Functions — JEE Maths Practice Questions & Solutions

hard

If

f:(0,\pi/n)\to\mathbb{R}

defined by

f(x)=\displaystyle\sum_{k=1}^{n}[1+\sin kx]

, where

[x]

denotes the integral part of

x

, then range of

f(x)

is:

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hard

If

f(x)+f(y) = f\!\left(\dfrac{x+y}{1-xy}\right)

for all

x, y \in \mathbb{R}

with

xy \ne 1

, and

\displaystyle\lim_{x\to 0}\dfrac{f(x)}{x} = 2

, then the value of

\dfrac{15\,f(\sqrt{3})}{\pi\,f'(-2)}

is.

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hard

If the domain of

f(x) = \dfrac{\sin^{-1}(\sin x)}{\sqrt{-\log_{\frac{x+4}{2}}\log_2\!\left(\frac{2x-1}{3+x}\right)}}

is

(a,b)\cup(c,\infty)

, then the value of

a+b+3c

.

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hard

Let

f(x) = [\sec\{x\}]

where

[x]

and

\{x\}

denote greatest integer and fractional parts of

x

respectively, and

g(x) = 2x^2 - 3x(k+1) + k(3k+1)

. Find the number of integral values of

k

such that

g(f(x)) < 0

for all

x \in \mathbb{R}

.

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hard

If

f(x) = \dfrac{2010x+163}{165x-2010}

,

x > 0

,

x \neq \dfrac{2010}{165}

, then the least value of

f(f(x)) + f\!\left(f\!\left(\dfrac{4}{x}\right)\right)

is.

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hard

Let

f(n)

denote the square of the sum of the digits of natural number

n

, where

f^2(n) = f(f(n))

,

f^3(n) = f(f(f(n)))

, and so on. Then the value of

\dfrac{f^{2011}(2011)-f^{2010}(2011)}{f^{2013}(2011)-f^{2012}(2011)}

.

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hard

If the range of

f(x) = \sqrt{24\{x\}-5-16\{x\}^2}

, where

\{\cdot\}

denotes fractional part, is

[a,b]

, then the value of

b^2-a

.

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medium

Domain of the definition

f(x)=\dfrac{\log_{4}(5-[x-1]-[x]^{2})}{x^{2}+x-2}

(where

[\cdot]

denotes greatest integer function) is:

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medium

The range of the function

f(x)=4\cos^{3}x-8\cos^{2}x+1

is:

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medium

The domain of the function

\log_{e}\!\left(\text{sgn}(9-x^{2})\right)+\sqrt{[x]^{3}-4[x]}

(where sgn is signum function and

[\cdot]

is step function) is:

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medium

If

[x]

denotes the integral part of

x

, then the domain of

f(x)=\cos^{-1}(x+[x])

is:

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medium

Range of

f(x)={}^{16-x}C_{2x-1}+{}^{20-3x}C_{4x-5}

is:

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medium

The domain of the function

y=\sqrt\dfrac{2x-1}{2x^{3}+3x^{2}+x}+\sqrt{\sin^{-1}(\log_{2}x)}

is:

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medium

The domain of the function

y=\sqrt{\sin x+\cos x}+\sqrt{7x-x^{2}-6}

is:

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medium

Let

f(x) = \dfrac{x+1}{x-1}

for all

x \neq 1

. Let

f^{-1}(x) = f(x)

,

f^2(x) = f(f(x))

, and

f^n(x) = f(f^{n-1}(x))

for

n > 1

. Let

P = f^1(2)\cdot f^2(3)\cdot f^3(4)\cdot f^4(5)

. The number of divisors of

P

.

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medium

If

f(x) + f\!\left(1-\dfrac{1}{x}\right) = 1+x

for all

x \in \mathbb{R} - \{0,1\}

,The value of

4f(2)

.

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medium

If

f(x) = \cos\!\left(2010\{x^3\}(2011^{[x^2]}+2012x)\right)

,

x \in \mathbb{R}

, where

\{\cdot\}

denotes fractional part and

[\cdot]

denotes greatest integer function, then the value of

f_{\max}

.

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medium

Let

f

be defined on the natural numbers as follows:

f(1) = 1

and for

n > 1

,

f(n) = f(f(n-1)) + f(n - f(n-1))

. then the value of

\dfrac{1}{30}\displaystyle\sum_{r=1}^{20} f(r)

.

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medium

Let

f: \mathbb{R} \to \mathbb{R}

be defined as

f(x) = x^3 + x + 1

,

1 \leq x \leq 2

. The graph of

y = g(x)

is the reflection of the graph of

y = f(x)

through the line

y = x

. If the domain of

g(x)

is

[a, b]

, then

|a-b|

is.

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medium

Number of solutions of the equation

f(x-1)+f(x+1) = \sin\alpha

,

0 < \alpha < \dfrac{\pi}{2}

, where

f(x) = \begin{cases} 1-|x|; & |x| \leq 1 \\ 0; & |x| > 1 \end{cases}

is.

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medium

If

0\le x\le\dfrac{\pi}{3}

, then range of

f(x)=\sec\left(\dfrac{\pi}{6}-x\right)+\sec\left(\dfrac{\pi}{6}+x\right)

is:

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medium

A = \{1,2,3\}

,

B = \{1,3,5,7,9\}

. Then

\dfrac{\text{No. of one-one functions}}{\text{No. of strictly monotonic functions}}

.

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medium

The number of linear functions satisfying

f[x+f(x)] = x+f(x)

for all

x \in \mathbb{R}

is.

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medium

f(x) = \dfrac{ax+b}{cx+d}

for all

x \in \mathbb{R}-\left\{-\dfrac{d}{c}\right\}

. If

f(5) = 5

,

f(13) = 13

, and

f(f(x)) = x

for all

x

, then the range of

f(x) = \mathbb{R}-\{x\}

. Then

x

.

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medium

If

n(A) = 4

,

n(B) = 5

and the number of functions from

A

to

B

such that range contains exactly 3 elements is

k

, then

k/60

is.

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medium

If

a

and

b

are constants,

f(x) = a\sin x + bx\cos x + 2x^2

. If

f(2) = 15

, then

f(-2)

is.

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medium

If

f(x) = 3[2x]+1000\displaystyle\sum_{r=1}^{2008}\dfrac{x+r-[x+r]}{2008}

, then

f(3)

(where

[\cdot]

is GIF) is.

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medium

The number of solutions of

[\sin x]+\left[\dfrac{x}{2\pi}\right]+\left[\dfrac{2x}{5\pi}\right] = \dfrac{9x}{10\pi}

in the interval

(30,40)

(where

[\cdot]

is GIF).

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easy

If

[x]

denotes the integral part of

x

, then domain of the function

f(x)=\dfrac{\sqrt{3-x}}{(x-1)(x-2)(x-3)}+\sin^{-1}\!\left[\dfrac{3x-2}{2}\right]

is:

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easy

The range of the function

y=[x^{2}]-[x]^{2}

,

x\in[0,2]

(where

[\cdot]

denotes the integral part) is:

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easy

The number of integers in the domain of

f(x) = \dfrac{1}{\sqrt{\ln\cos^{-1}x}}

.

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Functions — JEE Maths Practice Questions & Solutions

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