The function f : R → - 1 2 , 1 2 defined as f x = x 1 + x 2 , is:

Neither injective nor surjective

Let f x be a non-constant polynomial with real coefficients such that f 1 2 = 100 & f x ≤ 100 for all real x . Which of the following statements is NOT necessarily true?

The coefficient of the highest degree term in f x is negative

f x has at least two real roots

At least one of the coefficients of f x is bigger than 50

If [ x ] denotes the greatest integer ≤ x , then the system of linear equations [ s i n θ ] x + [ - c o s θ ] y = 0 , [ c o t θ ] x + y = 0

have infinitely many solutions if θ ∈ π 2 , 2 π 3 and has a unique solution if θ ∈ π , 7 π 6

has a unique solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

have infinitely many solution if θ ∈ π 2 , 2 π 3 ∪ π , 7 π 6

has a unique if θ ∈ π 2 , 2 π 3 and have infinitely many solutions if θ ∈ π , 7 π 6

Let A = { x ∈ R : x is not a positive integer} . Define a function f : A → R as f x = 2 x x - 1 , then f is:

Neither injective nor surjective

EASY

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A rational number minus a rational number is always:

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Important Questions on Relations and Functions

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

The minimum value of

f (x) = \max \{x, 1 + x, 2 - x\}

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

a \in R

and the equation

- 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0

(where

[x]

denotes the greatest integer

\leq

x

) has no integral solution, then all possible values of

a

lie in the interval

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

The function

f : R \to [- \frac{1}{2}, \frac{1}{2}]

defined as

f (x) = \frac{x}{1 + x^{2}},

is:

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

f : R \to R

be defined by

f (x) = \frac{|x| - 1}{|x| + 1},

then

f

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

The number of solutions of the equation

\log_{4} (x - 1) = \log_{2} (x - 3)

is ______.

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

f (x)

be a non-constant polynomial with real coefficients such that

f (\frac{1}{2}) = 100 & f (x) \leq 100

for all real

x .

Which of the following statements is NOT necessarily true?

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let $R$ be the set of real numbers and $f : R \to R$ be defined by $f (x) = \frac{\{x\}}{1 + {[x]}^{2}},$ where $[x]$ is the greatest integer less than or equal to $x,$ and $\{x\} = x - [x] .$ Which of the following statement are true?

I. The range of $f$ is closed interval
II. $f$ is continuous on $R$
III. $f$ is one-one on $R$

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

[x]

denotes the greatest integer

\leq x,

then the system of linear equations

[s i n θ] x + [- c o s θ] y = 0

[c o t θ] x + y = 0

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

The function

f : N \to I

defined by

f (x) = x - 5 [\frac{x}{5}]

, where

N

is the set of natural numbers and

[x]

denotes the greatest integer less than or equal to

x

, is:

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

4^{x} - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2 x - 1}

then value of

x

is equal to

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

f : (- \frac{π}{2}, \frac{π}{2}) \to R

be given by

f (x) = {(\log (\sec x + \tan x))}^{3} .

Then

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

f

and

g

be differentiable functions on

R

such that

f o g

is the identity function. If for some

a, b \in R, g^{'} (a) = 5

and

g (a) = b,

then

f^{'} (b)

is equal to:

EASY

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

For a real number

r

we denote by

[r]

the largest integer less than or equal to

r

. If

x, y

are real numbers with

x, y \geq 1

then which of the following statements is always true?

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

If the function

f : R - \{1, - 1\} \to A

defined by

f (x) = \frac{x^{2}}{1 - x^{2}},

is surjective, then

A

is equal to

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

[t]

denote the greatest integer

\leq t

and

\lim_{x \to 0} x [\frac{4}{x}] = A .

Then the function,

f (x) = [x^{2}] \sin (π x)

is discontinuous, when

x

is equal to:

EASY

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

S

be the set of all functions

f : [0, 1] \to R

which are continuous on

[0, 1]

and differentiable on

(0, 1)

. Then for every

f

S

, there exists

c \in (0, 1)

depending on

f

, such that.

EASY

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

Let

A =

{

x \in R : x

is not a positive integer}

.

Define a function

f : A \to R

f (x) = \frac{2 x}{x - 1}

, then

f

is:

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

g

is the inverse of a function

f

and

f^{'} (x) = \frac{1}{1 + x^{5}},

then

g^{'} (x)

is equal to

HARD

Mathematics>Algebra>Relations and Functions>Special Functions and Their Graphs

For

x \in R,

Let

[x]

denotes the greatest integer

\leq x,

then the sum of the series

[- \frac{1}{3}] + [- \frac{1}{3} - \frac{1}{100}] + [- \frac{1}{3} - \frac{2}{100}] + . . . . . + [- \frac{1}{3} - \frac{99}{100}]

A rational number minus a rational number is always:

Important Questions on Relations and Functions

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