2021
JEE Mains
MCQ
JEE Main 2021 (Online) 20th July Morning Shift
Let [ x ] denote the greatest integer $\le$ x, where x $\in$ R. If the domain of the real valued function $f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}} $ is ($-$ $\infty$, a) $]\cup$ [b, c) $\cup$ [4, $\infty$), a < b < c, then the value of a + b + c is :
A.
8
B.
1
C.
$-$2
D.
$-$3
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Evening Shift
Let f : R $-$ {3} $ \to $ R $-$ {1} be defined by f(x) = ${{x - 2} \over {x - 3}}$.
Let g : R $ \to $ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f$-$1(x) + g$-$1(x) = ${{13} \over 2}$ is equal to :
Let g : R $ \to $ R be given as g(x) = 2x $-$ 3. Then, the sum of all the values of x for which f$-$1(x) + g$-$1(x) = ${{13} \over 2}$ is equal to :
A.
3
B.
5
C.
2
D.
7
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Morning Shift
The real valued function
$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$, where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :
$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$, where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :
A.
all real except integers
B.
all non-integers except the interval [ $-$1, 1 ]
C.
all integers except 0, $-$1, 1
D.
all real except the interval [ $-$1, 1 ]
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 18th March Morning Shift
If the functions are defined as $f(x) = \sqrt x $ and $g(x) = \sqrt {1 - x} $, then what is the common domain of the following functions :
f + g, f $-$ g, f/g, g/f, g $-$ f where $(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}$
f + g, f $-$ g, f/g, g/f, g $-$ f where $(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}$
A.
$0 \le x \le 1$
B.
$0 \le x < 1$
C.
$0 < x < 1$
D.
$0 < x \le 1$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 17th March Morning Shift
The inverse of $y = {5^{\log x}}$ is :
A.
$x = {5^{\log y}}$
B.
$x = {y^{{1 \over {\log 5}}}}$
C.
$x = {5^{{1 \over {\log y}}}}$
D.
$x = {y^{\log 5}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 16th March Morning Shift
The range of a$\in$R for which the
function f(x) = (4a $-$ 3)(x + loge 5) + 2(a $-$ 7) cot$\left( {{x \over 2}} \right)$ sin2$\left( {{x \over 2}} \right)$, x $\ne$ 2n$\pi$, n$\in$N has critical points, is :
function f(x) = (4a $-$ 3)(x + loge 5) + 2(a $-$ 7) cot$\left( {{x \over 2}} \right)$ sin2$\left( {{x \over 2}} \right)$, x $\ne$ 2n$\pi$, n$\in$N has critical points, is :
A.
[1, $\infty $)
B.
($-$3, 1)
C.
$\left[ { - {4 \over 3},2} \right]$
D.
($-$$\infty $, $-$1]
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 26th February Evening Shift
Let $A = \{ 1,2,3,....,10\} $ and $f:A \to A$ be defined as
$f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.$
Then the number of possible functions $g:A \to A$ such that $gof = f$ is :
$f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right.$
Then the number of possible functions $g:A \to A$ such that $gof = f$ is :
A.
55
B.
105
C.
5!
D.
10C5
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Evening Shift
A function f(x) is given by $f(x) = {{{5^x}} \over {{5^x} + 5}}$, then the sum of the series $f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right)$ is equal to :
A.
${{{39} \over 2}}$
B.
${{{19} \over 2}}$
C.
${{{49} \over 2}}$
D.
${{{29} \over 2}}$
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Evening Shift
Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $\times$ B. Then :
A.
2y = 273x
B.
y = 91x
C.
2y = 91x
D.
y = 273x
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 25th February Morning Shift
Let f, g : N $ \to $ N such that f(n + 1) = f(n) + f(1) $\forall $ n$\in$N and g be any arbitrary function. Which of the following statements is NOT true?
A.
If g is onto, then fog is one-one
B.
f is one-one
C.
If f is onto, then f(n) = n $\forall $n$\in$N
D.
If fog is one-one, then g is one-one
2021
JEE Mains
MCQ
JEE Main 2021 (Online) 24th February Morning Shift
Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) =
${{x - {1 \over 2}} \over {x - 1}}$.
Then the composition function f(g(x)) is :
A.
one-one but not onto
B.
onto but not one-one
C.
both one-one and onto
D.
neither one-one nor onto
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 27th July Morning Shift
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $\to$ S
such that f(m . n) = f(m) . f(n) for every m, n $\in$ S and m . n $\in$ S is equal to _____________.
such that f(m . n) = f(m) . f(n) for every m, n $\in$ S and m . n $\in$ S is equal to _____________.
Correct Answer: 490
Explanation:
F(mn) = f(m) . f(n)
Put m = 1 f(n) = f(1) . f(n) $\Rightarrow$ f(1) = 1
Put m = n = 2
$f(4) = f(2).f(2)\left\{ \matrix{ f(2) = 1 \Rightarrow f(4) = 1 \hfill \cr or \hfill \cr f(2) = 2 \Rightarrow f(4) = 4 \hfill \cr} \right.$
Put m = 2, n = 3
$f(6) = f(2).f(3)\left\{ \matrix{ when\,f(2) = 1 \hfill \cr f(3) = 1\,to\,7 \hfill \cr \hfill \cr f(2) = 2 \hfill \cr f(3) = 1\,or\,2\,or\,3 \hfill \cr} \right.$
f(5), f(7) can take any value
Total = (1 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7) + (1 $\times$ 1 $\times$ 3 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7)
= 490
Put m = 1 f(n) = f(1) . f(n) $\Rightarrow$ f(1) = 1
Put m = n = 2
$f(4) = f(2).f(2)\left\{ \matrix{ f(2) = 1 \Rightarrow f(4) = 1 \hfill \cr or \hfill \cr f(2) = 2 \Rightarrow f(4) = 4 \hfill \cr} \right.$
Put m = 2, n = 3
$f(6) = f(2).f(3)\left\{ \matrix{ when\,f(2) = 1 \hfill \cr f(3) = 1\,to\,7 \hfill \cr \hfill \cr f(2) = 2 \hfill \cr f(3) = 1\,or\,2\,or\,3 \hfill \cr} \right.$
f(5), f(7) can take any value
Total = (1 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7) + (1 $\times$ 1 $\times$ 3 $\times$ 1 $\times$ 7 $\times$ 1 $\times$ 7)
= 490
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 22th July Evening Shift
Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A $\to$ A such that f(1) + f(2) = 3 $-$ f(3) is equal to
Correct Answer: 720
Explanation:
f(1) + f(2) = 3 $-$ f(3)
$\Rightarrow$ f(1) + f(2) = 3 + f(3) = 3
The only possibility is : 0 + 1 + 2 = 3
$\Rightarrow$ Elements 1, 2, 3 in the domain can be mapped with 0, 1, 2 only.
So number of bijective functions.
$\left| \!{\underline {\, 3 \,}} \right. $ $\times$ $\left| \!{\underline {\, 5 \,}} \right. $ = 720
$\Rightarrow$ f(1) + f(2) = 3 + f(3) = 3
The only possibility is : 0 + 1 + 2 = 3
$\Rightarrow$ Elements 1, 2, 3 in the domain can be mapped with 0, 1, 2 only.
So number of bijective functions.
$\left| \!{\underline {\, 3 \,}} \right. $ $\times$ $\left| \!{\underline {\, 5 \,}} \right. $ = 720
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 18th March Evening Shift
If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to ___________.
Correct Answer: 0
Explanation:
Given, p(x) = f(x3) + xg(x3)
We know, x2 + x + 1 = (x $-$ $\omega$) (x $-$ $\omega$2)
Given, p(x) is divisible by x2 + x + 1. So, roots of p(x) is $\omega$ and $\omega$2.
As root satisfy the equation,
So, put x = $\omega$
p($\omega$) = f($\omega$3) + $\omega$g($\omega$3) = 0
= f(1) + $\omega$g(1) = 0 [$\omega$3 = 1]
= f(1) + $\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)$ g(1) = 0
$ \Rightarrow $ f(1) $-$ ${{g(1)} \over 2} + i\left( {{{\sqrt 3 g(1)} \over 2}} \right)$ = 0 + i0
Comparing both sides, we get
f(1) $-$ ${{g(1)} \over 2}$ = 0
and ${{{\sqrt 3 } \over 2}g(1) = 0}$ $ \Rightarrow $ g(1) = 0
So, f(1) = 0
Now, p(1) = f(1) + 1 . g(1) = 0 + 0 = 0
We know, x2 + x + 1 = (x $-$ $\omega$) (x $-$ $\omega$2)
Given, p(x) is divisible by x2 + x + 1. So, roots of p(x) is $\omega$ and $\omega$2.
As root satisfy the equation,
So, put x = $\omega$
p($\omega$) = f($\omega$3) + $\omega$g($\omega$3) = 0
= f(1) + $\omega$g(1) = 0 [$\omega$3 = 1]
= f(1) + $\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)$ g(1) = 0
$ \Rightarrow $ f(1) $-$ ${{g(1)} \over 2} + i\left( {{{\sqrt 3 g(1)} \over 2}} \right)$ = 0 + i0
Comparing both sides, we get
f(1) $-$ ${{g(1)} \over 2}$ = 0
and ${{{\sqrt 3 } \over 2}g(1) = 0}$ $ \Rightarrow $ g(1) = 0
So, f(1) = 0
Now, p(1) = f(1) + 1 . g(1) = 0 + 0 = 0
2021
JEE Mains
Numerical
JEE Main 2021 (Online) 24th February Evening Shift
If a + $\alpha$ = 1, b + $\beta$ = 2 and $af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0$, then the value of the expression ${{f(x) + f\left( {{1 \over x}} \right)} \over {x + {1 \over x}}}$ is __________.
Correct Answer: 2
Explanation:
$af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x}$ ........(i)
Replace $x $ with $ {1 \over x}$
$af\left( {{1 \over x}} \right) + af(x) = {b \over x} + \beta x$ ..... (ii)
(i) + (ii)
$(a + \alpha )\left[ {f(x) + f\left( {{1 \over x}} \right)} \right] = \left( {x + {1 \over x}} \right)(b + \beta )$
${{f(x) + f\left( {{1 \over x}} \right)} \over {x + {1 \over x}}} = {{\beta + b} \over {a + \alpha }} = {2 \over 1} = 2$
Replace $x $ with $ {1 \over x}$
$af\left( {{1 \over x}} \right) + af(x) = {b \over x} + \beta x$ ..... (ii)
(i) + (ii)
$(a + \alpha )\left[ {f(x) + f\left( {{1 \over x}} \right)} \right] = \left( {x + {1 \over x}} \right)(b + \beta )$
${{f(x) + f\left( {{1 \over x}} \right)} \over {x + {1 \over x}}} = {{\beta + b} \over {a + \alpha }} = {2 \over 1} = 2$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Evening Slot
For a suitably chosen real constant a, let a
function, $f:R - \left\{ { - a} \right\} \to R$ be defined by
$f(x) = {{a - x} \over {a + x}}$. Further suppose that for any real number $x \ne - a$ and $f(x) \ne - a$,
(fof)(x) = x. Then $f\left( { - {1 \over 2}} \right)$ is equal to :
function, $f:R - \left\{ { - a} \right\} \to R$ be defined by
$f(x) = {{a - x} \over {a + x}}$. Further suppose that for any real number $x \ne - a$ and $f(x) \ne - a$,
(fof)(x) = x. Then $f\left( { - {1 \over 2}} \right)$ is equal to :
A.
$ {1 \over 3}$
B.
–3
C.
$ - {1 \over 3}$
D.
3
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 6th September Morning Slot
If f(x + y) = f(x)f(y) and $\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$ , x, y $ \in $ N, where N is the set of all natural number, then the
value of
${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :
A.
${2 \over 3}$
B.
${1 \over 9}$
C.
${1 \over 3}$
D.
${4 \over 9}$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 2nd September Evening Slot
Let f : R $ \to $ R be a function which satisfies
f(x + y) = f(x) + f(y) $\forall $ x, y $ \in $ R. If f(1) = 2 and
g(n) = $\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $, n $ \in $ N then the value of n, for which g(n) = 20, is :
f(x + y) = f(x) + f(y) $\forall $ x, y $ \in $ R. If f(1) = 2 and
g(n) = $\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $, n $ \in $ N then the value of n, for which g(n) = 20, is :
A.
20
B.
9
C.
5
D.
4
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 9th January Evening Slot
Let a – 2b + c = 1.
If $f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|$, then:
If $f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|$, then:
A.
Æ’(50) = 1
B.
ƒ(–50) = –1
C.
ƒ(50) = –501
D.
ƒ(–50) = 501
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Evening Slot
Let Æ’ : (1, 3) $ \to $ R be a function defined by
$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$ , where [x] denotes the greatest integer $ \le $ x. Then the range of Æ’ is
$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$ , where [x] denotes the greatest integer $ \le $ x. Then the range of Æ’ is
A.
$\left( {{2 \over 5},{1 \over 2}} \right) \cup \left( {{3 \over 4},{4 \over 5}} \right]$
B.
$\left( {{3 \over 5},{4 \over 5}} \right)$
C.
$\left( {{2 \over 5},{4 \over 5}} \right]$
D.
$\left( {{2 \over 5},{3 \over 5}} \right] \cup \left( {{3 \over 4},{4 \over 5}} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 8th January Morning Slot
The inverse function of
f(x) = ${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$, x $ \in $ (-1, 1), is :
f(x) = ${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$, x $ \in $ (-1, 1), is :
A.
${1 \over 4}{\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$
B.
${1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$
C.
${1 \over 4}\left( {{{\log }_8}e} \right){\log _e}\left( {{{1 + x} \over {1 - x}}} \right)$
D.
${1 \over 4}{\log _e}\left( {{{1 + x} \over {1 - x}}} \right)$
2020
JEE Mains
MCQ
JEE Main 2020 (Online) 7th January Morning Slot
If g(x) = x2 + x - 1 and
(goÆ’) (x) = 4x2 - 10x + 5, then Æ’$\left( {{5 \over 4}} \right)$ is equal to:
(goÆ’) (x) = 4x2 - 10x + 5, then Æ’$\left( {{5 \over 4}} \right)$ is equal to:
A.
${1 \over 2}$
B.
${3 \over 2}$
C.
-${1 \over 2}$
D.
-${3 \over 2}$
2020
JEE Mains
Numerical
JEE Main 2020 (Online) 6th September Evening Slot
Suppose that a function f : R $ \to $ R satisfies
f(x + y) = f(x)f(y) for all x, y $ \in $ R and f(1) = 3.
If $\sum\limits_{i = 1}^n {f(i)} = 363$ then n is equal to ________ .
f(x + y) = f(x)f(y) for all x, y $ \in $ R and f(1) = 3.
If $\sum\limits_{i = 1}^n {f(i)} = 363$ then n is equal to ________ .
Correct Answer: 5
Explanation:
f(x + y) = f(x) f(y)
put x = y = 1
$ \therefore $ f(2) = (Æ’(1))2 = 32
put x = 2, y = 1
$ \therefore $ f(3) = (Æ’(1))3 = 33
Similarly f(x) = 3x
$ \Rightarrow $ f(i) = 3i
Given, $\sum\limits_{i = 1}^n {f(i)} = 363$
$ \Rightarrow $ 3 + 32 + 33 +.... + 3n = 363
$ \Rightarrow $ ${{3\left( {{3^n} - 1} \right)} \over {3 - 1}}$ = 363
$ \Rightarrow $ 3n - 1 = ${{363 \times 2} \over 3}$ = 242
$ \Rightarrow $ 3n = 243 = 35
$ \Rightarrow $ n = 5
put x = y = 1
$ \therefore $ f(2) = (Æ’(1))2 = 32
put x = 2, y = 1
$ \therefore $ f(3) = (Æ’(1))3 = 33
Similarly f(x) = 3x
$ \Rightarrow $ f(i) = 3i
Given, $\sum\limits_{i = 1}^n {f(i)} = 363$
$ \Rightarrow $ 3 + 32 + 33 +.... + 3n = 363
$ \Rightarrow $ ${{3\left( {{3^n} - 1} \right)} \over {3 - 1}}$ = 363
$ \Rightarrow $ 3n - 1 = ${{363 \times 2} \over 3}$ = 242
$ \Rightarrow $ 3n = 243 = 35
$ \Rightarrow $ n = 5
2020
JEE Mains
Numerical
JEE Main 2020 (Online) 5th September Evening Slot
Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the
number of elements in the set
C = {f : A $ \to $ B | 2 $ \in $ f(A) and f is not one-one} is ______.
C = {f : A $ \to $ B | 2 $ \in $ f(A) and f is not one-one} is ______.
Correct Answer: 19
Explanation:
The desired functions will contain either one
element or two elements in its codomain of
which '2' always belongs to f(A).
Case 1 : When 2 is the image of all element of set A.
Number of ways this is possible = 1
Case 2 : When one image is 2 and other one image is one of {1, 3, 4}.
Number of ways we can choose one of {1, 3, 4} is = 3C1.
Now divide 3 elements {a, b, c} of set A into two parts.
We can do this ${{3!} \over {2!1!}}$ ways.
Now map one part of set A into the element 2 of set B and map other part of set A into one of {1, 3, 4} of set B.
We can do that 2! ways.
So number of functions in this case
= 3C1 $ \times $ ${{3!} \over {2!1!}}$ $ \times $ 2! = 18
$ \therefore $ Total number of functions = 1 + 18 = 19
Case 1 : When 2 is the image of all element of set A.
Number of ways this is possible = 1
Case 2 : When one image is 2 and other one image is one of {1, 3, 4}.
Number of ways we can choose one of {1, 3, 4} is = 3C1.
Now divide 3 elements {a, b, c} of set A into two parts.
We can do this ${{3!} \over {2!1!}}$ ways.
Now map one part of set A into the element 2 of set B and map other part of set A into one of {1, 3, 4} of set B.
We can do that 2! ways.
So number of functions in this case
= 3C1 $ \times $ ${{3!} \over {2!1!}}$ $ \times $ 2! = 18
$ \therefore $ Total number of functions = 1 + 18 = 19
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 12th April Morning Slot
For x $ \in $ (0, 3/2), let f(x) = $\sqrt x $ , g(x) = tan x and h(x) = ${{1 - {x^2}} \over {1 + {x^2}}}$. If $\phi $ (x) = ((hof)og)(x), then $\phi \left( {{\pi \over 3}} \right)$
is equal to :
A.
$\tan {{7\pi } \over {12}}$
B.
$\tan {{11\pi } \over {12}}$
C.
$\tan {\pi \over {12}}$
D.
$\tan {{5\pi } \over {12}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Morning Slot
Let f(x) = ex – x and g(x) = x2 – x, $\forall $ x $ \in $ R. Then the set of all x $ \in $ R, where the function h(x) = (fog) (x) is increasing, is :
A.
[0, $\infty $)
B.
$\left[ { - 1, - {1 \over 2}} \right] \cup \left[ {{1 \over 2},\infty } \right)$
C.
$\left[ { - {1 \over 2},0} \right] \cup \left[ {1,\infty } \right)$
D.
$\left[ {0,{1 \over 2}} \right] \cup \left[ {1,\infty } \right)$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th April Morning Slot
Let f(x) = x2
, x $ \in $ R. For any A $ \subseteq $ R, define g (A) = { x $ \in $ R : f(x) $ \in $ A}. If S = [0,4], then which one of the
following statements is not true ?
A.
g(f(S)) $ \ne $ S
B.
f(g(S)) = S
C.
f(g(S)) $ \ne $ f(S)
D.
g(f(S)) = g(S)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Evening Slot
The domain of the definition of the function
$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$ is
$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$ is
A.
(-1, 0) $ \cup $ (1, 2) $ \cup $ (2, $\infty $)
B.
(-2, -1) $ \cup $ (-1,0) $ \cup $ (2, $\infty $)
C.
(1, 2) $ \cup $ (2, $\infty $)
D.
(-1, 0) $ \cup $ (1,2) $ \cup $ (3, $\infty $)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Morning Slot
Let $\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)} $ where the function
Æ’ satisfies
Æ’(x + y) = Æ’(x)Æ’(y) for all natural numbers x, y and Æ’(1) = 2. then the natural number 'a' is
Æ’(x + y) = Æ’(x)Æ’(y) for all natural numbers x, y and Æ’(1) = 2. then the natural number 'a' is
A.
2
B.
16
C.
4
D.
3
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th April Morning Slot
If the function ƒ : R – {1, –1} $ \to $ A defined by
Æ’(x) = ${{{x^2}} \over {1 - {x^2}}}$ , is surjective, then A is equal to
Æ’(x) = ${{{x^2}} \over {1 - {x^2}}}$ , is surjective, then A is equal to
A.
R – (–1, 0)
B.
R – {–1}
C.
R – [–1, 0)
D.
[0, $\infty $)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Evening Slot
Let Æ’(x) = ax
(a > 0) be written as
Æ’(x) = Æ’1 (x) + Æ’2 (x), where Æ’1 (x) is an even function of Æ’2 (x) is an odd function.
Then ƒ1 (x + y) + ƒ1 (x – y) equals
Æ’(x) = Æ’1 (x) + Æ’2 (x), where Æ’1 (x) is an even function of Æ’2 (x) is an odd function.
Then ƒ1 (x + y) + ƒ1 (x – y) equals
A.
2Æ’1
(x)Æ’1
(y)
B.
2Æ’1
(x + y)Æ’1
(x – y)
C.
2Æ’1
(x)Æ’2
(y)
D.
2Æ’1
(x + y)Æ’2
(x – y)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 8th April Morning Slot
If $f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$, $\left| x \right| < 1$ then $f\left( {{{2x} \over {1 + {x^2}}}} \right)$ is equal to
A.
2f(x2)
B.
2f(x)
C.
(f(x))2
D.
-2f(x)
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
Let a function f : (0, $\infty $) $ \to $ (0, $\infty $) be defined by f(x) = $\left| {1 - {1 \over x}} \right|$. Then f is :
A.
not injective but it is surjective
B.
neiter injective nor surjective
C.
injective only
D.
both injective as well as surjective
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Evening Slot
The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3,
whenever k is a multiple of 4, is :
A.
65 $ \times $ (15)!
B.
56 $ \times $ 15
C.
(15)! $ \times $ 6!
D.
5! $ \times $ 6!
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Let fk(x) = ${1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$ for k = 1, 2, 3, ... Then for all x $ \in $ R, the value of f4(x) $-$ f6(x) is equal to
A.
${1 \over 4}$
B.
${5 \over {12}}$
C.
${{ - 1} \over {12}}$
D.
${1 \over {12}}$
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 11th January Morning Slot
Let f : R $ \to $ R be defined by f(x) = ${x \over {1 + {x^2}}},x \in R$. Then the range of f is :
A.
$\left[ { - {1 \over 2},{1 \over 2}} \right]$
B.
$R - \left[ { - {1 \over 2},{1 \over 2}} \right]$
C.
($-$ 1, 1) $-$ {0}
D.
R $-$ [$-$1, 1]
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 10th January Evening Slot
Let N be the set of natural numbers and two functions f and g be defined as f, g : N $ \to $ N such that
f(n) = $\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$;
and g(n) = n $-$($-$ 1)n.
Then fog is -
f(n) = $\left\{ {\matrix{ {{{n + 1} \over 2};} & {if\,\,n\,\,is\,\,odd} \cr {{n \over 2};} & {if\,\,n\,\,is\,\,even} \cr } \,\,} \right.$;
and g(n) = n $-$($-$ 1)n.
Then fog is -
A.
neither one-one nor onto
B.
onto but not one-one
C.
both one-one and onto
D.
one-one but not onto
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Evening Slot
Let A = {x $ \in $ R : x is not a positive integer}.
Define a function $f$ : A $ \to $ R as $f(x)$ = ${{2x} \over {x - 1}}$,
then $f$ is :
Define a function $f$ : A $ \to $ R as $f(x)$ = ${{2x} \over {x - 1}}$,
then $f$ is :
A.
not injective
B.
neither injective nor surjective
C.
surjective but not injective
D.
injective but not surjective
2019
JEE Mains
MCQ
JEE Main 2019 (Online) 9th January Morning Slot
For $x \in R - \left\{ {0,1} \right\}$, Let f1(x) = $1\over x$, f2 (x) = 1 – x
and f3 (x) = $1 \over {1 - x}$ be three given
functions. If a function, J(x) satisfies
(f2 o J o f1) (x) = f3 (x) then J(x) is equal to :
and f3 (x) = $1 \over {1 - x}$ be three given
functions. If a function, J(x) satisfies
(f2 o J o f1) (x) = f3 (x) then J(x) is equal to :
A.
f1 (x)
B.
$1 \over x$ f3 (x)
C.
f2 (x)
D.
f3 (x)
2018
JEE Mains
MCQ
JEE Main 2018 (Online) 15th April Evening Slot
Let f : A $ \to $ B be a function defined as f(x) = ${{x - 1} \over {x - 2}},$ Where A = R $-$ {2} and B = R $-$ {1}. Then f is :
A.
invertible and ${f^{ - 1}}(y) = $ ${{3y - 1} \over {y - 1}}$
B.
invertible and ${f^{ - 1}}\left( y \right) = {{2y - 1} \over {y - 1}}$
C.
invertible and ${f^{ - 1}}\left( y \right) = {{2y + 1} \over {y - 1}}$
D.
not invertible
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 9th April Morning Slot
The function f : N $ \to $ N defined by f (x) = x $-$ 5 $\left[ {{x \over 5}} \right],$ Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :
A.
one-one and onto
B.
one-one but not onto.
C.
onto but not one-one.
D.
neither one-one nor onto.
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
Let f(x) = 210.x + 1 and g(x)=310.x $-$ 1. If (fog) (x) = x, then x is equal to :
A.
${{{3^{10}} - 1} \over {{3^{10}} - {2^{ - 10}}}}$
B.
${{{2^{10}} - 1} \over {{2^{10}} - {3^{ - 10}}}}$
C.
${{1 - {3^{ - 10}}} \over {{2^{10}} - {3^{ - 10}}}}$
D.
${{1 - {2^{ - 10}}} \over {{3^{10}} - {2^{ - 10}}}}$
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
The function $f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$ defined as
$f\left( x \right) = {x \over {1 + {x^2}}}$, is
$f\left( x \right) = {x \over {1 + {x^2}}}$, is
A.
invertible
B.
injective but not surjective.
C.
surjective but not injective
D.
neither injective nor surjective.
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
Let $a$, b, c $ \in R$. If $f$(x) = ax2 + bx + c is such that
$a$ + b + c = 3 and $f$(x + y) = $f$(x) + $f$(y) + xy, $\forall x,y \in R,$
then $\sum\limits_{n = 1}^{10} {f(n)} $ is equal to
$a$ + b + c = 3 and $f$(x + y) = $f$(x) + $f$(y) + xy, $\forall x,y \in R,$
then $\sum\limits_{n = 1}^{10} {f(n)} $ is equal to
A.
165
B.
190
C.
255
D.
330
2016
JEE Mains
MCQ
JEE Main 2016 (Online) 9th April Morning Slot
For x $ \in $ R, x $ \ne $ 0, Let f0(x) = ${1 \over {1 - x}}$ and
fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . .
Then the value of f100(3) + f1$\left( {{2 \over 3}} \right)$ + f2$\left( {{3 \over 2}} \right)$ is equal to :
fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . .
Then the value of f100(3) + f1$\left( {{2 \over 3}} \right)$ + f2$\left( {{3 \over 2}} \right)$ is equal to :
A.
${8 \over 3}$
B.
${5 \over 3}$
C.
${4 \over 3}$
D.
${1 \over 3}$
2016
JEE Mains
MCQ
JEE Main 2016 (Offline)
If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$
A.
is an empty set.
B.
contains exactly one element.
C.
contains exactly two elements.
D.
contains more than two elements.
2011
JEE Mains
MCQ
AIEEE 2011
The domain of the function f(x) = ${1 \over {\sqrt {\left| x \right| - x} }}$ is
A.
$\left( {0,\infty } \right)$
B.
$\left( { - \infty ,0} \right)$
C.
$\left( { - \infty ,\infty } \right) - \left\{ 0 \right\}$
D.
$\left( { - \infty ,\infty } \right)$
2009
JEE Mains
MCQ
AIEEE 2009
Let $f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$
Statement - 1 : The set $\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}$.
Statement - 2 : $f$ is a bijection.
Statement - 1 : The set $\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}$.
Statement - 2 : $f$ is a bijection.
A.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1
B.
Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1
C.
Statement - 1 is true, Statement - 2 is false
D.
Statement - 1 is false, Statement - 2 is true
2009
JEE Mains
MCQ
AIEEE 2009
For real x, let f(x) = x3 + 5x + 1, then
A.
f is one-one but not onto R
B.
f is onto R but not one-one
C.
f is one-one and onto R
D.
f is neither one-one nor onto R
2008
JEE Mains
MCQ
AIEEE 2008
Let $f:N \to Y$ be a function defined as f(x) = 4x + 3 where
Y = { y $ \in $ N, y = 4x + 3 for some x $ \in $ N }.
Show that f is invertible and its inverse is
Y = { y $ \in $ N, y = 4x + 3 for some x $ \in $ N }.
Show that f is invertible and its inverse is
A.
$g\left( y \right) = {{3y + 4} \over 4}$
B.
$g\left( y \right) = 4 + {{y + 3} \over 4}$
C.
$g\left( y \right) = {{y + 3} \over 4}$
D.
$g\left( y \right) = {{y - 3} \over 4}$