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CBSE Questions for Class 12 Commerce Maths Relations And Functions Quiz 5 - MCQExams.com
CBSE
Class 12 Commerce Maths
Relations And Functions
Quiz 5
If
f
(
x
)
=
1
1
−
x
,
x
≠
0
,
1
then the graph of the function
y
=
f
[
f
{
f
(
x
)
}
]
for
x
>
1
is
Report Question
0%
a straight line
0%
a circle
0%
an ellipse
0%
a pair of straight lines
Explanation
Given
f
(
x
)
=
1
1
−
x
⇒
f
{
f
(
x
)
}
=
1
1
−
f
(
x
)
=
1
1
−
1
1
−
x
=
1
−
x
1
−
x
−
1
=
1
−
1
x
∴
f
[
f
{
f
(
x
)
}
]
=
1
1
−
f
{
f
(
x
)
}
=
1
1
−
1
+
1
x
=
x
, which is a straight line.
Let
f : R \rightarrow R, g : R \rightarrow R
be two function such that
f(x) = 2x- 3, g(x) = x^3 + 5
The function
(fog)^{1}(x)
is equal to.
Report Question
0%
\left ( \dfrac {x+7}{2} \right )^{1/3}
0%
\left (x- \dfrac {7}{2} \right )^{1/3}
0%
\left ( \dfrac {x-2}{7} \right )^{1/3}
0%
\left ( \dfrac {x-7}{2} \right )^{1/3}
Explanation
Given
f(x) = 2x-3
and
g(x) = x^3+ 5
(fog)(x) = fg(x) = f(x^3+5) = 2(x^3+5) -3 = 2x^3 +7
Let
(fog)(x) = y = 2x^3+7
y-7 = 2x^3
x = (\dfrac{y-7}{2})^{\dfrac{1}{3}}
\therefore (fog)^{-1}(x) = (\dfrac{x-7}{2})^{\dfrac{1}{3}}
If
f(x) = 3x - 5, \,\,then\,\, f^{-1}(x)
.
Report Question
0%
is given by
\dfrac {1}{3x-5}
0%
is given by
\dfrac {x+5}{3}
0%
does not exist because f is not one-one
0%
does not exist because f is not onto
Explanation
Given,
f(x)= 3x-5
Let
y=f^{ -1 }\left( x \right)
\Rightarrow x=f\left( y \right) =3y-5
\Rightarrow y=\dfrac { x+5 }{ 3 }
If
f : [0, \Pi ] \rightarrow [-1, 1]
, f(x) = cosx, then f is.
Report Question
0%
one-one
0%
onto
0%
one-one onto
0%
none of these
Explanation
f takes all values from
[-1,1]
while travelling from
[0,\pi]
. None of the values are repeated either.
Hence it is one-one and onto.
If
f(x) = \left\{\begin{matrix} 1&x \in Q \\ 0 &x \notin Q\end{matrix}\right.
then
fof(\sqrt 3 )
is equal to
Report Question
0%
0
0%
1
0%
\sqrt 3
0%
none of these
Explanation
We know
\sqrt{3}
is an irrational number
Thus
f(\sqrt{3}) =0
and
0
is a rational number
Hence
fof(\sqrt 3 )= f(0) = 1
If functions
f\left ( x \right )
and
g\left ( x \right )
are defined on
R\rightarrow R
such that
f(x)=x+3, x
\in
rational
=4x, x
\in
irrational
g(x)=x+\sqrt{5}
, x
\in
irrational
=-x, x
\in
rational
then
\left ( f-g \right )\left ( x \right )
is
Report Question
0%
one-one & onto
0%
neither one-one nor onto
0%
one-one but not onto
0%
onto but not one-one
Explanation
f\left( x \right) =x+3, x\epsilon
rational
=4x, x\epsilon
irrational
g\left( x \right) =x+\sqrt { 5 } , x \epsilon
irrational
= -x,x\epsilon
rational
(f-g)(x)=2x+3, x\ \epsilon
rational
+3x-\sqrt { 5\quad } x\quad \epsilon
irrational
For one-one;
We know that one one function is a
{ f }^{ \underline { n } }
for which every element of the range of the function corresponds to exactly one element of the domain.
But in this case this is not true as for all rational nos. the
{ f }^{ \underline { n } }
is one one but for irrational
{ f}^{ \underline { n } }
, every elements of the range does not corresponds to exactly one element of the domain.
(f-g)(x)\neq (f-g)({ x }^{ \prime })
when
x \epsilon
rational and
x \epsilon
irrational
for onto:
The
{ f }^{ \underline { n } }(f-g)(x)
does not cover the whole range of the function.
If
\displaystyle f\left ( x \right )=\left ( 1-x^{3} \right )^{\frac{1}{3}}
, then find
fof(x)
Report Question
0%
\dfrac1x
0%
x
0%
x^2
0%
x^3
Explanation
Given,
\displaystyle f\left ( x \right )=\left ( 1-x^{3} \right )^{\frac{1}{3}}
Therefore,
\displaystyle fof\left ( x \right )=f\left [ f\left ( x \right ) \right ]
\displaystyle =f\left ( \left ( 1-x^{3} \right )^{\frac{1}{3}} \right )=\left [ 1-\left \{ \left ( 1-x^{3} \right )^{\frac{1}{3}} \right \}^{3} \right ]^{\frac{1}{3}}
\displaystyle =\left [ 1-\left ( 1-x \right )^{3} \right ]^{\frac{1}{3}}=\left ( x^{3} \right )^{\frac{1}{3}}=x
Let
f(x) =\frac {ax+b} {cx+d}
. Then fof(x) = x provided that.
Report Question
0%
d =- a
0%
d = a
0%
a = b = c = d = 1
0%
a = b = 1
Explanation
f(f(x))=\dfrac{af(x) + b}{cf(x)+d}
f(f(x))=\dfrac{a\dfrac{ax+b}{cx+d} + b}{c\dfrac{ax+b}{cx+d}+d}
f(f(x))=\dfrac{a^2x + ab+bcx+bd}{acx+bc+d^2+dcx}=x
\Rightarrow a^2x+ab+bcx+bd=acx^2+bcx+d^2x+dcx^2
Given
a=-d
, the above equation can also be verified
If
\displaystyle f(x)=2x^{3}+7x-5\, and\, g(x)=f^{-1}(x)
then g' (4) is equal to
Report Question
0%
\displaystyle \frac{1}{13}
0%
\displaystyle \frac{1}{103}
0%
\displaystyle \frac{1}{4}
0%
non existent
Explanation
Given,
f(x)=2x^3+7x-5
\therefore f'(x)= 6x ^2+7
g(x) =f^{-1}(x)\Rightarrow g\{f(x)\}=x
Differentiating both sides, w.r.t
x
\displaystyle {g}'\left ( f(x) \right )=\frac{1}{f'\left ( x \right )}=\frac{1}{6x^{2}+7}
when
x = 1,f(x) = 4
\therefore \displaystyle g'\left ( 4 \right )=\frac{1}{13}
If
\displaystyle R=R^{-1}
then the relation R is ________
Report Question
0%
reflexive
0%
symmetric
0%
anti-symmetric
0%
transitive
Explanation
When a relation is equal to its inverse, then the relation is symmetric.
Let
\displaystyle f\left ( x \right )=\frac{3}{2}+\sqrt{x-\frac{3}{4}}
be a function and
g\left ( x \right )
be another function such that
g\left ( f\left ( x \right ) \right )=x,
then the value of
g\left ( 20 \right )
will be
Report Question
0%
333
0%
335
0%
338
0%
343
Explanation
g\left ( x \right )
is inverse of
f\left ( x \right )
\Rightarrow
\displaystyle x=\frac{3}{2}+\sqrt{y-\frac{3}{4}}
\Rightarrow
\displaystyle y=\left ( x-\frac{3}{2} \right )^2+\frac{3}{4}=f^{-1}(x)=g(x)
\therefore
\displaystyle g\left ( 20 \right )=\left ( 20-\frac{3}{2} \right )^{2}+\frac{3}{4}=343
If f(x) + f(1-x) = 10 then the value of
\displaystyle f\left ( \frac{1}{10} \right )+f\left ( \frac{2}{10} \right )+.........+f\left ( \frac{9}{10} \right )
Report Question
0%
is 45
0%
is 50
0%
is 90
0%
Cannot be determined
Explanation
f(\dfrac {1}{10}) + f(\dfrac {2}{10}) + f(\dfrac {3}{10}) + f(\dfrac {4}{10}) + f(\dfrac {5}{10}) + f(\dfrac {6}{10}) + f(\dfrac {7}{10}) + f(\dfrac {8}{10}) + f(\dfrac {9}{10})
= [f(\dfrac {1}{10}) + f(\dfrac {9}{10}) ] + [ f(\dfrac {2}{10}) + + f(\dfrac {8}{10}) ] + [ f(\dfrac {3}{10}) + + f(\dfrac {7}{10}) ] + [ f(\dfrac {4}{10}) + f(\dfrac {6}{10}) ] + f(\dfrac {5}{10}) ]
= 10 + 10 + 10 + 10 + 5 = 45
If
f (x) = 2x - 1
and
g (x) = 3x + 2
, then find
(fog) (x)
:
Report Question
0%
2 (3x + 1)
0%
2 ( 3x + 2)
0%
3 (2x + 1 )
0%
3 ( 3x + 1 )
Explanation
fog(x)=2(3x+2)-1=6x+4-1=6x+3=3(2x+1)
If
f(x) = -x^2+1, g(x) = -\sqrt[3]{x}
then (gofogofogogog) (x) is.
Report Question
0%
an odd function
0%
an even function
0%
a polynomial function
0%
an identity function
Explanation
If
f(x) = -x^2+1
......
f(x) = -x^2+1, g(x) = -\sqrt[3]{x}
........
\Rightarrow
f
is even and
g
is odd
\therefore
gogog
is even
\Rightarrow
fogogog
is even
\Rightarrow
gofogofogogog
is even.
If the function
f(x) = x^3 + e^{x/2}
and
g(x) = f^{-1}(x)
; then the value of g'(1) is
Report Question
0%
2
0%
-2
0%
1
0%
0
Explanation
f(x) =x^3+e^{x/2}
\displaystyle f'(x) = 3x^2+\frac{1}{2}e^{x/2}
Given g is inverse of f
\Rightarrow g(f(x))=x
Differentiating both sides w.r.t
x
\displaystyle g'(f(x)).f'(x) =1\Rightarrow g'(f(x))=\frac{1}{f'(x)}
Clearly
\displaystyle f'(0)=1\therefore g'(1)=g'(f(0))=\frac{1}{f'(0)}=2
If f(x)=2x-1 and g(x)=3x+2 then find (fog) (x)
Report Question
0%
2(3x+1)
0%
2(3x+2)
0%
3(2x+1)
0%
3(3x+1)
Explanation
(fog)(x) = f(g(x)) = 2(3x+2) -1 = 6x+4-1 = 6x+3 = 3(2x+1)
If f(x) = 2x+1 and g(x) = 3x-5 then find
\left ( fog \right )^{-1}\left ( 0 \right )
Report Question
0%
5/3
0%
3/2
0%
2/3
0%
3/5
Explanation
(fog)(x) = f(g(x)) = 2(3x-5) + 1 = 6x-10 + 1 = 6x-9
Let
(fog)(x)= y
Then, to find
(fog)^{-1} (x)
, we find
x
in terms of
y
So,
6x-9 = y
=> x = \dfrac {y+9}{6}
=> (fog)^{-1} (x)= \dfrac {y+9}{6}
=> (fog)^{-1} (0)= \dfrac {0+9}{6} = \dfrac {3}{2}
If X = {2,3,5,7,11} and Y = {4,6,8,9,10} then find the number of one-one functions from X to Y
Report Question
0%
720
0%
120
0%
24
0%
12
Explanation
Both sets X and Y have the same number of elements.
When two sets have same number of
n
elements, then the number of one to one functions from one set to the other is
n!
So, number of one to one functions from X to Y
= 5 ! = 5 \times 4 \times 3 \times 2 \times 1 = 120
Find
\left( f\circ g \right) \left( 3 \right)
when
f\left( x \right) =7x-6
and
g\left( x \right) =5{ x }^{ 2 }-7x-6
.
Report Question
0%
-36
0%
1014
0%
-90
0%
120
Explanation
We have,
f(x)=7x-6
and
g(x)=5x^2-7x-6
Then,
(f\circ g)(x)
=f(g(x))
=f(5x^2-7x-6)
=7(5x^2-7x-6)-6
=35x^2-49x-42-6
=35x^2-49x-48
\therefore (f\circ g)(3)
=35(3)^2-49(3)-48
=315-147-48
=120
So,
\text{D}
is the correct option.
If
R
is a relation from a set
A
to the set
B
and
S
is a relation from
B
to
C,
then the relation
SoR
Report Question
0%
is from
C
to
A
0%
does not exist
0%
is from
A
to
C
0%
None of these
Explanation
Since
R\subseteq A\times B
and
S\subseteq B\times C
, we have
So
R\subseteq A\times C
.
\therefore
So R is a relation from
A
to
C.
If f = {(1,3) (2,1) (3,4) (4,2)} and g = {(1,2) (2,3) (3,4) (4,1)} then find n(fog)
Report Question
0%
12
0%
16
0%
4
0%
5
Explanation
we can find
n(fog)
by finding the range of the domain of
f
In
(1,3)
as the range
3
is mapped to
4
in
g
, the ordered pair here for
fog
is
(1,4)
In
(2,1)
as the range
1
is mapped to
2
in
g
, the ordered pair here for
fog
is
(2,2)
In
(3,4)
as the range
4
is mapped to
1
in
g
, the ordered pair here for
fog
is
(3,1)
In
(4,2)
as the range
2
is mapped to
3
in
g
, the ordered pair here for
fog
is
(4,3)
As we have
4
ordered pairs for
fog
, we have
n(fog) = 4
What is the relation for the following diagram?
Report Question
0%
R =\{(2, 5), (2, 6), (3, 7)\}
0%
R = \{(1, 5), (2, 6), (3, 7)\}
0%
R =\{(1, 5), (2, 6), (1, 7)\}
0%
R = \{(1, 5), (2, 6), (2, 7)\}
Explanation
R:A\to B
A\times B = \{(1,5),(2,6),(3,7)\}
Which of the following do(es) not belong to
A \times B
for the sets
A = \{1, 2\}
and
B =\{0, 2\}
?
Report Question
0%
R = \{(1, 0), (2, 2)\}
0%
R = \{(1, 1), (2, 1)\}
0%
R = \{(1, 0), (1, 2)\}
0%
R = \{(1, 2), (2, 2)\}
Explanation
A relation for the sets
A
and
B
from
A
to
B
will be a subset of the
A\times B.
Now the ordered pair belonging to
A\times B
are
\{1,2\}\times\{0,2\}
=\{(1,0),(1,2),(2,0),(2,2)\}
Hence,
\{(1,1)\}
and
\{(2,1)\}
do not belong to
A\times B
.
What is the relation for the set
A =\{-1, 0, 3\}
and
B =\{1, 2, 3\}
?
Report Question
0%
R =\{(-1, 1), (1, 2), (3, 3)\}
0%
R = \{(-1, 1), (0, 2), (3, 3)\}
0%
R = \{(-1, 1), (2, 2), (3, 3)\}
0%
R = \{(-1, 1), (0, 2), (-1, 3)\}
Explanation
A\times B=\{(-1,1),(-1,2),(-1,3),(0,1),(0,2),(0,3),(3,1),(3,2),(3,3)\}
R is subset of
A\times B
So option B is ans.
If
f: R\rightarrow R
be given by
f(x) = (3 - x^{3})^{ {1}/{3}}
, then find
f(f (x))
is
Report Question
0%
x^{{1}/{3}}
0%
x^{3}
0%
x
0%
3 - x^{3}
Explanation
f(x)= (3-x^3)^{\dfrac{1}{3}}\\ f(f(x))=f\left((3-x^3)^{\frac{1}{3}}\right)
= (3-((3-x^3)^{\frac{1}{3}})^3)^{\frac{1}{3}}=x
If
f:R\rightarrow R
and
g:R\rightarrow R
are defined by
f\left( x \right) =\left| x \right|
and
g\left( x \right) =\left[ x-3 \right]
for
x\in R
, then
g\left( f\left( x \right) \right) :\left\{ -\dfrac { 8 }{ 5 } < x < \dfrac { 8 }{ 5 } \right\}
is equal to
[.] is Greatest integer function
Report Question
0%
\left\{ 0,1 \right\}
0%
\left\{ 1,2 \right\}
0%
\left\{ -3,-2 \right\}
0%
\left\{ 2,3 \right\}
Explanation
Given that,
f\left( x \right) =\left| x \right|
and
g\left( x \right) =\left[ x-3 \right]
For
-\dfrac { 8 }{ 5 } < x < \dfrac { 8 }{ 5 } , 0\le f\left( x \right) < \dfrac { 8 }{ 5 }
Now, for
0 < f \left( x \right) < 1
,
g\left( f\left( x \right) \right) = \left[ f\left( x \right) -3 \right]
=-3\quad \because -3\le f\left( x \right) -3 < -2
Again, for
1 < f\left( x \right) < 1.6
g\left( f\left( x \right) \right) =-2
\because -2\le f\left( x \right) -3 < -1.4
Hence, required set is
\left\{ -3,-2 \right\}
If
R
be the set of all real numbers and
f:R\rightarrow R
is given by
f\left( x \right)=3{ x }^{ 2 }+1
. Then, the set
f^{ -1 }\left( \left[ 1,6 \right] \right)
is
Report Question
0%
\left\{ -\sqrt { \dfrac { 5 }{ 3 } } ,0,\sqrt { \dfrac { 5 }{ 3 } } \right\}
0%
\left[ -\sqrt { \dfrac { 5 }{ 3 } } ,\sqrt { \dfrac { 5 }{ 3 } } \right]
0%
\left[ -\sqrt { \dfrac { 1 }{ 3 } } ,\sqrt { \dfrac { 1 }{ 3 } } \right]
0%
\left( -\sqrt { \dfrac { 5 }{ 3 } } ,\sqrt { \dfrac { 5 }{ 3 } } \right)
Explanation
Given,
f\left( x \right) =3{ x }^{ 2 }+1
Let
y=3{ x }^{ 2 }+1
\Rightarrow 3{ x }^{ 2 }=y-1\Rightarrow { x }^{ 2 }=\dfrac { y-1 }{ 3 }
\Rightarrow x=\pm \sqrt { \dfrac { y-1 }{ 3 } }
\therefore f^{ -1 }\left( x \right) =\pm \sqrt { \dfrac { x-1 }{ 3 } }
When
x\in \left[ 1,6 \right]
, t
hen,
f^{ -1 }\left( x \right) \in \left[ -\sqrt { \dfrac { 5 }{ 3 } } ,\sqrt { \dfrac { 5 }{ 3 } } \right]
Let
R
be the set of real numbers and the functions
f: R \rightarrow R
and
g: R\rightarrow R
be defined by
f(x) = x^{2} + 2x - 3
and
g(x) = x + 1
. Then the value of
x
for which
f(g(x)) = g(f(x))
is
Report Question
0%
-1
0%
0
0%
1
0%
2
Explanation
According to the question,
f(g(x)) = g(f(x))
\Rightarrow f(x + 1) = g(x^{2} + 2x - 3)
\Rightarrow (x + 1)^{2} + 2 (x + 1) - 3 = x^{2} + 2x - 3 + 1
\Rightarrow x^{2} + 1 + 2x + 2x + 2 - 3 = x^{2} + 2x - 2
\Rightarrow x^{2} + 4x = x^{2} + 2x - 2
\Rightarrow x^{2} + 4x - x^{2} - 2x + 2 = 0
\Rightarrow 2x + 2 = 0
\Rightarrow 2x = -2
\Rightarrow x = -1
Let
f:R\rightarrow R
be such that
f
is injective and
f(x)f(y)=f(x+y)
for all
x,y\in R
, if
f(x), f(y)
and
f(z)
are in GP, then
x,y
and
z
are in
Report Question
0%
AP always
0%
GP always
0%
AP depending on the values of
x,y
and
z
0%
GP depending on the values of
x,y
and
z
Explanation
Let the funtion
f(x)={a}^{kx}
which define in
f:R\rightarrow R
and injective also.
Now, we have
f(x)f(y)=f(x+y)
\Rightarrow
{a}^{kx}.{a}^{ky}={a}^{k(x+y)}
\Rightarrow
{a}^{k(x+y)}={a}^{k(x+y)}
\because
f(x), f(y)
and
f(z)
are in GP
\therefore
f({y}^{2})=f(x).f(z)
\Rightarrow
{a}^{2ky}={a}^{kx}.{a}^{kz}
\Rightarrow
{e}^{2ky}={e}^{k(x+z)}
On comparing, we get
2ky=k(x+z)
\Rightarrow
2y=x+z
\Rightarrow
x,y
and
z
are in AP
Find the number of binary operations on the set
\left \{a, b\right \}
Report Question
0%
10
0%
16
0%
20
0%
8
Explanation
Considering 'n' elements in a set, the no. of binary operations on that set will be
=n^{n^{2}}
Applying this concept to the above set gives us
=2^{2^{2}}
...................
\because n=2
=2^{4}
=16
.
Let Q be the set of all rational numbers in [0, 1] and
f : [0, 1]\rightarrow [0, 1]
be defined by
f(x)=\begin{cases}x&for&x\in Q\\ 1-x&for&x\notin Q\end{cases}
Then the set
S=\{x\in [0, 1]: (f\, o \, f)(x)=x\}
is equal to
Report Question
0%
[0, 1]
0%
Q
0%
[0, 1] - Q
0%
(0, 1)
Explanation
Let
x\in Q
, then,
f(x)=x
where
x\in Q
So,
fof(x)=f(f(x))=f(x)=x
as
x\in Q
\therefore fof(x)=x
when
x \in Q
Now,
Let
x\notin Q
then
f(x) =1-x
\therefore fof(x)=1-(1-x) = x
as
1-x\notin Q
as
x\notin Q
where
x\notin Q
fof(x)=\begin{cases}x \,where\, x\in Q \,\&\, x\in [0, 1] \\ x\, where x\notin Q\, \&\, x\in [0, 1] \end{cases}
\therefore
the set
S = [0, 1]
If
f(x)={2}^{100}x+1, g(x)={3}^{100}x+1
, then the set of real numbers
x
such that
f\left\{ g(x) \right\} =x
is
Report Question
0%
empty
0%
a singleton
0%
a finite set with more than one element
0%
infinite
Explanation
Given
f(x)={2}^{100}x+1, g(x)={3}^{100}x+1
Now
fo{g(x)}=x
\Rightarrow
f({3}^{100}.x+1)=x\\
\Rightarrow
{2}^{100}({3}^{100}.x+1)+1=x\\
\Rightarrow
{6}^{100}.x+{2}^{100}+1=x\\
\Rightarrow
x(1-{6}^{100})=(1+{2}^{100})
\Rightarrow
x=\cfrac{1+{2}^{100}}{1-{6}^{100}}
Hence
fog(x)=x
represent a singleton set.
In three element group
\{e, a, b\}
where
e
is the identity,
a^5b^4
is equal to
Report Question
0%
a
0%
e
0%
ab
0%
b
Explanation
Given group
G\equiv\{e,a,b\}
It is given that
e
is an identity.
Therefore only possibilities we have are
Either
a\cdot a = e
and
b\cdot b = e
.... (i)
Or
a\cdot b = b\cdot a = e
...(ii)
Case I:
a\cdot a = e
and
b\cdot b = e
a^5b^4 = a\cdot a\cdot a\cdot a\cdot a\cdot b\cdot b\cdot b\cdot b
\Rightarrow a^5b^4 = e\cdot a\cdot a\cdot a\cdot b\cdot b\cdot e
\Rightarrow a^5b^4 = e\cdot a\cdot e
\Rightarrow a^5b^4 = a
Case II:
a\cdot b = b\cdot a = e
a^5b^4 = a\cdot a\cdot a\cdot a\cdot a\cdot b\cdot b\cdot b\cdot b
\Rightarrow a^5b^4 = a\cdot a\cdot a\cdot a\cdot e\cdot b\cdot b\cdot b
\Rightarrow a^5b^4 = a\cdot a\cdot a\cdot e\cdot b\cdot b
\Rightarrow a^5b^4 = a\cdot a\cdot e\cdot b
\Rightarrow a^5b^4 = a\cdot e
\Rightarrow a^5b^4 = a
Hence, both cases resulted in
a
.
Find the value of
{f}^{-1}(1.5)
if
f(x)=\sqrt [ 3 ]{ { x }^{ 3 }+1 }
.
Report Question
0%
3.4
0%
2.4
0%
1.3
0%
1.5
Explanation
As per the condition,
1.5 = \sqrt[3]{x^3+1}
Taking cube roots on both the sides,
1.5^3=x^3+1
3.375 - 1 = x^3
2.375 = x^3
x \approx 1.3
If
f(x)\, =\, (p\, -\, x^n)^{1/n},\, p\, >\, 0
and
n
is a positive integer, then
f(f(x)) =
Report Question
0%
x
0%
x^n
0%
p^{1/n}
0%
p\, -\, x^n
Explanation
Given,
f(x)=(p-x^{n})^{\dfrac{1}{n}}
\therefore f(f(x))=(p-((p-x^{n})^{\dfrac{1}{n}})^{n})^{\dfrac{1}{n}}
=(p-(p-x^{n}))^{\dfrac{1}{n}}
=(x^{n})^{\dfrac{1}{n}}
=x
Which of the following is a subgroup of the group
G = \left \{1, 2, 3, 4, 5, 6\right \}
under
\otimes_{7}
.
(
\otimes_7
: under multiplication modulo 7)
Report Question
0%
\left \{2, 6, 1\right \}
0%
\left \{1, 2, 4\right \}
0%
\left \{5, 4, 2\right \}
0%
\left \{2, 3, 1\right \}
Explanation
(c) can not be a subgroup as identity
'1'
is not present
(d) can not be a subgroup of
2\otimes_{7} 3 = 6\not {\epsilon} \left \{1, 2, 3\right \}
(a) can not be a subgroup as
2\otimes_{7} 6 = 5\not {\epsilon} \left \{2, 6, 1\right \}
\therefore
(b) is a subgroup
If
f: R\rightarrow R^{+}
and
g: R^{+} \rightarrow R
are such that
g(f(x)) = |\sin x|
and
f(g(x)) = (\sin \sqrt {x})^{2}
, then a possible choice for f and g is
Report Question
0%
f(x) = x^{2} , g(x) = \sin \sqrt {x}
0%
f(x) = \sin x, g(x) = |x|
0%
f(x) = \sin^{2}x, g(x) = \sqrt {x}
0%
f(x) = x^{2}, g(x) = \sqrt {x}
Explanation
As f returns only positive values and takes any real number it can be mod or square hence option 2 is eliminated
For g only positive values has to be given and it will return any real number
so on trying the option only option
A
and
C
results in the given equation ie
g(f(x)) = |sin x|
Now function g should take positive values and return real values which is satisfied by only option
A
as in option
C
square root of positive number will always result in positive number while
sin\sqrt{x}
will give -ve real numbers as well.
If
\displaystyle { Q }_{ 1 }
is the set of all relations other than
1
with the binary operation
\displaystyle \ast
defined by
\displaystyle a\ast b=a+b-ab
for all
a, b \in \displaystyle Q_1
, then the identity in
\displaystyle { Q }_{ 1 }
with respect to
\displaystyle { Q }_{ 1 }
is
Report Question
0%
1
0%
0
0%
-1
0%
2
Explanation
Let
b
be the identity
a*b=a
a+b-ab=a
b-ab=0
b(1-a)=0
b=0
and
a=1
However,
1
is excluded.
Hence
b=0
Hence
0
is the identity element
In
Z
, the set of all integers, the inverse of
-7
w.r.t. defined by
a\times b=a+b+7
for all
a, b, \in Z
is :
Report Question
0%
-14
0%
7
0%
14
0%
-7
Explanation
Consider the identity element of 'b' for any element
a\in Z
a\times b=a
a+b+7=a
b+7=0
b=-7
.
Hence identity element is -7.
Hence
e=-7
.
Now for the inverse element
a\times a^{-1}=e
a+a^{-1}+7=-7
...
(e=-7)
a^{-1}=-14-a
Hence
7^{-1}=-14-(-7)
=-7
.
f(x) = x^{2} + d
and
g(x) = 2x^{2}
, where d is a constant. If
\dfrac {f(g(2))}{f(2)} = 4
, find the value of
d
.
Report Question
0%
16
0%
5
0%
22
0%
18
Explanation
Given,
f(x)=x^2+d
and
g(x)=2x^2
g(2)=2(2^{2})=8
Hence,
f(g(2))=f(8)=64+d
Therefore
\dfrac{f(g(2))}{f(2)}=\dfrac{64+d}{4+d}=4
Hence,
64+d=16+4d
\Rightarrow 48=3d
\Rightarrow d=16
If
f(g(a)) = 0
where
g(x) = \dfrac {x}{4} + 2
and
f(x) = |x^{2} - 3|
, find the possible value of
a.
Report Question
0%
-8+4\sqrt{3}
0%
-(8+4\sqrt{3})
0%
6
0%
18
Explanation
Given
g(a)=\dfrac{a}{4}+2,f(x)=|x^2|-3
and
f(g(a))=0
Now,
f(g(a))=|g(a)^{2}-3|=|(\dfrac{a}{4}+2)^{2}-3|
=|\dfrac{a^{2}}{16}+a+4-3|=|\dfrac{a^{2}}{16}+a+1|=\dfrac{|a^{2}+16a+16|}{16}
but
f(g(a))=0
\Rightarrow\dfrac{|a^2+16a+16|}{16}=0
\Rightarrow a^{2}+16a+16=0
[\because |x|=0\Rightarrow x=0]
\Rightarrow a= \dfrac{-16\pm\sqrt{16^2-4(1)(64)}}{2}
\Rightarrow a=\dfrac{-16\pm\sqrt{256-64}}{2}=\dfrac{-16\pm8\sqrt{3}}{2}
Hence,
a=-(8+4\sqrt{3})
and
a=-8+4\sqrt{3}
.
If
p(x) = \dfrac{x}{x-2}
and
q(x) = \sqrt{9-x}
, find the value of
(p\circ q)(5)
Report Question
0%
0
0%
\dfrac{8}{7}
0%
2
0%
Undefined
Explanation
Given,
p = \dfrac{x}{x-2}
q=\sqrt{9-x}
For
(p\circ q)
, substitute the value
p
in
q
ie.,
\dfrac{\sqrt{9-x}}{\sqrt{9-x}-2}(5)
is a undefined.
If
f(x) =
\sqrt{x}
and
g(x) =
\sqrt{x^2+4}
, calculate the value of
f(g(2))
.
Report Question
0%
0
0%
1.41
0%
1.68
0%
2.45
0%
2.83
Explanation
Given functions are
f(x)=\sqrt {x}
and
g(x)=\sqrt {x^2+4}
.
Now
g(2)=\sqrt{2^2+4}=2\sqrt2
Then
f(g(2))=\sqrt { g(2) } = \sqrt { 2\sqrt { 2 } } ={ 2 }^{ 3/4 }=1.681
If
f(x) = 2x
and
f(f(x)) = x + 1
, then the value of
x
is
Report Question
0%
\dfrac {1}{3}
0%
1
0%
2
0%
3
0%
5
Explanation
Given
f(x)=2x
and
f(f(x))=x+1
Then
f(f(x))=x+1
\Rightarrow 2[f(x)]=x+1
\Rightarrow 2(2x)=x+1
\Rightarrow 4x=x+1
\Rightarrow x=\dfrac {1}{3}
Find the value of
g(f(2))
, if
f(x) = e^{x}
and
g(x) = \dfrac {x}{2}
Report Question
0%
2.7
0%
3.7
0%
4.2
0%
5.4
0%
6.1
Explanation
Given,
f(x)=e^x, g(x)=\dfrac {x}{2}
\Rightarrow f(2) = { e }^{ 2 }=7.389
\Rightarrow g(f(2)) = g(7.389) =\dfrac { 7.389}{2} = 3.7
If
h(x)={x}^{3}+x
and
g(x)=2x+3
, then calculate
g(h(2))
.
Report Question
0%
7
0%
10
0%
17
0%
19
0%
23
Explanation
Given,
h(x)=x^3+x
\therefore h(2)=2^3+2
\Rightarrow h(2)=8+2=10
Also given,
g(x)=2x+3
\Rightarrow g(h(2))=g(10)
\because h(2)=10
Then the value of
g(10)=2\times 10+3
\Rightarrow g(10)=20+3=23
If
f(x) = x^{2} - 10
and
g(x) = 4x + 3
, calculate the value of
f(g(2))
.
Report Question
0%
-24
0%
-21
0%
12
0%
27
0%
111
Explanation
Given,
f(x)=x^2-10
and
g(x)=4x+3
\therefore g(2) = 4 \times 2 + 3 = 8+3 = 11
Then,
f(g(2)) = f(11) = { 11 }^{ 2 }-10=121-10=111
If
f(x) =
x^2
and
g(x) = 2x
, calculate the value of
f(g(-3))-g(f(-3))
.
Report Question
0%
54
0%
18
0%
0
0%
-18
0%
-54
Explanation
Given
f(x)=x^2
and
g(x)=2x
.
Then
f(-3)={(-3)}^2=9
and
g(-3)=2(-3)=-6
So,
f(g(-3))={[g(-3)]^2}={[-6]}^2=36
and
g(f(-3))=2*f(-3)=2*9=18
The value of
f(g(-3))-g(f(-3))
is
36-18=18
The above figure shows the graph of the function
f(x)
, the value of
f(f(3))
is:
Report Question
0%
-4
0%
-2
0%
0
0%
1
0%
3
Explanation
From the graph shown, we need to find
f(f(3))
The value of the function at
x = 3
is given by
f(3) = -2
Next,
f(f(3)) = f(-2) = 0
Find the correct
expression for
\displaystyle f\left( g\left( x \right) \right)
if
\displaystyle f(x)=4x+1
and
\displaystyle g\left( x \right) ={ x }^{ 2 }-2
Report Question
0%
\displaystyle -{ x }^{ 2 }+4x+1
0%
\displaystyle { x }^{ 2 }+4x-1
0%
\displaystyle 4{ x }^{ 2 }-7
0%
\displaystyle 4{ x }^{ 2 }-1
0%
\displaystyle 16{ x }^{ 2 }+8x-1
Explanation
f(x)=4x+1
and
g(x)=x^2-2
\Rightarrow f(g(x))=f(x^2-2)
\Rightarrow (4(x^2-2)+1
\Rightarrow 4x^2-8+1
\Rightarrow 4x^2-7
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Practice Class 12 Commerce Maths Quiz Questions and Answers
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