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 ( x ) = sec log x + 1 + x 2 is

neither an odd nor an even function

The function f x = log x + x 2 + 1 is

Neither an even nor an odd function

EASY

Earn 100

The derivative of an even function is always an odd function.

50% studentsanswered this correctly

Important Questions on Functions

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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>Even and Odd 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>Even and Odd 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?

EASY

Mathematics>Differential Calculus>Functions>Even and Odd 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>Even and Odd Functions

Which of the following functions is neither even nor odd?

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd 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>Even and Odd 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>Even and Odd 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>Even and Odd Functions

The real valued function

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

defined on

R \ {0}

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The even function of the following is

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The function

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

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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>Differential Calculus>Functions>Even and Odd 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>Even and Odd 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>Even and Odd 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>Differential Calculus>Functions>Even and Odd 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>Even and Odd Functions

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

8 .

MEDIUM

Mathematics>Differential Calculus>Functions>Even and Odd Functions

The function

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

EASY

Mathematics>Differential Calculus>Functions>Even and Odd Functions

f : R \to R,

such that

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

then

f

HARD

Mathematics>Differential Calculus>Functions>Even and Odd 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

The derivative of an even function is always an odd function.

Important Questions on Functions

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