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

The function f ( x ) = sec log x + 1 + x 2 is

neither an odd nor an even function

EASY

Earn 100

Define the smallest integer function or ceiling function. Find the range of ceiling function by using graph of ceiling function.

Important Questions on Relations and Functions

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

8 .

MEDIUM

Mathematics>Algebra>Relations and 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>Algebra>Relations and 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>Algebra>Relations and 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>Algebra>Relations and 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>Algebra>Relations and Functions>Special Functions

The equation

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

, where

[x]

denotes the greatest integer function, has:

HARD

Mathematics>Algebra>Relations and 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}

HARD

Mathematics>Algebra>Relations and Functions>Special Functions

Let

f (x) = a^{x} (a > 0)

be written as

f (x) = f_{1} (x) + f_{2} (x),

where

f_{1} (x)

is an even function and

f_{2} (x)

is an odd function. Then

f_{1} (x + y) + f_{1} (x - y)

equals:

MEDIUM

Mathematics>Algebra>Relations and 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>Algebra>Relations and Functions>Special Functions

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

is equal to, where

[\cdot]

represents greatest integer function

HARD

Mathematics>Algebra>Relations and 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>Algebra>Relations and 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

MEDIUM

Mathematics>Algebra>Relations and Functions>Special Functions

The function

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

HARD

Mathematics>Algebra>Relations and Functions>Special Functions

x, y

are positive real numbers such that

x \cdot [x] = 36

and

y \cdot [y] = 71

, then

x + y

equals

HARD

Mathematics>Algebra>Relations and Functions>Special Functions

P (x)

be a polynomial with real coefficients such that

P ({s i n}^{2} x) = P ({c o s}^{2} x),

for all

x \in [0, \frac{π}{2}] .

Consider the following statements:

I.

P (x)

is an even function.

II.

P (x)

can be expressed as a polynomial in

{(2 x - 1)}^{2}

III.

P (x)

is a polynomial of even degree. Then

EASY

Mathematics>Algebra>Relations and Functions>Special Functions

Which of the following functions is neither even nor odd?

EASY

Mathematics>Algebra>Relations and Functions>Special Functions

f : R \to R,

such that

f (x) = \frac{e^{x} + e^{- x}}{e^{x} - e^{- x}},

then

f

EASY

Mathematics>Algebra>Relations and 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>Algebra>Relations and 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>Algebra>Relations and 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?

Define the smallest integer function or ceiling function. Find the range of ceiling function by using graph of ceiling function.

Important Questions on Relations and Functions

Learn and Prepare for any exam you want !