RGPV Question Papers BE

Engineering Mathematics 2nd Sem Jan/Feb 2006

Note  : (i) Answer all questions.

(ii) Total six question are to be attempted

(iii) There is internal choice within given below.

(iv) Answer of all objective question should be at one place in the beginning

(v) Assume suitable data wherever necessary.

1. (a) Attempt any ten questions out of fifteen questions.

(a) In the fourier series for f(x) =  |sin x| in (-π, π) the value of bn is :

(i) 1      (ii) (-1)n/n2          (iii)      (iv) 0

(b) Laplace  transform of t3  e-3t is :

(i)         (ii)      (iii)     (iv)

(c ) Laplace transform of   is :

(i)      log        (ii) ½ log       (iii) ½ log     (iv)  log

(d) J_ ½ (x) is :

(i)     (ii)      (iii)     (iv)

(e) Bessel’s  equation of order zero is :

(i) xy1 + y = 0                  (ii) xy2 + xy1 + y = 0

(iii) xy2 + y1 + xy = 0       (iv) xy2 + xy1 + xy = 0

(f) The solution of heat equation       is :

(i) u = (c1 epx + c2 e-px)     ec 2 p 2 t

(ii)  u = c3 + c4 x

(iii) u=( c5 cos px + c6 sin px)e –c2 p2 t

(iv) u= c1 + c2 x

(g) The equation   + 2xy (2 +  =5 is of order and degree :

(i) One and Two   (ii) Two and one   (iii) Two and two  (iv) None of these :

(h) If u = x2 + t2 is a solution of c2  = , then c is:

(i) One   (ii) Two  (iii) Zero (iv) None of these

(i) The completementary function of r – 7s + 6t = ex+y is :

(i) f1(y-x)  + f2(y-6x)    (ii) f1(y+x)  + f2(y+6x)

(iii) f1(y+2x)  + f2(y+3x) (iv) f1(y+3x)  + f2(y-4x)

(j) If =  xi + yj + z k and |  | = r, then div   is :

(i) 2  (ii) 3   (iii) -3    (iv) -2

(k) If  = xy2 I + 2yx2z j – 3yz2, then curl   is at the point (1, -1, 1) :

(i) – (j+2k)    (ii) (i+3k) (iii) – (i+2k)  (iv) (i+2j+k)

(I) If = xi + yj + zk  and r= ||, then ϕ (r) is :

(i) ϕ’(r )        (ii)     (iii)      (iv) None of these

(m) In the poisson distribution if 2P(x=2), then the variance is :

(i) 0    (ii) -1  (iii) 4      (iv) 2

(n) The mean deviation from the mean of the normal distribution is :

(i)       (ii)     (iii)    (iv)

(o) Poisson distribution with unit mean , mean-deviatian about the mean is :

(i)       (ii)       (iii)       (iv)

2. (a) Define the Dirac-Delta function or unit impulse function and evaluate :

L-1     log  s2-1/s2

(b) Find the fourier series for the function  f(x) = x + x2 in the interval  -π < x < π. Hence show that :

1 + +  + ……….

OR

(a) Find the inverse Laplace transform of :

S2 +6/(S2 + 1) (S2+4)

(B) Given that :

F(x)  =       kx          0 x l /2

K(l-x)     l /2  x  l

Expand f(x) as a fourier series .

Solve x2   x  – y=0, given that   x +   is one integral.

(b) Show that :

Missing some

(a) Show that when n is a positive integer :

J-n(x) = (-1)n  Jn(x)

(b)(i)   Solve y’’ -6y’ + 9y = e3x/x2  by using method variation of parameter .

(ii) Solve :

Y’’ –(1+ 4ex) y’+ 3e2xy  = Exp (2(x+ ex))

By using the method of changing the independent variable .

4. (a) From the partial differential equation by eliminating the arbiratery function from :

Z = y2 + 2f (1/x + log y)

(b) A string is stretched between the fixed points (0,0) and (l,0) and released at rest from the initial deflection given by  :

0 <x< /2

F(x) =          (-x)        < x <

Find the deflection of the string at any time .

OR

(a ) Solve the following equation:

(x2 –y2-z2)  p+ 2xyq = 2xz

(b)(i)  Solve the complete integral of the equation :

2(z+ xp+ yq)  = yp2

(ii) Solve (D2– DD’ – 2D’) Z= (Y-1) ex

5. (a) Given that :

(t) =2i  – j + 2k  at t = 2

= 4i -2j +3k at t = 3

Show that  :

r. dr/dt. = dt = 10

(b) Evaluate   dS, where = 18zi – 12j + 3yk and S is the part of the plane 2x + 3y + 6z = 12 which is an the first octant.

OR

(a) If     = (x+y+1)  i + j – (x+y) k, then show that :

curl   =0

(b) Verify Stoke’s theorem for the function    = x2i +  xyz integrated round the square whose sides are x=0 , y=0, x=a  and y=a in the plane z=0.

6. (a) The probability that a bomb dropped from a plane will strike the target  is 1/5. If six bombs are dropped , Find the probability that :

(i) Exactly two will strike the largest.

(ii) At least two will strike the largest

(b) (i) Fit a second degree parabola to the following :

(ii) Find the mean and variance for poisson distribution .

OR

(a) Write a short notes on ‘Theory of Reilabilbillity’.

(b) (i) A car-hire firm has two cars, which it hires out day-by-day . The number of demands for a car on each day is distributed as a poission distribution with mean 1.5 Calculate the proportion of days on which neither car is used and the proportion of days on which some demand is refused . (e-1.5 = 0.2231)

(ii) Prepare Index number for 1904 on the basis of 1902 where the following information is given.