University of Mumbai 2007-1st Sem B.E FE Applied Maths Old - Question Paper

CUj

2hf-Nov-Nk-07 -52

Con. 4950-07.

(OLD COURSE)

( 3 Hours)

N.B. (1) Question No. 1 is compulsory.

(2)    Attempt any four questions out of remaining six questions.

(3)    Figures to the right indicate marks.

1. (a) Solve the equation 17 coshx + 18 sinhx = 1, for real values of x.

(b) If r = acost i + asint j + at tana k then prove that

dr d¹ r

X

dt dt²

(c) Find the n^,h derivative of

(d) If u = (l-2xy + y²) ² then

dU dU o 3 Prove that x--y =y* u* .

dx dy

1 1 +

(ii)

1+2w 2+w 1+w

(b)    Using DeMoivres theorem solve the equation

x⁷ + x⁴ + ix³ + i = 0.

(c)    If cos (x + iy) = cosa + i sina then P.T. sina = + sin²x = + sin h²y.

3. (a) Find the equations of the osculating plane and rectifying plane to the curve x ss 2 log t,    y = 4t, z = 2t² + 1 at t = 1

* A f iB , considering the principal values proves that

^{tan 1} IT)⁼ f^- and A* ^{+ B}* ^{= e B}

(c) Show that

[TURN OVER

iiinwvivvi// '03

' ^l> _vx .. .

Con. 4950-CD-6237-07.    2    

4.    (a) Find the curvature, torsion, radius of curvature and radius of torsion for the curve

r = 3t i +3t² j + 2t³k .

(b)    Apply Taylors theorem to expand

x⁵ - x⁴ +- x³ - x² + x - 1 in powers of (x - 1)

(c)    State Rolles Theorem and verify the same for f(x) = x² -(1 - x)² in 0 < x < 1.

5.    (a) State and prove Euler's Theorem for function of two variables and verify the same for

x

y

, . _CC6 X

(a) P.T. e =e²

2 3

(b)    Expand y = cos⁷x in cosines of multiples of x and then find y_n.

(c)    Find the values of a, b if

lim a sinhx + bsinx _ 5 x->0    x³    - 3 '

7. (a) If z = f(x, y) ; x = e^u cos v_f y = e^u sin v then Prove that

dz dz __2u dz

(i) x + y -- = e ~ 3v du dy

' dz' 2 . 5x , +

' dz = e"2u I ay

(b)    If y = - 18 cos (log x) + 17 sin (log x)    6 Prove that x² y_n + (2n + 1) xy_{n +} 1 + (n² + 1) y = o.

(c)    In calculating the volume of a cone errors of 2% and 1% are found in measuring height 6 and base radius respectively. Find the percentage error in calculating volume.

1

(a) Find the cube roots of unity. If w is a complex cube root of unity then prove that

2

(i) 1 + w + w^? = 0

Attachment: university-of-mumbai-2007-b-e-50943-1.pdf

Earning:   Approval pending.