Relations and Functions Test

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Relations and Functions Test - 35

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Weekly Quiz Competition

Question 1
1 / -0

The number of one-one functions that can be defined from $$A=\{4,8,12,16\}$$ to $$B$$ is $$5040,$$ then $$n(B)=$$

A
$$7$$

B
$$8$$

C
$$9$$

D
$$10$$

Solution

The first element $$4$$ of $$A$$ can be mapped to any of the $$n$$ elements in $$B$$
Similarly second element $$8$$ of $$A$$ can be mapped to any of $$(n-1)$$ elements in $$B$$
$$\Rightarrow$$ Total no of one - one functions is
$$n(n-1)(n-2)(n-3)=5040$$
$$\Rightarrow n=10$$
Question 2
1 / -0

If $$f:R^+\rightarrow [1,\infty)$$, defined by $$f\left( x \right) ={ x }^{ 2 }+1$$, then the value of $${ f }^{ -1 }\left( 17 \right) $$ and $${ f }^{ -1 }\left( -3 \right) $$ respectively are

A
$$\phi ,\left\{ 4,-4 \right\}$$

B
$$\left\{ 3,-3 \right\},\phi $$

C
Not invertible

D
$$\left\{ 4 \right\},\phi $$

Solution

Given, $$f(x)=x^{2}+1$$
Put $$f(x)=y$$
$$y=x^{2}+1$$
$$\Rightarrow x=\pm \sqrt{y-1}$$
$$\Rightarrow f^{-1}(y)=\pm \sqrt{y-1}$$
So, $$f^{-1}(17)=\pm 4$$
$$f^-{1} (17) = 4$$ [ As $$-4 \notin$$ domain $$R^+$$]

and $$f^{-1}(-3)=\phi $$
$$f^{-1}(-3)$$ is not defined in $$R^+$$ as square root of negative number is not defined in $$R^+$$
Option D is correct
Question 3
1 / -0

The number of injections that are possible from $$A$$ to itself is $$720,$$ then $$n (A) =$$

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$

Solution

first element of $$A$$ in domain can be mapped to
any of the $$n$$ elements of $$A$$ in range.
$$2^{nd}$$ element of $$A$$ in domain can be mapped to any of $$(n-1)$$ elements of $$A$$ in range.
$$\therefore$$ Total no of injections possible is
$$n\times (n-1)\times (n-2)...\times 3\times 2\times 1=n!$$
$$\therefore n!=720$$
$$\Rightarrow n=6$$
Question 4
1 / -0

The number of one-one functions that can be defined from $$A = \left \{ 1,2,3 \right \} $$ to $$ B = \left \{ a,e,i,o,u \right \}$$ is

A
$$3^{5}$$

B
$$5^{3}$$

C
$${_{}}^{5}P_{3}$$

D
$$5!$$

Solution

You can correlate this to permutation and combination problems.

We have to arrange 3 people on 5 places.

So total no of permutations $$={_{}}^{5}P_{3}$$
Question 5
1 / -0

The number of non-surjective mappings that can be defined from $$A = \left \{ 1,4,9,16 \right \} $$ to$$ B=\left \{ 2,8,16,32,64 \right \}$$ is

A
$$1024$$

B
$$20$$

C
$$505$$

D
$$625$$

Solution

Here $$n(B)>n(A)$$, no function can be a surjective.
No of functions $$=$$ No of non-surjectives

Any elements of $$A$$ can be mapped to any of $$5$$ elements in $$B$$
So non surjective mapping from $$A$$ to $$B$$
$$=5\times5\times5\times5$$
$$=625$$
Question 6
1 / -0

If $$ f:A\rightarrow B $$ is a constant function which is onto then $$B$$ is

A
a singleton set

B
a null set

C
an infinite set

D
a finite set

Solution

$$f(x)$$ is a constant fucntion $$\Rightarrow$$ Range of $$f(x)$$ is a singleton set.
For $$f$$ to be an onto function, Co domain $$B$$ should be equal to Range.
$$\therefore B $$ is a singleton set.
Question 7
1 / -0

If $$ f:A\rightarrow B $$ is a bijection then $$ f^{-1} of = $$

A
$$fof^{-1}$$

B
$$f$$

C
$$f^{-1}$$

D
an identity

Solution

Since $$f$$ is bijective, its inverse will also be bijective.
$$f(x)=y$$
$$x=f^{-1}(y)$$
$$f(x)=f^{-1}(y)$$
$$y=f(f)^{-1}(y)=f(x)=y$$
Hence, it is an identity.
Question 8
1 / -0

The number of bijection that can be defined from $$A = \left \{ 1,2,8,9 \right \} $$ to $$ B = \left \{ 3,4,5,10 \right \}$$ is

A
$$4^{4}$$

B
$$4^{2}$$

C
$$24$$

D
$$18$$

Solution

There are 4 inputs $$\{1, 2, 8, 9\}$$ and 4 outputs $$\{3, 4, 5, 10\}$$. Hence function will be bijective if and only if each output is connected with only one input.
Therefore number of bijective function is $$4! = 24$$
Question 9
1 / -0

The number of possible surjection from $$A=\{1,2,3,...n\}$$ to $$B = \{1,2\}$$ (where $$n \geq 2)$$ is $$62$$, then $$n=$$

A
$$5$$

B
$$6$$

C
$$7$$

D
$$8$$

Solution

Each element in $$A$$ can be mapped onto any of two elements of $$B$$
$$\therefore$$ Total possible functions are $$2^{n}$$
For the $$f^{{n}'s}$$ to be surjections , they shouldn't be mapped alone to any of the two elements.
$$\therefore$$ Total no of surjections $$= 2^{n}-2$$
$$2^{n}-2=62$$
$$\Rightarrow n=6$$
Question 10
1 / -0

The number of injections possible from $$A=\{1,3,5,6\}$$ to $$B =\{2,8,11\}$$ is

A
$$8$$

B
$$64$$

C
$$2^{12}$$

D
$$0$$

Solution

co-domain $$n(B) <$$ domain $$n(A)$$
There can't be any one-one function from $$A$$ to $$B.$$

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Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 35

Result

Relations and Functions Test - 35

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SHARING IS CARING

Question 1

$$7$$

$$8$$

$$9$$

$$10$$

Solution

Question 2

$$\phi ,\left\{ 4,-4 \right\}$$

$$\left\{ 3,-3 \right\},\phi $$

Not invertible

$$\left\{ 4 \right\},\phi $$

Solution

Question 3

$$5$$

$$6$$

$$7$$

$$8$$

Solution

Question 4

$$3^{5}$$

$$5^{3}$$

$${_{}}^{5}P_{3}$$

$$5!$$

Solution

Question 5

$$1024$$

$$20$$

$$505$$

$$625$$

Solution

Question 6

a singleton set

a null set

an infinite set

a finite set

Solution

Question 7

$$fof^{-1}$$

$$f$$

$$f^{-1}$$

an identity

Solution

Question 8

$$4^{4}$$

$$4^{2}$$

$$24$$

$$18$$

Solution

Question 9

$$5$$

$$6$$

$$7$$

$$8$$

Solution

Question 10

$$8$$

$$64$$

$$2^{12}$$

$$0$$

Solution

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