Relations and Functions Test

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Relations and Functions Test - 54

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Weekly Quiz Competition

Question 1
1 / -0

Find the number of binary operations on the set $$\left \{a, b\right \}$$

A
$$10$$

B
$$16$$

C
$$20$$

D
$$8$$

Solution

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

$$=2^{2^{2}}$$ ................... $$\because n=2$$

$$=2^{4}$$
$$=16$$.
Question 2
1 / -0

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

A
empty

B
a singleton

C
a finite set with more than one element

D
infinite

Solution

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.
Question 3
1 / -0

In three element group $$\{e, a, b\}$$ where $$e$$ is the identity, $$a^5b^4$$ is equal to

A
$$a$$

B
$$e$$

C
$$ab$$

D
$$b$$

Solution

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 $$a$$.
Question 4
1 / -0

Find the value of $${f}^{-1}(1.5)$$ if $$f(x)=\sqrt [ 3 ]{ { x }^{ 3 }+1 } $$.

A
$$3.4$$

B
$$2.4$$

C
$$1.3$$

D
$$1.5$$

Solution

As per the condition, $$1.5 = \sqrt[3]{x^3+1}$$
Taking cube roots on both the sides,
$$1.5^3=x^3+1$$
$$3.375 - 1 = x^3$$
$$2.375 = x^3$$
$$x \approx 1.3$$
Question 5
1 / -0

If $$f(x)\, =\, (p\, -\, x^n)^{1/n},\, p\, >\, 0$$ and $$n$$ is a positive integer, then $$f(f(x)) =$$

A
$$x$$

B
$$x^n$$

C
$$p^{1/n}$$

D
$$p\, -\, x^n$$

Solution

Given, $$f(x)=(p-x^{n})^{\dfrac{1}{n}}$$

$$\therefore f(f(x))=(p-((p-x^{n})^{\dfrac{1}{n}})^{n})^{\dfrac{1}{n}}$$

$$=(p-(p-x^{n}))^{\dfrac{1}{n}}$$

$$=(x^{n})^{\dfrac{1}{n}}$$

$$=x$$
Question 6
1 / -0

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

A
$$1$$

B
$$0$$

C
$$-1$$

D
$$2$$

Solution

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
Question 7
1 / -0

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)

A
$$\left \{2, 6, 1\right \}$$

B
$$\left \{1, 2, 4\right \}$$

C
$$\left \{5, 4, 2\right \}$$

D
$$\left \{2, 3, 1\right \}$$

Solution

(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
Question 8
1 / -0

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

A
$$f(x) = x^{2} , g(x) = \sin \sqrt {x}$$

B
$$f(x) = \sin x, g(x) = |x|$$

C
$$f(x) = \sin^{2}x, g(x) = \sqrt {x}$$

D
$$f(x) = x^{2}, g(x) = \sqrt {x}$$

Solution

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.
Question 9
1 / -0

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 :

A
$$-14$$

B
$$7$$

C
$$14$$

D
$$-7$$

Solution

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
$$7^{-1}=-14-(-7)$$
$$=-7$$.
Question 10
1 / -0

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

A
[0, 1]

B
Q

C
[0, 1] - Q

D
(0, 1)

Solution

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 $$S = [0, 1]$$

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Relations and Functions Test - 54

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Relations and Functions Test - 54

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SHARING IS CARING

Question 1

$$10$$

$$16$$

$$20$$

$$8$$

Solution

Question 2

empty

a singleton

a finite set with more than one element

infinite

Solution

Question 3

$$a$$

$$e$$

$$ab$$

$$b$$

Solution

Question 4

$$3.4$$

$$2.4$$

$$1.3$$

$$1.5$$

Solution

Question 5

$$x$$

$$x^n$$

$$p^{1/n}$$

$$p\, -\, x^n$$

Solution

Question 6

$$1$$

$$0$$

$$-1$$

$$2$$

Solution

Question 7

$$\left \{2, 6, 1\right \}$$

$$\left \{1, 2, 4\right \}$$

$$\left \{5, 4, 2\right \}$$

$$\left \{2, 3, 1\right \}$$

Solution

Question 8

$$f(x) = x^{2} , g(x) = \sin \sqrt {x}$$

$$f(x) = \sin x, g(x) = |x|$$

$$f(x) = \sin^{2}x, g(x) = \sqrt {x}$$

$$f(x) = x^{2}, g(x) = \sqrt {x}$$

Solution

Question 9

$$-14$$

$$7$$

$$14$$

$$-7$$

Solution

Question 10

[0, 1]

Q

[0, 1] - Q

(0, 1)

Solution

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