University of Mumbai 2008-1st Sem B.E FE Applied Maths I Old - Question Paper

'^,-4-08-Nk-Ex. 228

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Con. 2940-08. (\ppUQ.ol n1a(7h<;    - CQ-fcfcQg

1 I (OLD COURSE) ^J    Sl/o\ ^>

(3 Hours)    [TotalMarks : 100

N.B.(1) Question No. 1 is compulsory.    ..    A-"

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

(3)    Figures to the right indicate marks.

1. (a) Prove that sin ¹ ix = 2n tc + i log (x -r yj 1 + x^a j.

(b)    If y = sin² x cos³ x find y_n .

(c)    If z = f(x, y). x = e^u + e - \ y = e~ ^u - e ^v. prove that: .

- x - -² 3u 3v dx t?y

(d)    Show that the vector:

(axb)x(cxd) + (bxc )x(axd ) + (c x a ) x (b x d ) is a vector parallel to 3 .

2. (a) Considering only the principal value, if (1 + I tan a)^{1 + ltar}>P is real, prove that its value is q (sec ar^cH

(b) Use De'Moivres theorem to show that    6

5tan8 - 10tan³0 +tan^Ge

tan50 =

1 i0tan⁵0 + 5tan⁴0

(c) Show that-

tan

Q> 1 a- 1- o CO [ a 4 bi)

az -b

3. (a) For the curve r =cost i + sint j + t k , prove that 2 (k^? + t²)

(b)    Prove that tan h*¹ ( sin 0) = cos h^-1 (sec 0)

71 . . 7t

(c)    !f x_r = cos -Tf + I sin , prove that :

O    o

^xr^x2*^x3

x₀-x_rx_a

_x3 5 7

4. (a) Prove that tan¹x=x+ ---- +.........

O O t

Hence expand log (1 + x²) in powers of x.

(b)    Find the equatons of the osculating plane and normal plane to the curve, x = 2t³, y = 3t. z = 6t at t = 1.

(c)    Examine the validity of the conditions and the conclusion of Lagrarge's M.V.T. for the function

Vx² -4 on [ 2, 3 ] .

[TURN OVER

Con. 2940-00-5102-08.

x + y - z

cos

5* (^a) ^,f H " , x² + y²+z^{2 w<} Vx + $ + Vz _a then find the value of

du du 6u x + y +z dx dy dz *

(b) Find the stationary value of xy ( 3 - x - y) .

6. (a) If y = [log( x+ /l+x² )

Prove that (1 + x³) y ₂ + (2n + 1) x y_n , + n^J y_n = 0. Hence deducs that y_{n + 2} (0) = - n² y_n(0).

x r~2-2

+ V x 4 y

U J

flu OU

OU OU J22

Prove that ^xx ^H= "v* +y'

(c) Apply Taylor's theorem to expand

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

7. (a) if u = e^x*⁷ f [ ), prove that-

du du _ x + z =2xyz u dx dz

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Attachment: university-of-mumbai-2008-b-e-50932-1.pdf

Earning:   Approval pending.