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

EASY

Earn 100

Range of a signum function is

100% studentsanswered this correctly

Important Questions on Functions

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

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

[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>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 :-

EASY

Mathematics>Differential Calculus>Functions>Special Functions

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

, then the complete set of values of

x

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 :

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?

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]

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)

EASY

Mathematics>Differential Calculus>Functions>Special Functions

Let

f : [- 5, 5] \to R

be a continuous function such that

f (x)

assumes only irrational values. If

f (3) = \sqrt{3}

, then

f (\sqrt{3} + 1)

is equal to

HARD

Mathematics>Differential Calculus>Functions>Special Functions

f (x)

and

g (x)

are two polynomials such that the polynomial

P (x) = f (x^{3}) + x g (x^{3})

is divisible by

x^{2} + x + 1

, then

P (1)

is equal to ___ .

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:

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

p, q, r

are any real numbers, then

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:

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

f (10 - x) = 3 x^{2} + 4 x - 5

and

f (x) = p x^{2} + q x + r

then

p + q + r =

MEDIUM

Mathematics>Differential Calculus>Functions>Special Functions

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

8 .

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

EASY

Mathematics>Differential Calculus>Functions>Special Functions

| 3 x - 5 | \leq 2

then

EASY

Mathematics>Differential Calculus>Functions>Special 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.

Range of a signum function is

Important Questions on Functions

Learn and Prepare for any exam you want !