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.

\(n(p(A))=512,n(p(B))=32,n(A\cup B)=16,\) find \(n(A\cap B) \)  ___________

(a)

2

(b)

9

(c)

4

(d)

5

If A and B are any two finite sets having m and n elements respectively then the cardinality of the power set of A \(\times\) B is ___________

(a)

2^m

(b)

2ⁿ

(c)

mn

(d)

2^mn

The value of log₁₀⁸ + log₁₀⁵- log₁₀⁴ = ___________

(a)

\({ log }_{ 10 }^{ 9 }\)

(b)

\({ log }_{ 10 }^{ 36 }\)

(c)

1

(d)

-1

Everybody in a room shakes hands with everybody else. The total number of handshakes is 91. The total number of persons in the room is _________

(a)

11

(b)

14

(c)

13

(d)

12

Choose the incorrect pair:

(a)

\(\frac{d}{dx}(sin x)\) - cos x

(b)

\(\frac{d}{dx}(tanx)\) - sec²x

(c)

\(\frac{d}{dx}(cos x)\) - sin x

(d)

\(\frac{d}{dx}(logx)\) - \(\frac{1}{x}\)

Equation of the straight line that forms an isosceles triangle with coordinate axes in the I-quadrant with perimeter 4 + 2\(\sqrt{2}\) is

(a)

x + y + 2 = 0

(b)

x + y - 2 = 0

(c)

\(x+y-\sqrt{2}=0\)

(d)

\(x+y+\sqrt{2}=0\)

The lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0(a ≠ b ≠ c ≠ 1) are concurrent, then the value of \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\) ______________

(a)

-1

(b)

1

(c)

0

(d)

abc

The projection of \(\overrightarrow { b } \) on \(\overrightarrow { a } \) is ________ .

(a)

\(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \right) \overrightarrow { b } \)

(b)

\(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \)

(c)

\(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \)

(d)

\(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \right) \)

\(lim_{\theta\rightarrow0}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

(a)

1

(b)

-1

(c)

0

(d)

2