(iii) +2

+ y = x

2cos

2x (3+5+4.5)

OR

(i) (D2+1) y = sinxsin2x (ii) - = (1+x) secy (6+6.5)

Q3 (a) Find the Fourier series for the function given by f(x) = xsinx, - .

Deduce that - +………

(b) If f(x) = then show that

f(x) = Also, find the sum of 1+ + +………..

(6+6.5)

OR

(a) Find the half range cosine series for (x-1)2, 0<x<1. Deduce that

= 1+ + +………..



IITM HELPING HAND

(b) If f(x) = then show that

f(x) = + . Deduce that

= 1+ + +……….. (6+6.5)

Q4 (a) Prove that = - 2

+ (6)

(b) Verify the Green’s theorem in the plane for over C,

where C is the boundary bounded by y = , y=x2. (6.5)

OR

(a) Using green’s theorem evaluate over C, where C is the

boundary bounded by y2 = 8x, x=2. (6.5)

(b) Prove that (fg) = f g + 2 + g f. (6)

Q8 (a) Express sin6

in terms of cosines of multiples of . (5.5)

(b) Show that the sequence <xn> given by x1=1, xn+1= , n 1 is convergent to .

(7)

OR

(a) If Z = cos +isin , then prove that Zn - = sin (n ). (5.5)

(b) Test the convergence of the series n (7)

IITM HELPING HAND




Q1 (a) Find the 5th

root of (-32).

(b) Separate cos h(x +iy) into real and imaginary parts.

(c) Is the sequence <an> defined an = 0, if n is odd, an = n, if n is even convergent?

(d) Test for the convergence of series ..................

(e) Explain the convergence of logarithmic series.

(f) Give the physical interpretation of divergence or curl.

(g) If = 2x3y

2z

4 find .

(h) Is the series a ............. a Fourier series?

(i) What is the order and degree of the differential equation = 5.

(j) Find the complementary function of the differential equation

- 4 + 8 - 8 + 4y = 0. (10 2.5=25)

Unit-I

Q2 (a) Solve the equation x6 + x

3 + 1 = 0 (6)

(b) Teat for convergence the series whose nth term is xn. (6.5)

OR

(a) Prove that -128sin6

cos2

= cos8 -4cos4 + 4cos2 - 5. (6)

(b) Prove that the sequence <Sn> defined by the recursion formula Sn+1 = , S1 =

converges to the positive root of x2 – x – 7 = 0. (6.5)

Unit-II

Q3 (a) For what value of the constant a will the vector

A = (axy – z3)i + (a – z)x

2j + (1-a)xz

2k be irrotational. (3)

(b) For the function f(x, y) = , find the value of the direction derivative making an angle

300 with the positive axis at the point (0, 1). (3.5)

(c) Evaluate by Green’s theorem where C is the circle

x2 + y

2 = 1. (6)

OR

(a) Show that gradient field describing a motion is irrotational. (4)

(b) If a force F = 2 y +3xy displaces a particle in the xy-plane from (0, 0) to (1, 4) along a

curve y = 4x3. Find the work done. (6)

(c) Calculate 2

where = 4x2 + 9y

2 + z

2 + 1. (2.5)

Unit-III Q4 (a) Obtain the Fourier series for the function



IITM HELPING HAND

f(x) = x + x2 in [-1, 1] (6)

(b) Show that for all values of x in [- ] when k is not an integer,

coskx = . Deduce that cotk =

(6.5)

OR

(a) Find the Fourier series of the function f(x) =

(6.5)

(b) Find the Fourier half- range even expansion of the function f(x) = (-x/l) +1, 0 x 1.

(6)

Unit-IV Q5 (a) Solve by the method of undetermined coefficients (D

2 – 2D +3) y = x

3 +sinx. (6)

(b) Solve - 4y = xsinhx. (6.5)

OR

(a) Solve x2

– 3x + 5y = x2 sinlogx (6.5)

(b) Is the equation (x4y

4 + x

2y

2 + xy) ydx + (x

4y

4 + x

2y

2 + xy) xdy = 0 exact? If not, reduce it

to exact equation and hence solve. (6)

IITM HELPING HAND




Q1 (a) Prove that sinh-1

x = log(x + ).

(b) Show that (1+i )8 + (1+i )

8 = -2

8.

(c) What are the dirichlet’s conditions for Fourier series?

(d) Prove that for every field , div (curl ) =0.

(e) Show that curl of vector field is connected with rotational properties of the vector field

and justifies the same rotation for curl.

(f) Examine the convergence of .

(g) Represent the following function by a Fourier series f(x) = x, 0<x<2 .

(h) Discuss the convergence and divergence of P- series

(i) Solve the following differential equation x4

+ x3y = -sec (xy)

(j) Find the PI of (D2- 5D+6) = e

xcos2x. (10 2.5=25)

Q2 (a) Show that the function z is not analytic any where (3)

(b) Evaluate (3)

(c ) Use De Moivre’s to solve the differential equation x4 – x

3 + x

2 –x + 1 =0 (6.5)

OR

(a) Show that

+i (6.5)

(b) If n is positive integer, prove that (i+ )n + (i+ )

n = 2

n+1cos( ). (3)

(c) Write log(x+ iy) in the form of a +ib. (3)

Q3 Solve:

(a) (6.5)

(b) (D2+5D+6)y = sec

2x(1+2tanx) (6)

OR

(a) + 2 +2y = e-x

sec3x (6.5)

(b) If + 2ytanx = sinx, and y=0 for x = , show that maximum value of y is 1/8. (6)

Q4 (a) Obtain the Fourier series for the function

f(x) = x2, - <x< , hence deduce (6.5)



IITM HELPING HAND

(b) Show that = (2xy+z3)i+x

2j+3xz

2k is a conservative field. Find its scalar potential

such that = (grad ). Find the work done by the force in moving particle from (1, -2, and 1)

to (3, 1, and 4). (6)

OR

(a) Verify the green’s theorem in the plane for over C,

where C is the boundary bounded by y = , y=x2. (6.5)

(b) Represent the following function by a Fourier sine series

f(x) = (6)

Q8 (a) Test the series 1+ + +…………… (6.5)

(b) Test the series 2 3

+………… (6)

OR

(a) Find the directional derivative of div( ) at the point (1, 2, 2) in the direction of outer

normal of sphere x2+y

2+z

2=9 for ( = x

4i+y

4j+z

4k. (3.5)

(b) Find the value of n for which rn

is solenoidal, (3)

(c) Test the following series for divergence and convergence

(i) Exponential series

(ii) Logarithmic series

(iii) Binomial series (2 3=6)

IITM HELPING HAND




1. (a) Convert 12 300 to the rectangular form. (2.5 X 10)

(b) Expand cos4 in power of cos ansd sin

(c) Discuss the convergence of the series 2 -2 + 2- 2+..........

(d) Define absolute convergence

(e) Show that lim(n)1/n

=1

n→

(f) If A is a vector function of a scalar t, prove that d/dt(A ×dA/dt) = A ×d2A/dt

2

(g) If = x + y + z , show that div = 3.

(h) If f(x) is an even function in [-π, π] then shoe that the fourier series of f(x) is f(x) =

a0/2 +

(i) Solve sec2xtanydx + sec

2y tanxdy = 0

(j)Find the particular integral of (D -3)4y = e

3x

2. (a) If cosα + cosβ + cos = 0, sinα + sinβ + sin = 0 prove that cos3α + cos3β + cos3 =

3cos(α+β+ ) and sin3α + sin3β + sin3 = 3sin(α+β+ ) (6.5)

(b) Discuss the convergence of 1 + a + a(a+1)/1.2 + a(a+1)(a+2)/1.2.3+.........a<0 (6)

OR

3. (a) If tan(x+iy) = i, where x and y are real, prove that x is indeterminate and y is infinite.

(6) (b) Show that <xn> defined by xn = 1 +1/6 + 1/11+ ...........1/(5n-4) is not convergent(6.5)

4. (a) The temperature of the points in a soace is given by T(x, y, z) = x2 + y –z . A

mosquito located at (1, 1, 2) desire to fly in such a direction that it will get warm as soon

as possible. In what direction should it move? (3)

(b) Find the magnitude of velocity and acceleration of a particle which moves along the

curve x = 2sin3t, y = 2cos3t, z = 8t, t>0. (3.5)

(c) Evaluate by Greeen’s theorem over C , where

C : x2 + y

2 =1 (6)

OR

5. (a) Find the constant so that the vector = (6xy + z3) + (3x

2 – z) + (3xz

2 –αy) is

irrotational. (4.5)

(b) If a force = 2x2y + 3xy displaces a partiocle in the xy plane from A(0, 0) to

B(1,4) along the curve y = 4x2 Find the work done? (4)

(c) If & are irrotational , prove that is solenoidal (4)

6. (a) Obtain the fourier series of f(x) x sinx, 0 < x < 2 (7.5)



IITM HELPING HAND

(b) Obtain the half range cosine series for f(x) = (5)

OR

7. (a) Obtain the half range cosine series for f(x) =sinx in [0, ] and hence show that

= ½ (5)

(b) Find the fourier series for the function f(x) = x+ x2 in the interval - < x < (7.5)

8. (a) Solve (x2 +1) dy/dx + 2xy = x

2 (3)

(b) If the differential equation (ay2 + x + x

8 )dx + (y

8 – y + bxy)dy = 0 is exact then show

that b =2a (3)

(c) Solve x2 d

2y /dx

2 -2x dy/dx -4y =x

4 (6.5)

OR

9. (a) Solve using the method of undetermined coefficients:

d2y/dx

2 + 3dy/dx+ 2y = 3e

-x + cosx (6.5)

(b) Solve (D2 -4D + 4) y = x

2 + e

x +cos2x (6)

IITM HELPING HAND

IITM HELPING HAND

iitm helping handiitmhelpinghand.webs.com/papers/mathematics-iii 2006-12.pdf... expand sin7 cos3 in...

Documents