Relations and Functions Test

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Relations and Functions Test - 31

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Weekly Quiz Competition

Question 1
1 / -0

In the group $$G = \left \{1, 5, 7, 11\right \}$$ under $$\otimes_{12}$$ the value of $$7\otimes_{12} 11^{-1}$$ is equal to
($$\otimes_{12}$$: under multiplication modulo $$12$$)

A
$$5$$

B
$$7$$

C
$$11$$

D
$$1$$

Solution

Clearly $$11^{-1} = 11(\because 11\otimes_{12} 11 = 1)$$
$$\therefore 7 \otimes_{12} 11^{-1} = 7\otimes_{12} 11 = 5$$
Question 2
1 / -0

What is the relation for the statement "A is taller than B"?

A
is taller than

B
A is taller

C
B is taller

D
is less than

Solution

A is taller than B.
A is related to B by 'taller than'.
Question 3
1 / -0

x varies directly as y and inversely as the square of z. When y = 4 and z is 14 x = 10. If y = 16 and z = 7 what is x?

A
180

B
160

C
280

D
200

Solution

Given x varies directly as y and inversely as the square of z. When y = 4 and z is 14 x = 10. If y = 16 and z = 7
Then $$x=k\frac{y}{z^{2}}$$
$$\Rightarrow 10=k\times \frac{4}{14\times14}$$
$$\Rightarrow k=490$$
Thus $$x=490\times \frac{16}{7\times 7}=160$$
Question 4
1 / -0

Find the correct expression for $$\displaystyle f\left( g\left( x \right) \right) $$ given that $$\displaystyle f\left( x \right) =4x+1$$ and $$\displaystyle g\left( x \right) ={ x }^{ 2 }-2$$

A
$$\displaystyle -{ x }^{ 2 }+4x+1$$

B
$$\displaystyle { x }^{ 2 }+4x-1$$

C
$$\displaystyle 4{ x }^{ 2 }-7$$

D
$$\displaystyle 4{ x }^{ 2 }-1$$

E
$$\displaystyle 16{ x }^{ 2 }+8x-1$$

Solution

Given, $$g(x)=x^2-2$$
$$f(g(x))=f(x^2-2)$$
Also given, $$f(x)=4x+1$$
$$\therefore f(x^2-2)=4(x^2-2)+1$$
$$\Rightarrow 4x^2-8+1$$
$$\Rightarrow 4x^2-7$$
Question 5
1 / -0

If $$a,b\in A, a*b\in A$$ then

A
$$*$$ is a unary operation in $$A$$

B
$$a * b = b * a$$

C
$$*$$ is a binary operation in $$A$$

D
$$a * b \neq b * a$$

Solution

If $$a,b\in A,a*b\in A,$$ then $$*$$ is a binary operation in A.
Question 6
1 / -0

If $$f(x) = \sqrt {x^{2} - 3x + 6}$$ and $$g(x) = \dfrac {156}{x +17}$$, find the value of the composite function $$g(f(4))$$.

A
$$5.8$$

B
$$7.4$$

C
$$7.7$$

D
$$8.2$$

E
$$10.3$$

Solution

$$f(x)=\sqrt { x^{ 2 }-3x+6 } ,\quad g(x)=\frac{ 156 }{ x+17 }\\ g(f(4))=\quad ?\\ f(4)=\sqrt { 10 } \\ g(f(4))=g(\sqrt { 10 } )=\quad 3.16\\ g(f(4))=7.7$$
Question 7
1 / -0

Squaring a given number is a

A
relation in some set

B
relation

C
unary operation

D
binary operation

Solution

Squarring a given number is a Binary operation beacuse squarring two numbers is multiplying the same number twice.
Question 8
1 / -0

If $$*$$ is a binary operation in $$A$$ then

A
$$A$$ is closed under $$*$$

B
$$A$$ is not closed under $$*$$

C
$$A$$ is not closed under $$+$$

D
$$A$$ is closed under $$-$$

Solution

If $$ *$$ is a binary in $$A$$,then A is closed under $$*$$
Question 9
1 / -0

$$+$$ is

A
binary operation on $$R$$

B
not a binary operation on $$R$$

C
a binary operation in $$Q^c$$

D
not a binary operation in $$ E$$
Solution
Addition is a binary operation on R since it satisfies the following properties:
It is closed in R, i.e; sum of two real numbers is also real.
It follows Commutativity and Associativity.
Existence of Identity element 0, which when added to a number gives back the same number.
Existence of Additive Inverse. The negative of a number added to that number gives the identity element 0.
Question 10
1 / -0

$$*$$ is said to be commutative in $$A$$ for all $$a,b\in A$$

A
$$a + b = b + a$$

B
$$a * b = b * a$$

C
$$a - b = b - a$$

D
$$a * b \neq b * a$$

Solution

$$(B) a*b = b*a$$
here $$*$$ is commutative.

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Relations and Functions Test - 31

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Relations and Functions Test - 31

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SHARING IS CARING

Question 1

$$5$$

$$7$$

$$11$$

$$1$$

Solution

Question 2

is taller than

A is taller

B is taller

is less than

Solution

Question 3

180

160

280

200

Solution

Question 4

$$\displaystyle -{ x }^{ 2 }+4x+1$$

$$\displaystyle { x }^{ 2 }+4x-1$$

$$\displaystyle 4{ x }^{ 2 }-7$$

$$\displaystyle 4{ x }^{ 2 }-1$$

$$\displaystyle 16{ x }^{ 2 }+8x-1$$

Solution

Question 5

$$*$$ is a unary operation in $$A$$

$$a * b = b * a$$

$$*$$ is a binary operation in $$A$$

$$a * b \neq b * a$$

Solution

Question 6

$$5.8$$

$$7.4$$

$$7.7$$

$$8.2$$

$$10.3$$

Solution

Question 7

relation in some set

relation

unary operation

binary operation

Solution

Question 8

$$A$$ is closed under $$*$$

$$A$$ is not closed under $$*$$

$$A$$ is not closed under $$+$$

$$A$$ is closed under $$-$$

Solution

Question 9

binary operation on $$R$$

not a binary operation on $$R$$

a binary operation in $$Q^c$$

not a binary operation in $$ E$$

Solution

Question 10

$$a + b = b + a$$

$$a * b = b * a$$

$$a - b = b - a$$

$$a * b \neq b * a$$

Solution

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