Cochin University of Science and Techology (CUST) 2011-2nd Sem B.Tech Computer Science and Engineering B tech I & ester(combined) 101 Engineering Mathematics I - Question Paper
BTS(C)-I All- II -024-B
B. Tech Degree I & II Semester (Combined) Examination June 2011
IT/CS/EC/CE/ME/SE/EB/El/EE/FT 101 ENGINEERING MATHEMATICS I
(2006 Scheme)
Time: 3 Hours Maximum Maries: 100
PART-A (Answer ALL questions)
(8x5=40)
I (a) Solve sec2x tanydx+s&y tanxdy0.
(b) Solve += x3-3. dx x
(c) Examine the convergence of the series j=:---.
*< Vn + vw + 1
(d) If _y = sin(msin"1x),provethat (l-x2)<v2-jr+mV = 0!
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(e) If asin"1 * +y 1 x + y |
, prove that x~-+y~- = tanu. czr cy |
(0 If the horsepower required to propel a steamer varies as the cube of the velocity and square of its length. Prove that a 3% increase in velocity and 4% increase in length will require an increase of about 17% in HP.
(g) Find the length of an arch of the cycloid xa(/-sinf). >' = o(l-cosi).
\Jtff Find the area of the cardioid r = a(l + cos 9).
PART-B
(4 x 15-60)
IL (a) Solve + o2y = sec ax = (8)
dx
(b) A condenser of capacity C discharged through an inductance L and resistance R in
series and the charge q at time Y satisfies the equation L-%-+R-+ = 0 where
dr at C
L = 1 henry, R = 400 ohms, C = 16x10** farads. Find q in terms of/. (7)
OR
III. (a) Solve the differential equations
+v = sin/ dt *
dy
-f-+x = cos / dt
given x *= 2, y = 0 when t 0 (8)
(b) Solve = (7)
//r X
IV. (a) Determine the interval in which the following series is convergent x2 x3 x4
x~T?in;'-~ (8)
(b) Find the Taylor series expansion of logxaboutx= I and hence evaluate Iog(M) correct to four decimal places.
(p.T.aj
OR 1 '
If = [x + >/r*+7] , prove that
(i) (x2 + \)y1xyy-m7y = 0
00 ynrtH*? ~ml)yn=0 at * = 0
Examine the convergence of the series * -t ?+.
3 -I 4-1 5-1
If u* / (r),x - r cos 0 and;/= rsin#, then prove that + = f'(r)+f\r).
dx dy r
If u is n homogeneous function of degree ln' inx and_y, then show dial
..2 d2u . d2u . ..2 d*u . t%.. x ~r+2xyr+y tt=rKn~l)u dx2 dxdy dy2
If z=r,+y3-3axy .find and .
& & dx2 dy2
OR
Find the maximum and minimum distances of the point (3,4, 12) from the sphere x*+y2 +z2 =1.
If x ~ a(l - v), y = icv, J = afev and J' = , then prove that JJ' = I.
d(u,v) d(x,y)
Prove that /?(m,n) = jiLE-.
Im+n
The arc of the curve x% + y = in the first quadrant revolves about the x - axis. Find the area of the surface generated.
OR
-y
Evaluate J j - dx dy by changing the order of integration. q x y
a x x*y
Evaluate f J J *y*s dx dy dz .
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Attachment: |
| Earning: Approval pending. |
