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CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 3 - MCQExams.com
CBSE
Class 12 Commerce Maths
Relations And Functions
Quiz 3
Let
S
be set of all rational numbers. The functions
f
:
R
→
R
,
g
:
R
→
R
are
defined as
f
(
x
)
=
{
0
,
x
∈
S
1
,
x
∉
S
g
(
x
)
=
{
−
1
x
∈
S
0
x
∉
S
then,
(
f
o
g
)
(
π
)
+
(
g
o
f
)
(
e
)
=
Report Question
0%
−
1
0%
0
0%
1
0%
2
Explanation
f
(
x
)
=
{
0
,
x
is rational
1
,
x
is irrational
g
(
x
)
=
{
−
1
x
is rational
0
x
is irrational
f
o
g
(
π
)
=
f
(
g
(
π
)
)
=
f
(
0
)
=
0
so
g
o
f
(
e
)
=
g
(
f
(
e
)
)
=
g
(
1
)
=
−
1
∴
Set
A
has
n
elements. The number of functions that can be defined from
A
into
A
is:
Report Question
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n^2
0%
n!
0%
n^n
0%
n
Explanation
The number of functions that can be defined from a set into another set can be found by
{ \text{(Number of elements in co-domain)} }^{ \text{Number of elements in domain} }
Since, set
A
has
n
elements, we have
Number of elements in co-domain
=
Number of elements in domain
= n (A) = n
.
Therefore, number of functions that can be defined from
A
into
A
=n(A)^{n(A)}= n^{n}
.
If
n\geq 1
is any integer,
\mathrm{d}(n)
denotes the number of positive factors of
n
, then for any prime number
\mathrm{p},\ \mathrm{d}(\mathrm{d}(\mathrm{d}(\mathrm{p}^{7})))=
Report Question
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1
0%
2
0%
3
0%
4
Explanation
For a number being
p^{n}
the total number of positive factors excluding
1
and the number itself will be
n-1
.
Therefore total number of positive factors will be
n-1+2
=n+1
Hence for
p^{7}
we will have
8
factors.
Now for
8=2^{3}
.
Hence
8
will have
4
factors.
Now
4=2^{2}
, hence the number of factors will be
3
.
Let
\displaystyle f\left( x \right)={ x }^{ 2 }-x+1,x\ge \left( \frac { 1 }{ 2 } \right)
then the solution of the equation
f(x)=f^{-1}(x)
is
Report Question
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x=1
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x=2
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\displaystyle x=\frac{1}{2}
0%
None of these
Explanation
\displaystyle y={ x }^{ 2 }-x+1\Rightarrow { x }^{ 2 }-x+\left( 1-y \right) =0
Here,
a=1,b=-1
and
c=1-y
\displaystyle x=\frac { 1\pm \sqrt { 1-4\left( 1-y \right) } }{ 2 } \quad \quad \left[ \because x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a } \right]
\displaystyle \because x>\frac { 1 }{ 2 } ,\quad \therefore x=\frac { 1 }{ 2 } +\sqrt { y-\frac { 3 }{ 4 } }
\displaystyle \Rightarrow f^{ -1 }\left( x \right) =\frac { 1 }{ 2 } +\sqrt { x-\frac { 3 }{ 4 } }
Now,
\displaystyle { x }^{ 2 }-x+1=\frac { 1 }{ 2 } +\sqrt { x-\frac { 3 }{ 4 } }
Since the graphs of the original and inverse functions can intersect only on the straight line
y=x,
therefore
x=f\left( x \right) \quad \quad \Rightarrow x={ x }^{ 2 }-x+1
\Rightarrow { x }^{ 2 }-2x+1=0
\Rightarrow { \left( x-1 \right) }^{ 2 }=0
\Rightarrow x=1
lf
f:[-6,6]\rightarrow \mathbb{R}
is defined by
f(x)=x^{2}-3
for
x\in \mathbb{R}
then
(fofof)(-1)+(fofof)(0)+(fofof)(1)=
Report Question
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f(4\sqrt{2})
0%
f(3\sqrt{2})
0%
f(2\sqrt{2})
0%
f(\sqrt{2})
Explanation
fofof(-1)=fof(-2)=f(1)=-2
fofof(1)=fof(-2)=f(1)=-2
fofof(0)=fof(-3)=f(0)=33
\therefore fofof(-1)+fofof(1)+fofof(0)=29
=32-3
=(4\sqrt{2})^{2}-3
\Rightarrow f(4\sqrt{2})=(4\sqrt{2})^{2}-3
lf
f
:
R\rightarrow R
is defined by
f(x)=\left\{\begin{array}{l}x+4 & x<-4\\3x+2 & -4\leq x<4\\x-4 & x\geq 4\end{array}\right.
then the correct matching of list I to List II is.
List - I
List - II
\mathrm{A}) f(-5)+f(-4)=
\mathrm{i}) 14
\mathrm{B}) f(|f(-8)|)=
ii
) 4
\mathrm{C}) f(f(-7)+f(3))=
\mathrm{i}\mathrm{i}\mathrm{i})-11
\mathrm{D}) f(f(f(f(0)))+1=
\mathrm{i}\mathrm{v})-1
v)
1
vi)
0
Report Question
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A-iii , B-vi , C-ii , D- v
0%
A-iii , B-iv , C-ii , D- vi
0%
A-iv , B-iii , C-ii , D- i
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A-ii , B-vi , C-v , D- ii
Explanation
f(-5)=-5+4=-1 ; f(-4)=3(-4)+2=-10
\therefore f(-5)+f(4)=-11
f(|f(-8)|)=f((-8+4))=f(4)=4-4=0
f(f(-7)) tf(3) = f(-7+4)+3(3)+2
=3(-3)+2+3(3)+2
=4
f(f(f(f(0))))+1=f(f(f(2)))+1=f(f(8))+1
=f(4)+1=0+1=1
lf
g(f(x)) =|\sin \mathrm{x}|,f(g(x)) =(\sin\sqrt{\mathrm{x}})^{2}
, then
Report Question
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{f}({x})=\sin^{2} {x},{g}({x})=\sqrt{{x}}
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{f}({x})=\sin x,g({x})=|{x}|
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{f}({x})={x}^{2},{g}({x})=\sin\sqrt{{x}}
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{f}, {g}
cannot be determined
Explanation
g(f(x))=|\sin x|=\sqrt{\sin^{2}x}
.......(i)
f(g(x))=\sin^{2}\sqrt{x}
.......(ii)
Comparing
i
and
ii
\Rightarrow
g(f(x))=|\sin x|=\sqrt{\sin^{2}x}
\Rightarrow
g(\sin^{2}x)=\sqrt{\sin^{2}x}=g(f(x))
And
\Rightarrow
f(g(x))=\sin^{2}\sqrt{x}
\Rightarrow
f(\sqrt{x})=\sin^{2}\sqrt{x}=f(g(x))
Therefore
\Rightarrow
g(x)=\sqrt{x}
\Rightarrow
f(x)=\sin^{2}x
lf
f(x)=x-x^{2}+x^{3}-x^{4}+\ldots..\infty
when
|x|<1
, then the ascending order of the following is
a)
f(1/2)
b)
f^{-1}(1/2)
c)
f(-1/2)
d)
f^{-1}(-1/2)
Report Question
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a, b, c, d
0%
c, d, a, b
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b, a, d, c
0%
d, c, a, b
Explanation
f(x)=x-x^2+x^3-x^4
f(x)=\dfrac{x}{1+x}(x)< 1
f\left ( \dfrac{1}{2} \right )=\dfrac{1}{3}
f\left ( \dfrac{-1}{2} \right )=-1
f(x)=\dfrac{x}{1+x}
y=\dfrac{x}{1+x}
y+xy=x
\dfrac{y}{(u-1)}=x
f^{-1}x=\dfrac{-x}{x-1}
f^{-1}\left ( \dfrac{1}{2} \right )=1
f^{-1}\left ( \dfrac{-1}{2} \right )=\dfrac{-1}{3}
Let
f
be an injective function with domain
\{x, y, z\}
and range
\{1,2,3\}
such that exactly one of the follwowing statements is correct and the remaining are false :
{f}({x})=1,{f}({y})\neq 1
,
{f}({z})\neq 2
,
then the value of
{f}^{-1}(1)
is
Report Question
0%
x
0%
y
0%
z
0%
none
Explanation
It gives three cases
Case (1) When
f(x)=1
is true.
In this case remaining two are false
f(y)=1
and
f(z)=2
This means
x
and
y
have the same image, so
f(x)
is not an injective, which is a contradiction.
Case (2) when
f\left( y \right)\neq 1
is true
If
f\left( y \right)\neq 1
is true then the remaining statements are false.
\therefore f\left( x\right)\neq 1
and
f(z)=2
i.e., both
x
and
y
are not mapped to
1.
So, either both associate to
2
or
3,
Thus, it is not injective.
Case (3) When
f(z) \neq 2
is true
If
f(z) \neq2
is true then remaining statements are fales
\therefore
If
f(x) \neq1
and
f(y)=1
But
f
is injective
Thus, we have
f(x)=2, f(y)=1
and
f(z)=3
Hence,
\displaystyle f^{ -1 }\left( 1 \right)=y
If
f : R \rightarrow R
is defined by
f(x)=2x-2,
then
(f\circ f) (x) + 2 =
Report Question
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f(x)
0%
2f(x)
0%
3f(x)
0%
-f(x)
Explanation
Let
f : R \rightarrow R
is defined by
f(x)=2x-2
.
Then
(f\circ f)(x)+2= f[f(x)]+2
=f(2x-2)+2
=2(2x-2)-2+2
=2f(x)
If
f(x) =\displaystyle \frac{x}{\sqrt{1-x^2}}, g(x)=\frac{x}{\sqrt{1+x^2}}
then
(f\circ g)(x) =
Report Question
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\displaystyle \frac{x}{\sqrt{1-x^2}}
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\displaystyle \frac{x}{\sqrt{1+x^2}}
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\displaystyle \frac{1-x^2}{\sqrt{1-x^2}}
0%
x
Explanation
Let
x= \tan \theta
(f\circ g)(x)=f[g(x)] = f[g(\tan \theta)]
.
Since
\displaystyle g(x) = \frac{x}{\sqrt{1+x^2}}
, we have
f[g(\tan \theta)]=\displaystyle f \left [ \frac{\tan \theta}{\sqrt{1+\tan^2 \theta}} \right ] = f \left [ \frac{\tan \theta}{\sec \theta} \right ]=f (\sin \theta)
Also since
f(x)=\cfrac{x}{\sqrt{1-x^2}}
, we have
\displaystyle f (\sin \theta) = \frac{\sin \theta}{\sqrt{1-\sin^2 \theta}}
=\tan \theta =x
If
f(x)=\log x, g(x) = x^3
then
f[g(a)]+f[g(b)]=
Report Question
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f[g(a)+g(b)]
0%
f[g(ab)]
0%
g[f(ab)]
0%
g[f(a)+f(b)]
Explanation
Given that
g(x)=x^3
. Therefore,
f[g(a)]+f[g(b)]=f(a^3)+f(b^3)
.
Since
f(x)=\log x
, we have
f(a^3)+fb^3=\log a^3+\log b^3
=\log (a^3b^3)=\log ((ab)^{3})
=f[g(ab)]
If
X=\left \{ 1, 2, 3, 4, 5 \right \}
and
Y=\left \{ 1, 3, 5, 7, 9 \right \}
, determine which of the following sets represent a relation and also a mapping.
Report Question
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R_{1}=\left \{ (x, y):y=x+2, x\in X, y\in Y \right \}
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R_{2}=\left \{ (1, 1),(2, 1),(3, 3),(4, 3),(5, 5) \right \}
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R_{3}=\left \{ (1, 1),(1, 3),(3, 5),(3, 7),(5, 7) \right \}
0%
R_{4}=\left \{ (1, 3),(2, 5),(4, 7),(5, 9),(3, 1) \right \}
Explanation
R_1={(1,3),(2,4),(3,5),(4,6),(5,7)}
Since
4
and
6
do not belong to
Y
(2,4),(4,6)∉R_1
R_1={(1,3),(3,5),(5,7)}⊂A×B
Hence
R_1
is a relation but not a mapping as the elements
2
and
4
do not have any image.
R_2
: It is certainly a mapping and since every mapping is a relation, it is a relation as well.
R_3
: It is a relation being a subset of
A×B
but the elements
1
and
3
do not have a unique image and hence it is not mapping.
R_4
: It is both a mapping and a relation. Each element in A has a unique image. It is also one-one and onto mapping and hence a bijection
If
f:R \rightarrow R
and
g : R \rightarrow R
are defined by
f(x)=2x+3
and
g(x)=x^2+7
, then the values of
x
such that
g(f(x)) =8
are:
Report Question
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1, 2
0%
-1, 2
0%
-1, -2
0%
1, -2
Explanation
Since
f(x)=2x+3
,
g[f(x)]=8 \Rightarrow g(2x+3)=8
.
Also since
g(x)=x^2+7
,
g(2+3x)=8\Rightarrow (2x+3)^2+7=8
\Rightarrow 2x+3 = \pm 1
\Rightarrow x=-1
or
-2
.
If
f(x) = \dfrac{2x+5}{x^{2} + x + 5}
, then
f\left [ f(- 1 ) \right ]
is equal to
Report Question
0%
\dfrac{149}{155}
0%
\dfrac{155}{147}
0%
\dfrac{155}{149}
0%
\dfrac{147}{155}
Explanation
Given expression is
f(x)=\dfrac { 2x+5 }{ { x }^{ 2 }+x+5 }
\therefore f(-1)=\dfrac { 2\times (-1)+5 }{ (-1)^{ 2 }+(-1)+5 } =\dfrac { 3 }{ 5 }
\therefore f\left( f\left( -1 \right) \right) =\dfrac { 2\times \dfrac { 3 }{ 5 } +5 }{ \left( \dfrac { 3 }{ 5 } \right) ^{ 2 }+\dfrac { 3 }{ 5 } +5 } =\dfrac { 155 }{ 149 }
Set A has 3 elements and set B has 4 elements. The number of injections that can be defined from A into B is :
Report Question
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144
0%
12
0%
24
0%
64
Explanation
If
ABC\sim PQR
, then AB:PQ =
Report Question
0%
AC:PB
0%
AC:PR
0%
AB:PR
0%
AC:RQ
If
f : R \rightarrow R
and
g :R \rightarrow R
are defined by
f(x) = x -[x]
and
g(x) = [x]
for
x \in R
, where
[x]
is the greatest integer not exceeding
x
, then for every
x \in R, f(g(x)) =
Report Question
0%
x
0%
0
0%
f(x)
0%
g(x)
Explanation
Since
f(x)=x-[x]
, we have
f[g(x)] =g(x)-[g(x)]
Also since
g(x)=[x]
,we have
g(x)-[g(x)]=[x] - [[x]]
.
Here,
[x]
is the greatest integer not exceeding
x
. Therefore,
[[x]]=[x]
and hence,
[x]-[[x]]=[x]-[x]=0
If
y=f(x) = \dfrac{2x-1}{x-2}
, then
f(y)=
Report Question
0%
x
0%
y
0%
2y-1
0%
y-2
Explanation
\displaystyle f(y) = \frac{2y-1}{y-2} = \frac{2 \left ( \dfrac{2x-1}{x-2} \right ) -1}{\left ( \dfrac{2x-1}{x-2} \right ) -2}
=\displaystyle \frac{4x-2-x+2}{2x-1-2x+4} = \frac{3x}{3} = x
If
f(g(x))
is one-one function, then
Report Question
0%
g(x) must be one-one
0%
f(x) must be one-one
0%
f(x) may not be one-one
0%
g(x) may not be one-one
Explanation
It is fundamental concept that, If
f(g(x))
is one-one function then,
f(x)
must be one-one function and
g(x)
may be one-one or many one
Which of the following functions are one-one?
Report Question
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f:R\rightarrow R
given by
f(x)={ 2x }^{ 2 }+1
for all
\quad x\in R
0%
g:Z\rightarrow Z
given by
g(x)={ x }^{ 4 }
for all
\quad x\in R
0%
h:R\rightarrow R
given by
h(x)={ x }^{ 3 }+4
for all
\quad x\in R
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\phi :C\rightarrow C
given
\phi (z)={ 2z }^{ 6 }+4
for all
\quad x\in R
Explanation
For all even powered function,
f(x_{1})=f(x_{2})
where
x_{1}=\pm(x_{2})
Hence, the only possible one-one function is
f(x)=x^3+4
f(x)
is a strictly increasing function. So, it is one-one.
A mapping function
f:X\rightarrow Y
is one-one, if
Report Question
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f({ x }_{ 1 })\neq f({ x }_{ 2 })\
for all
{ x }_{ 1 },{ x }_{ 2 }\in X
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f({ x }_{ 1 })=f({ x }_{ 2 })\Rightarrow { x }_{ 1 }={ x }_{ 2 }
for all
{ x }_{ 1 },{ x }_{ 2 }\in X
0%
{ x }_{ 1 }={ x }_{ 2 }\Rightarrow f({ x }_{ 1 })=f({ x }_{ 2 })
for all
{ x }_{ 1 },{ x }_{ 2 }\in X
0%
none of these
Explanation
For a function to be one one
f(x_{1})=f(x_{2})
Implies that
x_{1}=x_{2}
Hence the answer is
option B
Let
\displaystyle f:R\rightarrow A=\left \{ y: 0\leq y< \dfrac{\pi}{2} \right \}
be a function such that
\displaystyle f(x)=\tan^{-1}(x^{2}+x+k),
where
k
is a constant. The value of
k
for which
f
is an onto function is
Report Question
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1
0%
0
0%
\displaystyle \frac{1}{4}
0%
none of these
Explanation
For
f
to be an onto function, Range and co-domain of
f
should be equal
\Rightarrow f(x)\ge 0 \forall x\in R
\Rightarrow \tan^{-1}(x^2+x+k)\ge 0
For the above equation to be valid for all
x
We must have, the discriminant of
x^2+x+k=0
, is zero
b^2-4ac=0
\Rightarrow 1-4k=0\Rightarrow k =\dfrac{1}{4}
If
f:R\rightarrow R
given by
f(x)={ x }^{ 3 }+({ a+2)x }^{ 2 }+3ax+5
is one-one, then
a
belongs to the interval
Report Question
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(-\infty ,1)
0%
(1 ,\infty)
0%
(1 ,4)
0%
(4 ,\infty)
Explanation
If
f(x)
is one one then
f'\left( x \right)
should be either positive or negative for all
x
.
f'\left( x \right) = 3{ x }^{ 2 }+2(a+2)x+3a
\Rightarrow D = 4{ (a+2) }^{ 2 }-4\times3\times3a<0
\Rightarrow
{ a }^{ 2 }-5a+4<0
\Rightarrow
(a-1)(a-4)<0
Hence,
a\in \left( 1,4 \right)
Let R be the relation in the set N given by
=\left \{ (a, b):a=b-2, b >6 \right \}
. Choose the correct answer.
Report Question
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(2, 4)\in R
0%
(3, 8)\in R
0%
(6, 8)\in R
0%
(8, 7)\in R
Explanation
(a,b)∈R
only if
a=b−2
and
b>6
(6,8)∈R
as
6=8−2
and
8 >6
Hence
(c)
is the correct alternative
Let
\displaystyle f:\left \{ x,y,z \right \}\rightarrow \left \{ a,b,c \right \}
be a one-one function and only one of the conditions
(i)f(x)\neq b, (ii)f(y)=b,(iii)f(z)\neq a
is true then the function
f
is given by the set
Report Question
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\displaystyle \left \{ (x,a),(y,b),(z,c)\right \}
0%
\displaystyle \left \{ (x,a),(y,c),(z,b)\right \}
0%
\displaystyle \left \{ (x,b),(y,a),(z,c)\right \}
0%
\displaystyle \left \{ (x,c),(y,b),(z,a)\right \}
Explanation
f:\left\{ x,y,z \right\} \rightarrow \left\{ a,b,c \right\}
is a one-one function
\Rightarrow
each element in
\left\{ x,y,z \right\}
will have exactly one image in
\left\{ a,b,c \right\}
and no two elements of
\left\{ x,y,z \right\}
will have same image in
\left\{ a,b,c \right\}
Coming to the given 3 conditions, only one is true.
1) if
f\left( x \right) \neq b
is true then
f\left( y \right) =b
is false which makes
f\left( z \right) \neq a
true
\Longrightarrow f\left( x \right) \neq b
is false.
2) if
f\left( y \right) =b
is true then
f\left( x \right) \neq b
will also be true
\Longrightarrow f\left( y \right) =b
is false
\therefore f\left( z \right) \neq a
is the true condition and remainig two are false conditions.
\therefore f\left( x \right) =b, f\left( y \right) =a, f\left( z \right) =c
hence
f=\left\{ \left( x,b \right) ,\left( y,a \right) ,\left( z,c \right) \right\}
Which of the following function is one-one?
Report Question
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f:R\rightarrow R
given by
f(x)=|x-1|
for all
x\in R
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g:\left[ -\dfrac{\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] \rightarrow R
given by
g(x)=|sinx|
for all
x\in \left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right]
0%
h:\left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right] \in R
given by
h(x)=sinx
for all
x\in \left[ \dfrac{ -\pi }{ 2 },\dfrac{ \pi }{ 2 } \right]
0%
\phi :R\rightarrow R
given by
f(x)={ x }^{ 2 }-4
for all
x\in R
Explanation
Option A - Consider
x = 2
and
x =0
, the value of
f(x)
is same. Hence it is not one-one
Option B - If we replace
x
by
-x
, then the value of
g(x)
remains the same. Hence it is not one-one
Option C -
h(x)
is an increasing function for the given values of
x
. Hence it is one-one function
Option D -
f(x)
is an even function. So it is not one-one for the given values of
x
If
f
and
g
are one-one functions from
R\to R
, then
Report Question
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f+g
is one-one
0%
fg
is one-one
0%
fog
is one-one
0%
none of these
Explanation
Let
{x_1},{x_2} \in R
be two distinct elements, then
g\left( {{x_1}} \right) \ne g\left( {{x_2}} \right)
, as
g
is one-one function. Similarly
f\left( {g\left( {{x_1}} \right)} \right) \ne f\left( {g\left( {{x_2}} \right)} \right)
as
f
is also one-one function.
Hence,
f\circ g
is one-one function.
Note that
f+g
and
f\cdot g
may not one-one functions even if
f
and
g
are one one. For example consider
f\left( x \right) = x
and
g\left( x \right) =- x
, then
f+g
and
f\cdot g
are not one-one.
If
f:R^+\rightarrow A
, Where
A=\{x:-5<x<\infty\}
be defined by
f(x)=x^2-5
. Then
f^{-1}(7)=
Report Question
0%
only
-2{\sqrt3}
0%
only
2{\sqrt3}
0%
both options A and B
0%
none
Explanation
The value
f^{-1}(7)
will be the roots of the equation
x^2-5= 7\Rightarrow x^2 = 12\Rightarrow x = \pm 2\sqrt{3}
But only positive.
\therefore x = 2\sqrt{3}
If the function
f:[1,\infty)\rightarrow [1,\infty)
is defined by
\displaystyle f(x)=3^{x(x-1)}
then
f^{-1}(x)
is
Report Question
0%
\displaystyle \left ( \frac{1}{2} \right )^{x(x-1)}
0%
\displaystyle \frac{1}{2}\left ( 1+\sqrt{1+4\log_{3}x}\right)
0%
\displaystyle \frac{1}{2}\left ( 1-\sqrt{1+4\log_{3}x}\right)
0%
not defined
Explanation
For the given domain, the function is one one.
Thus
,
f(x)=2^{x(x-1)}
y=2^{x^{2}-x}
lny=(x^{2}-x)ln2
log_{2}y=x^{2}-x
log_{2}y=(x-\dfrac{1}{2})^{2}-\dfrac{1}{4}
log_{2}y+\dfrac{1}{4}=(x-\dfrac{1}{2})^{2}
\dfrac{\pm 1}{2}(\sqrt{1+4log_{2}y})=x-\dfrac{1}{2}
Since
f:[1,\infty)\rightarrow[1,\infty)
Thus domain is
[1,\infty)
Hence domain is positive so negative sign will be ignored.
So
x-\dfrac{1}{2}=\dfrac{1}{2}(\sqrt{1+4log_{2}y})
x=\dfrac{1}{2}[1+\sqrt{1+4log_{2}y}]
Replacing x and y, gives us
f^{-1}(x)=\dfrac{1}{2}[1+\sqrt{1+4log_{2}x}]
If the function
f:R\rightarrow R
be such that
\displaystyle f(x)=x-[x],
where
[y]
denotes the greatest integer less than or equal to
y
, then
f^{-1}(x)
is
Report Question
0%
\displaystyle \frac{1}{x-[x]}
0%
\displaystyle [x]-x
0%
not defined
0%
none of these
Explanation
f\left( x \right) = x - \left[ x \right] = \left\{ x \right\}
is not one-one function from R to R. Hence
{f^{ - 1}}\left( x \right)
can not be defined.
If
\displaystyle f:\left ( 3,4 \right )\rightarrow \left ( 0,1 \right )
is defined by
\displaystyle f\left ( x \right )=x-\left [ x \right ]
where
\displaystyle [x]
denotes the greatest integer function then
f^{-1}(x)
is
Report Question
0%
\displaystyle \frac{1}{x-\left [ x \right ]}
0%
\displaystyle [x]-x
0%
\displaystyle x-3
0%
\displaystyle x+3
Explanation
Note that
\displaystyle f\left ( x \right )=x-\left [ x \right ] =\left\{ x \right\}
. Hence every number
x \in \left( {3,4} \right)
is mapped to its fractional part.
Hence, the inverse of the function can be written by adding the integer part of the number i.e.,
3
to each of the fraction part.
Hence the function
{f^{ - 1}}\left( x \right) = x + 3
.
Let
\displaystyle f:(-\infty,1]\rightarrow (-\infty,1]
such that
\displaystyle f(x)=x(2-x).
Then
f^{-1}(x)
is
Report Question
0%
1+\sqrt{1-x}
0%
1-\sqrt{1-x}
0%
\sqrt{1-x}
0%
none of these
Explanation
We have,
f(x)=x(2-x)
y=x(2-x)
-y=x(x-2)
-y=x^{2}-2x
1-y=x^{2}-2x+1
1-y=(x-1)^{2}
\pm\sqrt{1-y}=x-1
Now
f:(-\infty,1] \rightarrow (-\infty,1]
x-1=-\sqrt{1-y}
x=1-\sqrt{1-y}
Hence
f^{-1}(x)=1-\sqrt{1-x}
If
f(x)=ax+b
and
g(x)=cx+d
, then
f(g(x))=g(f(x))
implies
Report Question
0%
f(a)=g(c)
0%
f(b)=g(b)
0%
f(d)=g(b)
0%
f(c)=g(a)
Explanation
f(g(x))=a(cx+d)+b
=acx+ad+b
...(i)
g(f(x))=c(ax+b)+d
=ac(x)+bc+d
...(ii)
f(g(x))=g(f(x))
From (i) and (ii)
ac(x)+bc+d=acx+ad+b
cb+d=ad+b
g(b)=f(d)
If
\displaystyle f(x)=\frac{1}{1-x},x\neq 0,1
then the graph of the function
\displaystyle y=f\left \{ f(f(x)) \right \},x> 1,
is
Report Question
0%
a circle
0%
an ellipse
0%
a straight line
0%
a pair of straight lines
Explanation
f(f(x))=\dfrac{1}{1-\dfrac{1}{1-x}}
=-\dfrac{1-x}{x} =g(x)
f(g(x))=\dfrac{1}{1+\dfrac{1-x}{x}}
=\dfrac{x}{x+1-x}
=x
y=x
is an equation of straight line.
Hence, 'C' is correct.
If
\displaystyle f(x)=3x-5
then
f^{-1}(x)=
Report Question
0%
\displaystyle \frac{1}{3x-5}
0%
\displaystyle \frac{x+5}{3}
0%
does not exist because
f
is not one-one
0%
does not exist because
f
is not onto
Explanation
f(x)=3x-5
Since
f(x)
is a purely linear function (straight line), it is therefore a one-one function.
Hence
f(x)
is revertible for all real
x.
y=3x-5
y+5=3x
x=\dfrac{y+5}{3}
Replacing
x
by
y
, gives us
f^{-1}(x)=\dfrac{x+5}{3}
If
f
and
g
are two functions such that
\displaystyle \left ( fg \right )\left ( x \right )=\left ( gf \right )\left ( x \right )
for all
x
. Then
f
and
g
may be defined as
Report Question
0%
\displaystyle f\left ( x \right )=\sqrt{x}, g\left ( x \right )=\cos x
0%
\displaystyle f\left ( x \right )=x^{3}, g\left ( x \right )=x+1
0%
\displaystyle f\left ( x \right )=x-1, g\left ( x \right )=x^{2}+1
0%
\displaystyle f\left ( x \right )=x^{m}, g\left ( x \right )=x^{n}
where
m, n
are unequal integers
Explanation
A
)f(x)=
\sqrt{x},g(x)=cosx then (fg)(x) = \sqrt{\cos x} and (gf)(x) = \cos \sqrt{x}
.They are not equal.
B)
f(x)=x^{3},g(x)=x+1 then (fg)(x) = (x+1)^{3} and (gf)(x)= x^{3}+1
.They are not equal.
C)
f(x)=x1,g(x)=x^{2}+1 then (fg)(x)=x^{2} and (gf)(x)=x^{2}-2x
. They are not equal.
D)
f(x)=x^{m}, g(x)=x^{n} then (fg)(x)=x^{n+m} and (gf)(x)=x^{n+m}
.Hence
(fg)(x)= (gf)(x)
If
\displaystyle f(x)=x^{n},n\in N
and
(gof)(x)=ng(x)
then
g(x)
can be
Report Question
0%
n\:|x|
0%
3.\sqrt[3]{x}
0%
e^{x}
0%
\log\:|x|
Explanation
f(x)=x^{n}
g(f(x))=ng(x)
...
(i)
\log(f(x))=n\log(x)
...
(ii)
Taking
\log(|x|)
as
g(x)
the above expression is reduced to eq
i
.
Hence
g(x)=\log(|x|)
.
The composite mapping
fog
of the map
f: R\rightarrow R,f(x)=\sin x
and
g: R\rightarrow R, g(x)=x^2
is
Report Question
0%
x^2 \sin x
0%
(\sin x)^2
0%
\sin x^2
0%
\dfrac{ \sin x}{x^2}
Explanation
Composite mapping will be
f(g(x))
=f(x^2)
=\sin(x^2)
Hence option
'C'
is the answer.
Let
S
be a set containing
n
elements. Then the total number of binary operations on
S
is
Report Question
0%
n^n
0%
{2^n}^{2}
0%
n^{n^2}
0%
n^2
Explanation
No of binary operations imply
S\times S\rightarrow S
=n(S)^{[n(S\times S)]}
=n^{n\times n}
=n^{n^2}
Let
f: R\rightarrow R
be defined by
f(x)=3x-4
then
f^{-1}(x)
is
Report Question
0%
\dfrac{1}{3}(x+4)
0%
\dfrac{1}{3}(x-4)
0%
3x+4
0%
not defined
Explanation
f(x)
is invertible for all real values of
x
Hence
y=3x-4
y+4=3x
\dfrac{y+4}{3}=x
f^{-1}(x)=\dfrac{x+4}{3}
Let
\displaystyle f(x)=\frac{ax}{x+1}
, where
\displaystyle x\neq -1
. Then for what value of
\displaystyle a
is
\displaystyle f( f(x))=x
always true
Report Question
0%
\displaystyle \sqrt{2}
0%
\displaystyle -\sqrt{2}
0%
1
0%
-1
Explanation
\displaystyle f\left( {f\left( x \right)} \right) = \dfrac{{a\dfrac{{ax}}{{x + 1}}}}{{\dfrac{{ax}}{{x + 1}} + 1}} = \dfrac{{\dfrac{{{a^2}x}}{{x + 1}}}}{{\dfrac{{ax + x + 1}}{{x + 1}}}} = \dfrac{{{a^2}x}}{{ax + x + 1}}
Since,
\displaystyle f\left( {f\left( x \right)} \right) =x
, we have,
\displaystyle\dfrac{{{a^2}x}}{{ax + x + 1}}=x
.
Simplifying the equation we get,
{a^2}x = \left( {a + 1} \right){x^2} + x
\therefore \left( {a + 1} \right){x^2} + \left( {1 - {a^2}} \right)x = 0
or
\left( {a + 1} \right)x\left( {x + 1 - a} \right) = 0
Hence the only possible value is
a=-1
A function
y=f\left ( x \right )
is invertible only when
Report Question
0%
y=f\left ( x \right )
is monotonic increasing
0%
y=f\left ( x \right )
is bijective
0%
y=f\left ( x \right )
is monotonic decreasing
0%
invertible
Explanation
Invertible function is defined as, the function
f
applied to an input
x
gives a result of
y
, then applying its inverse function
g
to
y
gives the result
x
, and vice versa. i.e.,
f\left ( x \right )=y
\Rightarrow g\left ( y \right )
=
x
.
And bijective function has the same definition as that of an Invertible function
Let
f:R \rightarrow R
and
g:R \rightarrow R
be defined by
f(x)=x^2+2x-3,g(x)=3x-4
then
(gof) (x)=
Report Question
0%
3x^2+6x-13
0%
3x^2-6x-13
0%
3x^2+6x+13
0%
-3x^2+6x-13
Explanation
Given that,
f(x)=x^2+2x-3
and
g(x)=3x-4
(gof) (x)=g\left(f(x)\right)
=3(x^2+2x-3)-4
(gof) (x)=3x^2+6x-13
Hence, option A.
If
\displaystyle f:[1,+\infty ]\rightarrow [2,+\infty )
is given by
f(x)=x+\dfrac{1}{x}
then
f^{-1}(x)
equals
Report Question
0%
\displaystyle \frac{x+\sqrt{x^{2}+4}}{2}
0%
\displaystyle \frac{x}{1+x^{2}}
0%
\displaystyle \frac{x+\sqrt{x^{2}-4}}{2}
0%
\displaystyle 1+\sqrt{x^{2}-4}
Explanation
Let
y = x+\cfrac{1}{x}
\Rightarrow x^2-yx+1=0\Rightarrow x = \cfrac{y+\sqrt{y^2-4}}{2}=f^{-1}(y)
Hence
f^{-1}(x) = \cfrac{x+\sqrt{x^2-4}}{2}
Note: -ve term in step two is discarded using given domain and co-domain.
f:R\rightarrow R
is a function defined by
f(x)=10x-7
. If
g=f^{-1}
, then
g(x)
is equals
Report Question
0%
\dfrac{1}{10x-7}
0%
\dfrac{1}{10x+7}
0%
\dfrac{x+7}{10}
0%
\dfrac{x-7}{10}
Explanation
y=10x-7
It is a one one function over the given domain.
y+7=10(x)
x=\dfrac{y+7}{10}
Hence
f^{-1}
will be
g(x)
g(x)=\dfrac{x+7}{10}
Hence, option 'C' is correct.
Set
A
has
3
elements and set
B
has
4
elements. The number of injections that can be defined from
A
to
B
is
Report Question
0%
144
0%
12
0%
24
0%
64
Explanation
If
f:R\rightarrow R
such that
f(x)=log_3 x
then
f^{-1} x
is equal to
Report Question
0%
log x^3
0%
3^x
0%
3^{-x}
0%
3^{1/x}
Explanation
f(x)
is invertible for all positive
x
.
y=log_{3}(x)
Taking antilogarithm on both sides, we get
3^{y}=x
.
Hence,
f^{-1}(x)=3^{x}
Hence, option 'B' is correct.
The inverse of
\displaystyle f\left ( x \right )=\frac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}
is
Report Question
0%
\displaystyle \frac{1}{6}\log_{10}\left ( \frac{1+x}{1-x} \right )
0%
\displaystyle \frac{1}{6}\log_{10}\left ( \frac{x}{1-x} \right )
0%
\displaystyle \frac{1}{6}\log_{e}\left ( \frac{1+x}{1-x} \right )
0%
\displaystyle \frac{1}{6}\log_{e}\left ( \frac{1-x}{1+x} \right )
Explanation
\displaystyle y=\frac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}
\Rightarrow \displaystyle y=\frac{e^{6x}-1}{e^{6x}+1}
\Rightarrow \displaystyle \frac{y+1}{y-1}=\frac{e^{6x}-1+e^{6x}+1}{e^{6x}-1-e^{6x}-1}
\Rightarrow \displaystyle \frac{y+1}{y-1}=\frac{2e^{6x}}{-2}
\Rightarrow \displaystyle \frac{1+y}{1-y}=e^{6x}
Taking
\log_e
we get
\displaystyle \log_{e}\left(\frac{1+y}{1-y}\right)=6x
\displaystyle x=\frac{1}{6}\left(\log_{e}\left(\frac{1+y}{1-y}\right)\right)
Hence
f^{-1}
\displaystyle =\frac{1}{6}\left(\log_{e}\left(\frac{1+x}{1-x}\right)\right)
If
\displaystyle X= \left \{ 1,2,3,4,5 \right \}
and
\displaystyle Y= \left \{ 1,3,5,7,9 \right \}
then which of the following sets are relation from
X
to
Y
Report Question
0%
\displaystyle R_{1}= \left \{ (x,a):a= x+2,x\in X,a\in Y \right \}
0%
\displaystyle R_{2}= \left \{ (1,1),(2,1),(3,3),(4,3),(5,5) \right \}
0%
\displaystyle R_{3}= \left \{ (1,1),(1,3),(3,5),(3,7),(5,7) \right \}
0%
\displaystyle R_{4}= \left \{ (1,3),(2,5),(4,7),(5,9),(3,1) \right \}
Explanation
{ R }_{ 1 }=\left\{ \left( 1,3 \right) ,\left( 2,4 \right) ,\left( 3,5 \right) ,\left( 4,6 \right) ,\left( 5,7 \right) \right\}
since
4
and
6
do not belong to
y
.
\therefore \left( 2,4 \right) ,\left( 4,6 \right) \notin { R }_{ 1 }
\therefore { R }_{ 1 }=\left\{ \left( 1,3 \right) ,\left( 3,5 \right) ,\left( 5,7 \right) \right\} \subset X\times Y
Hence,
{ R }_{ 1 }
is a relation but not a mapping as the elements
2
and
4
do not have any image.
{ R }_{ 2 }:
It is certainly a mapping and since every mapping is a relation, it is a relation as well.
{ R }_{ 3 }:
It is a relation being a subset of
X\times Y
.
{ R }_{ 4 }:
It is both mapping and a relation. Each element in
X
has a unique image. It is also one-one and on-to mapping and hence a bijection.
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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