Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

(a)

T is an equivalence relation but S is not an equivalence relation

(b)

Neither S nor T is an equivalence relation

(c)

Both S and T are equivalence relation

(d)

S is an equivalence relation but T is not an equivalence relation.

Let f : Z➝Z be given by f(x) = \(\begin{cases} \frac { x }{ 2 } \quad if\quad x\quad is\quad even \\ 0\quad if\quad x\quad is\quad odd \end{cases}\). Then f is __________

(a)

one-one but not onto

(b)

onto but not one-one

(c)

one-one and onto

(d)

neither one-one nor onto

If \({ log }_{ \sqrt { x } }\) 0.25 = 4, then the value of x is

(a)

0.5

(b)

2.5

(c)

1.5

(d)

1.25

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

(a)

1

(b)

3

(c)

23

(d)

21

If cos pፀ + cos qፀ = 0 and if p ≠ q, then ፀ is equal to (n is any integer)

(a)

\(\frac { \pi (3n+1) }{ p-q } \)

(b)

\(\frac { \pi (2n+1) }{ p\pm q } \)

(c)

\(\frac { \pi (n\pm 1) }{ p\pm q } \)

(d)

\(\frac { \pi (n+2) }{ p+q } \)

If a is the arithmetic mean and g is the geometric mean of two numbers, then

(a)

a \(\le \) g

(b)

a \(\ge\) g

(c)

a = g

(d)

a > g

The equation of the line with slope 2 and the length of the perpendicular from the origin equal to \(\sqrt5\) is

(a)

x - 2y = \(\sqrt5\)

(b)

2x - y =\(\sqrt5\)

(c)

2x - y = 5

(d)

x - 2y - 5 = 0

If A and B are symmetric matrices of order n, where (A \(\neq\) B), then

(a)

A + B is skew-symmetric

(b)

A + B is symmetric

(c)

A + B is a diagonal matrix

(d)

A + B is a zero matrix

If A is skew-symmetric of order n and C is a column matrix of order n \(\times\) 1, then C^T AC is

(a)

an identity matrix of order n

(b)

an identity matrix of order 1

(c)

a zero matrix of order 1

(d)

an identity matrix of order 2

\(lim_{\alpha \rightarrow {\pi/4}}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

(a)

\(\sqrt{2}\)

(b)

\(1\over \sqrt{2}\)

(c)

1

(d)

2