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 t denote the greatest integer ≤ t . Then the equation in x , x 2 + 2 x + 2 - 7 = 0 has :

exactly four integral solutions

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

Neither injective nor surjective

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

HARD

Earn 100

Let $f (x) = s g n (x)$ where $s g n (x)$ denotes signum function of $x,$ then which one of the following statement is incorrect for $f (g (x)) ?$

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Important Questions on Functions

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f : R \to R

be defined by

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

then

f

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

The minimum value of

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

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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?

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

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

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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:

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

Let

f : R \to R

be defined as

f (x) = 3 x .

Then

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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>Differential Calculus>Functions>Types of Functions

A = {x | x \in N, x \leq 5}, B = \{x | x \in Z, x^{2} - 5 x + 6 = 0\},

then the number of onto functions from

A

B

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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>Differential Calculus>Functions>Types of Functions

[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

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

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>Differential Calculus>Functions>Types of Functions

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.

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

Let

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

be given by

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

Then

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

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:

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

Let

[t]

denote the greatest integer

\leq t

. Then the equation in

x, {[x]}^{2} + 2 [x + 2] - 7 = 0

has :

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

| 3 x - 5 | \leq 2

then

MEDIUM

Mathematics>Differential Calculus>Functions>Types of Functions

The function

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

defined as

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

is:

EASY

Mathematics>Differential Calculus>Functions>Types of Functions

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:

HARD

Mathematics>Differential Calculus>Functions>Types of Functions

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}]

Let fx=sgn x where sgn x denotes signum function of x, then which one of the following statement is incorrect for fgx?

Important Questions on Functions

Learn and Prepare for any exam you want !

Let $f (x) = s g n (x)$ where $s g n (x)$ denotes signum function of $x,$ then which one of the following statement is incorrect for $f (g (x)) ?$