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Maximum : 200 marks PARTI
Time : 1 1/2 hours

Write the definition of the following terms. Each question carries 4 marks.

1.Open interval (a.b).
2.Limit point of a subset of real numbers.
3.Orthogonal matrix.
4.Curl of a vector point function f(x, y, z).
5.Even function.
6.Monotonicaliy increasing sequence.
7.Hyperbolic function cosh x.
8.Symmetric relation.
9.Perfect number.

Finite set.
State True or False. Each question carries 4 marks.
11.Every convergent sequence is bounded.
12.Every continuous function is differentiable.
13.A set which is not open is always closed.
14.An analytic function always satisfies Cauchy-Riemann equations. 
15.x2 ~x is an odd function.
16.Sequence ((-if) is convergent.
17.Every Lipschitz function is uniformly continuous.
18.Uniform convergence or a sequence of functions does not imply pointwise convergence.
19.A group always satisfies commutativity.
20.Every field is an integral domain.

C. Choose the correct answer. Each question carries 4 marks.

21. The equation —-t--—*! represents
a" b'~
(a) parabola (c ) hyperbola
(b) ellipse (d) circle

22. The residue of f(z) =---- at its singular point is,
1 -z
(a)4
(b)—4
(c )1
(d) -1

23. Which of the following is not an entire function?
(a) e?
(c ) cos z
(b)sin z
(d)tan z

24.gives a result by which a volume integral is transformed into a surface
integral.
(a) Gauss’s divergence theorem (c ) Green’s theorem
(b) Stokes thoerem (d) Morera's theorem
25. Which of the following is true? (a) cosh2 x + sinh2 x = 1
(c )(cosh x)=-sinh x
dx
dx

26. Which of the following is a divergent sequence?
(a)(1.1,1..)(b) (-1, 1,-1, 1.)
111
(c )
[1,1/2,1/3,1/4............]
(d)
(1/3 2/ 3  3/4 ]
[ 2 ’ 3  4 .......J

27. Which of the following is a first order linear differential equation?
(a)dy v j —-+ y tan x = e y dx(b)d‘y — . + x dx2c/y -p- + y = sinx dx
(c )(dy  (d)dy x —+ vsinx=e ax
Laplace transform of cos at is,
(a)a 2 2(b)s 2 2
sr+as +a
(c )a s2 -a2(d)* s2-a2

29.The relation '<* (less than) is (a) reflexive (c ) transitive
----relation.
(b) symmetric (d) equivalence
dx

30. [sin"x</jc-
(a)
(b) (c )(d)
sin""1 x cos x
--------------+
n
-sin" 1 xcos x
n.
-sinn'1 xcos x n
sin*5"1 xcosx n
-—- fsin""2x dx n 
—- fain"‘2xdx n J
—- fsin*'2xdx n J
— [sin"-2 x dx
 


PART IT
Each question carries 10 marks.
31.Test for continuity the function
,, xsini for x*0 *
[0 for x = 0 at x-O.

32.If ] is a vector point function, 'prove that div(curlf)^0 i.e. V-(Vx /)=0.

33.Express the complex number l + i in modulus amplitude form (polar form).

34.Evaluate J| ^ |de where C is the unit circle in the left half plane.

35.Using Mathematical induction, prove that —— + —— ....—r~—r = n■ ■ ■ for all neN .
1-2 2-3 n(n +1) n + 1

36.Solve the differential equation, x (l + y2 )dx + y (l + x2 )dy = 0.

37.Find the equation of the line perpendicular to the line 2x + 3y < 7 = 0 and passing through the point (l, l).

38.Find the inverse of the matrix.