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

exactly four integral solutions

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

MEDIUM

Earn 100

Prove that the greatest integer function defined by $f (x) = [x]; 0 < x < 3$ is not differentiable at $x = 1$ and $x = 2$ .

Important Questions on Functions

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

8 .

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

Let $[x]$ be the greatest integer less than or equal to $x$ , for a real number $x$ . Then the following sum

$[\frac{2^{2020} + 1}{2^{2018} + 1}] + [\frac{3^{2020} + 1}{3^{2018} + 1}] + [\frac{4^{2020} + 1}{4^{2018} + 1}] + [\frac{5^{2020} + 1}{5^{2018} + 1}] + [\frac{6^{2020} + 1}{6^{2018} + 1}]$

is :-

HARD

Mathematics>Differential Calculus>Functions>Special Functions

For any real number

x,

let

[x]

denote the integer part of

x; {x}

be the fractional part of

x .

(\{x\} = x - [x])

. Let

A

denote the set of all real numbers

x

satisfying

\{x\} = \frac{x + [x] + [x + (\frac{1}{2})]}{20}

. If

S

is the sum of all numbers in

A,

find

[S]

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

Let

A = {x \in ℝ : [x + 3] + [x + 4] \leq 3}, B = \{x \in ℝ : 3^{x} {(\sum_{r = 1}^{\infty} \frac{3}{10^{r}})}^{x - 3} < 3^{- 3 x}\},

where

[t]

denotes greatest integer function. Then,

HARD

Mathematics>Differential Calculus>Functions>Special 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:

HARD

Mathematics>Differential Calculus>Functions>Special Functions

The equation

x^{2} - 4 x + [x] + 3 = x [x]

, where

[x]

denotes the greatest integer function, has:

HARD

Mathematics>Differential Calculus>Functions>Special Functions

Consider the sequence of numbers

[n + \sqrt{2 n} + \frac{1}{2}]

for

n \geq 1,

where

[x]

denotes the greatest integer not exceeding

x

. If the missing integers in the sequence are

n_{1} < n_{2} < n_{3} < \dots . .,

then find

n_{12}

MEDIUM

Mathematics>Differential Calculus>Functions>Special 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>Special Functions

Let

[t]

denote the greatest integer

\leq t

. Then the equation in

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

has :

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

\int_{0}^{2} [x^{2}] d x

is equal to, where

[\cdot]

represents greatest integer function

HARD

Mathematics>Differential Calculus>Functions>Special Functions

For a real number

r

let

[r]

denote the largest integer less than or equal to

r .

Let

a > 1

a real number which is not an integer and

k

be the smallest positive integer such that

[a^{k}] > {[a]}^{k}

Then which of the following statement is always true?

HARD

Mathematics>Differential Calculus>Functions>Special 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

HARD

Mathematics>Differential Calculus>Functions>Special 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>Special Functions

x, y

are positive real numbers such that

x \cdot [x] = 36

and

y \cdot [y] = 71

, then

x + y

equals

MEDIUM

Mathematics>Differential Calculus>Functions>Special 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>Special Functions

|3 x - 2| \leq \frac{1}{2}

, then the complete set of values of

x

EASY

Mathematics>Differential Calculus>Functions>Special Functions

| 3 x - 5 | \leq 2

then

EASY

Mathematics>Differential Calculus>Functions>Special Functions

The value of

\lim_{x \to 0^{+}} \frac{x}{p} [\frac{q}{x}]

is (where

[\cdot]

represents greatest integer function)

HARD

Mathematics>Differential Calculus>Functions>Special 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}]

HARD

Mathematics>Differential Calculus>Functions>Special 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?

Prove that the greatest integer function defined by fx=x; 0<x<3 is not differentiable at x=1 and x=2.

Important Questions on Functions

Learn and Prepare for any exam you want !

Prove that the greatest integer function defined by $f (x) = [x]; 0 < x < 3$ is not differentiable at $x = 1$ and $x = 2$ .