Periyar University 2006 B.Sc Electronics Allied I — MATHEMATICS — I - Question Paper

Electronics
Allied I — MATHEMATICS — I
Time : 3 hours Maximum : 100 marks
part A — (10 x two = 20 marks)
select the accurate answers :
1. the middle term of X - two / X 12 is
(a) 12C6212
(b) 12C626
(c) 12 C6
(d) 1.

2 one 3
2. The Rank of A = is
1 0 2
(a) 2 (b) 3
(c) 6 (d) 0.
3. The product of the roots of the formula
x5-x3+l = 0 is
(a) -1 (b) 1
(c) one / 5 (d) -1 / 5
4. x3 - 2x2 -3x - four = 0 has a root ranging from
(a) 0&1 (b) 1&2
(c) 2&3 (d) 3&4.
sin0 5045
5. If = the value 01 six is nearly
0 5046
(a) 1° (b) 3°
(c) 2° (d) 4°.
6. sinh 1(A:) is
(a) log/:*; + V*2-ll (b) loglx + V*2 +l)
(c) logx (d) log(l-*).
7, Radius of curvature is
(a) P= ^L1Z_ (b) ili£±_
(c) d+yp (d) a+y)2.
I 8. The curvature of a straight line at any of its point
is
01
(a) 1 (b) -
(c) 0 (d) 2.
9- f(x,y) attains a maximum value at (a, b) if
•q dx2 1S
J{a,b)
61
(a) -ve (b) + ve
(c) 0 (d) +1.
10. y2 + 2yx2 +4^-3 attains its maximum value 0 at
(a) f-f>2] OW (2,D I
(0 (*-±) (d) (ft I).
part B — (5 x six = 30 marks) ans the subsequent.
11. (a) obtain the sum to infinity of the series
,357 I
1+ — + — + —- + . ..<*>. I
2! 3! 4!
Or
(b) Examine the subsequent equations are
consistent or not
x+y+z=6 x + 2y - 2z = -3
2x+3y + z=ll> I

12. (a) Solve x4 + 2x2 -16x + 77 = 0 provided that is a root.
Or
(b) obtain by Newton's method, the positive root of (le formula x3 + 2x2 + 5x - 220 = 0.
|3. (a) Prove that
sin5 0 = —(sin50 -5sin30 +lOsin0).
16V ;
Or
(b) Separate into real and imaginary parts of
ian(x + iy).
114. (a) obtain the differential coefficient of (sinx)cosx .
Or
(b) obtain the radius of curvature at (x, y) for the
curve a2y =x ~a .
15. (a) Verify Euler's theorem when
u = x3 +y3 +z3 + three xyz .
Or
(b) obtain the maximum or minimum value of |
2U2-/)-x4+/.
part C — (5 x 10 = 50 marks) ans the subsequent.
16. (a) obtain the sum to infinity of the since
2-3 3-5 4-7 5-9
+ + + +
3! 4! 5! 6! Or
(b) obtain the eigen values and eigen vectors of
the matrix A =
I3 2,
17. (a) Solve the reciprocal formula
x5 ~5x4 +9x3 -9x2 +5x-l = 0.
Or
(b) obtain by Horner's method accurate to 3 decimal places, the root of the formula
x3 +3x2-8x-6 = 0.
18. (a) If sin(A + iB) = x +iy prove that
2 2 2 2
—iL_ + _4_- = l and -^"7 Z2~T = 1-
cosh^B sinh^B sin'A cos^A
Or
(b) (i) Express cosh6 x in terms of hyperbolic cosines of multiples of x.
¦ (ii) Express sinh5 x in terms of hyperbolic sines of multiples of*.
19. (a) If y = acos(logx) + bsin(logx). Show one that x2y2+xy1+y = Q and x2yn+2 +(2n +l)xyn+1 + , (/i2 + l)yn=0.
6
Or
(b) Prove that the radius of curvature at any )oint of the cycloid x =a(6 + sin six )y = a(l-cos 0) is
?! A 0 „ 4a cos — .
8 2
10. (a) (i) obtain — where u=x2+y2+z2, x=et,
dt
y = et sint f z -e* cost.
(ii) obtain & if x3+y3 =3 axy.
I dx
Or
(b) (i) Express cosh6 x in terms of hyperbolic cosines of multiples of x.
¦ (ii) Express sinh5 x in terms of hyperbolic sines of multiples of*.
19. (a) If y = acos(logx) + bsin(logx). Show one that x2y2+xy1+y = Q and x2yn+2 +(2n +l)xyn+1 + , (/i2 + l)yn=0.
6
Or
(b) Prove that the radius of curvature at any )oint of the cycloid x =a(6 + sin six )y = a(l-cos 0) is
?! A 0 „ 4a cos — .
8 2
10. (a) (i) obtain — where u=x2+y2+z2, x=et,
dt
y = et sint f z -e* cost.
(ii) obtain & if x3+y3 =3 axy.
I dx
Or
(b) (i) obtain the partial differential co-efficient of u = sinte + by + cz).
(n) Prove that ——- = if
axoy dydx
i J *+yz I
u = log — .
xy

Earning:   Approval pending.