Number Systems Test

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Number Systems Test - 34

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Weekly Quiz Competition

Question 1
1 / -0

Simplify the following using law of exponents for $$a=2, x=1,y=1,z=1$$
$$a^x\times a^y\times a^z$$

A
$$2$$

B
$$4$$

C
$$8$$

D
$$16$$

Solution

we know,

$$a^{x}*a^{y}*a^{z}=a^{x+y+z}$$

$$\therefore2^{1+1+1}$$

$$=2^{3}$$

$$=8$$
Question 2
1 / -0

Simplify the following using law of exponents.
$$(-3)^3\times (-3)^{10}\times (-3)^7$$

A
$$3^{20}$$

B
$$3^{40}$$

C
$$3^{60}$$

D
$$3^{10}$$

Solution

we know,

$$a^{m}*a^{n}*a^{p}=a^{m+n+p}$$

so,

$$(-3)^{3}*(-3)^{10}*(-3)^{7}=(-3)^{3+10+7}$$

$$=(-3)^{20}$$

$$=(3)^{20}$$
Question 3
1 / -0

What is the conjugate of the binomial surd $$5 + \sqrt {3}$$

A
$$5 - \sqrt {3}$$

B
$$3 + \sqrt {5}$$

C
$$\sqrt{5} + \sqrt {3}$$

D
$$\sqrt{5} - \sqrt {3}$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
The conjugate of $$5+\sqrt 3$$ is $$5-\sqrt 3$$.
Option A is correct.
Question 4
1 / -0

The conjugate of $$3 + \sqrt 5$$ is?

A
$$3 - \sqrt 5$$

B
$$\sqrt 3 + 5$$

C
$$\sqrt 3 +\sqrt 5$$

D
$$\sqrt 3 - 5$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
The conjugate of $$3+\sqrt 5$$ is $$3-\sqrt 5$$.
Option A is correct.
Question 5
1 / -0

The conjugate of $$\sqrt x - \sqrt y$$ is?

A
$$\sqrt x +y$$

B
$$\sqrt x + \sqrt y$$

C
$$x + \sqrt y$$

D
$$x - y$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
Thus the conjugate of $$\sqrt x-\sqrt y$$ is $$\sqrt x+\sqrt y$$.
Option B is correct.
Question 6
1 / -0

The product of $$\sqrt x + 2\sqrt y$$ with its conjugate is?

A
$$x + 4y$$

B
$$x - 4y$$

C
$$x+2y$$

D
$$x-2y$$

Solution

The sum and difference of two simple quadratic surds are said to be conjugate of each other.
The conjuate of $$\sqrt x+2\sqrt y$$ is $$\sqrt x-2\sqrt y$$
Use the algebric properties, $$(a-b)(a+b)=a^2-b^2$$
Product of $$\sqrt x+2\sqrt y$$ with $$\sqrt x-2\sqrt y$$ is equal to $$(\sqrt x)^2-(\sqrt {2y})^2=x-4y$$
Question 7
1 / -0

$$\dfrac {1}{2} + \sqrt {2}$$ is a conjugate of?

A
$$\dfrac {1}{2} + {2}$$

B
$${2} + \sqrt {2}$$

C
$$\dfrac {1}{2} - \sqrt {2}$$

D
$$\dfrac {1}{\sqrt2} + \sqrt {2}$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
The conjugate of $$\dfrac{1}{2}+\sqrt 2$$ is $$\dfrac{1}{2}-\sqrt 2$$.
Option C is correct.
Question 8
1 / -0

$$3\sqrt {7} + 7\sqrt {3}$$ is a conjugate of?

A
$$7 + 3\sqrt {7}$$

B
$$7\sqrt {3} - 3\sqrt {7}$$

C
$$7\sqrt {3} + 3\sqrt {7}$$

D
$$3\sqrt {7} - 7\sqrt {3}$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
The conjugate of $$3\sqrt 7+7\sqrt 3$$ is $$3\sqrt 7-7\sqrt 3$$.
Question 9
1 / -0

The conjugate of the binomial surd $$10\sqrt {2} + 3\sqrt {5}$$ is

A
$$10\sqrt {5} -3\sqrt {2}$$

B
$$10\sqrt {5} + 3\sqrt {2}$$

C
$$10\sqrt {2} - 3\sqrt {5}$$

D
$$-10\sqrt {5} + 3\sqrt {2}$$

Solution

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.
The conjugate of $$10\sqrt 2+3\sqrt 5$$ is $$10\sqrt 2-3\sqrt 5$$.
Question 10
1 / -0

What is the rationalizing factor of $$\dfrac {1}{\sqrt2 - 1}$$?

A
$$1-\sqrt2$$

B
$$2 + \sqrt2$$

C
$$\sqrt2+1$$

D
Both $$A$$ and $$B$$

Solution

When the denominator of an expression is a surd which can be reduced to an expression with the rational denominator, this process is called as rationalizing. When the product of two surds is a rational number, then each surd of the two surds is called as rationalizing factors of each other.

$$\dfrac{\sqrt a+\sqrt b}{(\sqrt a-\sqrt b)(\sqrt a+\sqrt b)}=\dfrac{\sqrt a+\sqrt b}{(\sqrt a)^2-(\sqrt b)^2}=\dfrac{\sqrt a+\sqrt b}{a-b}$$
So, the rationalizing factor will be $$\dfrac{\sqrt 2+1}{(\sqrt 2-1)(\sqrt 2+1)}=\dfrac{\sqrt 2+1}{1}$$

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Re-Attempt Weekly Quiz Competition

Number Systems Test - 34

Result

Number Systems Test - 34

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SHARING IS CARING

Question 1

$$2$$

$$4$$

$$8$$

$$16$$

Solution

Question 2

$$3^{20}$$

$$3^{40}$$

$$3^{60}$$

$$3^{10}$$

Solution

Question 3

$$5 - \sqrt {3}$$

$$3 + \sqrt {5}$$

$$\sqrt{5} + \sqrt {3}$$

$$\sqrt{5} - \sqrt {3}$$

Solution

Question 4

$$3 - \sqrt 5$$

$$\sqrt 3 + 5$$

$$\sqrt 3 +\sqrt 5$$

$$\sqrt 3 - 5$$

Solution

Question 5

$$\sqrt x +y$$

$$\sqrt x + \sqrt y$$

$$x + \sqrt y$$

$$x - y$$

Solution

Question 6

$$x + 4y$$

$$x - 4y$$

$$x+2y$$

$$x-2y$$

Solution

Question 7

$$\dfrac {1}{2} + {2}$$

$${2} + \sqrt {2}$$

$$\dfrac {1}{2} - \sqrt {2}$$

$$\dfrac {1}{\sqrt2} + \sqrt {2}$$

Solution

Question 8

$$7 + 3\sqrt {7}$$

$$7\sqrt {3} - 3\sqrt {7}$$

$$7\sqrt {3} + 3\sqrt {7}$$

$$3\sqrt {7} - 7\sqrt {3}$$

Solution

Question 9

$$10\sqrt {5} -3\sqrt {2}$$

$$10\sqrt {5} + 3\sqrt {2}$$

$$10\sqrt {2} - 3\sqrt {5}$$

$$-10\sqrt {5} + 3\sqrt {2}$$

Solution

Question 10

$$1-\sqrt2$$

$$2 + \sqrt2$$

$$\sqrt2+1$$

Both $$A$$ and $$B$$

Solution

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