If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

(a)

[-9,9]

(b)

R

(c)

[-3,3]

(d)

[0,9]

The function f:R➝R is defined by f(x)=\(\frac { \left( { x }^{ 2 }+cosx \right) \left( 1+{ x }^{ 4 } \right) }{ \left( x-sinx \right) \left( 2x-{ x }^{ 3 } \right) } +{ e }^{ -\left| x \right| }\) is

(a)

an odd function

(b)

neither an odd function nor an even function

(c)

an even function

(d)

both odd function and even function.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

(a)

{1, 4, 6, 9}

(b)

{4, 6, 9}

(c)

{1}

(d)

None of these

Which of the following is not an equivalence relation on z?

(a)

aRb ⇔ a+b is an even integer

(b)

aRb ⇔ a-b is an even integer

(c)

aRb ⇔ a

(d)

aRb ⇔ a=b

If A = {(x,y) : y = sin x, x ∈ R} and B = {(x,y) : y = cos x, x ∈ R} then A∩B contains

(a)

no element

(b)

infinitely many elements

(c)

only one element

(d)

cannot be determined

If f(x) = |x - 2| + |x + 2|, x ∈ R, then

(a)

\(f(x)=\left\{\begin{array}{lll} -2 x & \text { if } & x \in(-\infty,-2] \\ 4 & \text { if } & x \in(-2,2] \\ 2 x & \text { if } & x \in(2, \infty) \end{array}\right.\)

(b)

\(f(x)=\begin{cases}2x\ if\ x∈(-∞,-2] \\4x\ if \ x∈(-2,2]\\ - 2x\ if\ x∈(2,∞)\end{cases}\)

(c)

\(f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\-4x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}\)

(d)

\(f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\2x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}\)

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.

The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and 14% of the enrollment in Chemistry. The number of students take at least one of these two subjects, is

(a)

1120

(b)

1130

(c)

1100

(d)

insufficient data

If n((A \(\times\) B) ∩(A \(\times\) C)) = 8 and n(B ∩ C) = 2, then n(A) is

(a)

6

(b)

4

(c)

8

(d)

16

The range of the function \({1\over 1-2sinx}\) is

(a)

\((-∞,-1)\cup\left( {1\over 3},\infty\right)\)

(b)

\(\left( -1,{1\over 3}\right)\)

(c)

\(\left[ -1,{1\over 3}\right]\)

(d)

\((-∞,-1]\cup [\frac { 1 }{ 3 } ,∞)\)

The number of relations from a set containing 4 elements to a set containing 3 elements is:

(a)

2¹⁶

(b)

2⁵

(c)

2⁷

(d)

2¹²

Let X = {a, b,c},y = (1, 2, 3) then \(f:x\rightarrow y\) given by (a, 1) (b, 1) (c, 1) is called ___________

(a)

onto

(b)

constant function

(c)

one one

(d)

bijective

Which of the following functions is an even function?

(a)

\(f(x)={2^x+2^{-n}\over 2^x-2^{-x}}\)

(b)

\(f(x)={3^x+1\over 3^x-1}\)

(c)

\(f(x)={x.3^x-1\over 3^x+1}\)

(d)

\(f(x) = log (x +\sqrt{x^2 + 1})\)

If \(f(x)={1-x\over 1+x},(x\neq0)\) then f^-1(x) =

(a)

f(x)

(b)

\(1\over f(x)\)

(c)

-f(x)

(d)

-\(1\over f(x)\)

If A = {1, 2}, B = {1, 3} then n(A x B) = ___________

(a)

2

(b)

4

(c)

8

(d)

0

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

By taking suitable sets A, B, C, verify the following results:
(A\(\times\) B)\(\cap \)(B\(\times\)A) = (A\(\cap \)B) \(\times\) (B\(\cap \)A)

Show that the relation R on the set A = {1, 2, 3} given by R = {(1, 1) (2, 2) (3, 3) (1, 2) (2, 3)} is reflexive but neither symmetric nor transitive.

If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find \(n((A\cup B)\times(A\cap B)\times(A \triangle B))\)

Discuss the following relations for reflexivity, symmetricity and transitivity:
Let P denote the set of all straight lines in a plane. The relation R defined by "lRm if l is perpendicular to m".

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is equivalence.

Let U = {-2, -1, 0, 1, 2, 3, ... 10} A = {-2, 2, 3, 4, 5} and B = {1, 3, 5, 8, 9}. Verify the De Morgans' law \((A\cup B)'=A'\cap B'\).

Prove that \(((A\cup B'\cup C)\cap(A\cap B'\cap C'))\cup((A\cup B\cup C')\cap (B'\cap C'))=B'\cap C'.\)

If \(f:R-\{ -1,1\}\rightarrow R\) is defined by \(f(x)={x \over x^2-1},\) verify whether f is one-to-one or not.

Show that the relation R defined on the set A of all polygons as R = {(P₁ P₂) : P₁ and P₂ have same number of sides} is an equivalence relation.

The weight of the muscles of a man is a function of his body weight x and can be expressed as W(x) = 0.35x. Determine the domain of this function.