Number Theory Test 8

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Number Theory Test 8

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Weekly Quiz Competition

Question 1
1 / -0

The simplest form of the expression $$\dfrac {10 - \sqrt {-12}}{1 - \sqrt {-27}} $$ is

A
$$-\dfrac {2}{7}$$

B
$$\dfrac {28}{3}$$

C
$$-\dfrac {2}{7} + i\sqrt {3}$$

D
$$1 + i \sqrt {3}$$

Solution

We need to find simplest form of $$\dfrac{10-\sqrt{-12}}{1-\sqrt{-27}}$$

$$=\dfrac{10-2i\sqrt{3}}{1-3i\sqrt{3}}$$

Taking conjugate, we get

$$=$$ $$\dfrac{10-2\sqrt{-3}}{1-3\sqrt{-3}}\times \dfrac{1+3i\sqrt{3}}{1+3i\sqrt{3}}$$

$$=$$ $$\dfrac{\frac{10-2\sqrt{-3}}{1-3\sqrt{-3}}}{1^2-(3i\sqrt{3})^2}$$

$$=$$ $$\dfrac{10+30i\sqrt{3}-2i\sqrt{3}-18i^2}{28}$$

$$=$$ $$\dfrac{10+28i\sqrt{3}+18}{28}$$

$$=$$ $$\dfrac{28(1+i\sqrt{3})}{28}$$

$$=$$ $$1+i\sqrt{3}$$
Question 2
1 / -0

When $$(3-2i)$$ is subtracted from $$(4 + 7i)$$, then the result is

A
$$1 + 5i$$

B
$$1 + 9i$$

C
$$7 + 5i$$

D
$$7 + 9i$$

Solution

We need to subtract $$(3-2i)$$ from $$(4+7i)$$
$$\therefore 4 + 7i - (3 - 2i)$$
$$=4 + 7i - 3 + 2i $$
$$= 1 + 9i$$
Question 3
1 / -0

Fundamental theorem of arithmetic is when any .......... greater than $$1$$ is either a prime number or can be written as a unique product of prime numbers.

A
decimal number

B
fraction

C
irrational number

D
integer

Solution

Let us take an example, $$15$$ is not a prime number.
$$15 = 3 \times 5$$, where $$3$$ and $$5$$ are prime numbers.
We cannot get the number $$15$$ by multiplying any other prime numbers.
$$15 \neq 2 \times 5 \times 7.$$
It is only with one particular set of prime numbers.
Hence, it is said integers greater than $$1$$ are prime numbers or unique product of prime numbers.
Therefore, $$D$$ is the correct answer.
Question 4
1 / -0

Find the number of prime numbers between $$20$$ and $$40$$, both inclusive.

A
Three

B
Four

C
Five

D
Six

E
Seven

Solution

A number that is divisible by itself or $$1$$ is called prime number.
Prime number between $$20$$ and $$40$$ are $$23,29,31 ,37$$.
Then there are $$4$$ prime number between $$20$$ and $$40$$.
Question 5
1 / -0

If $$u = 3 - 5i$$ and $$v = -6 + i$$, then the value of $$(u+v)^2$$ is

A
$$-7 + 24i$$

B
$$9 + 8i$$

C
$$9 + 16i$$

D
$$25 - 24i$$

Solution

$$u=3-5i$$
$$v=-6+i$$
$$\therefore (u+v)^2=[3-5i+[-6+i]$$
                      $$=[-3-4i]^2$$
                      $$=[-3]^2+[-4i]^2+2*[-3]*[-4i]$$
                      $$=9+16i^2+24i$$
                      $$=9+16[-1]+24i$$
                      $$=-7+24i$$
Option A is correct.
Question 6
1 / -0

Every ........ number can be expressed as a product of prime numbers.

A
composite

B
prime

C
natural

D
irrational

Solution

For example, $$6 = 2\times 3$$, where $$2$$ and $$3$$ are prime numbers.
When we multiply two or more prime numbers we get the product as a composite number.
Therefore, $$A$$ is the correct answer.
Question 7
1 / -0

Addition of prime number

A
gives a prime number

B
gives a composite number

C
may give a prime number or composite number

D
none of these

Solution

$$2$$ and $$3$$ are prime numbers
$$7$$ and $$5$$ are prime numbers
$$2 + 3 = 5$$ (prime) but $$7 + 5 = 12$$ (composite).
So, option C is correct.
Question 8
1 / -0

The value of $$-i^{48}$$ is

A
$$-i$$

B
$$i$$

C
$$-1$$

D
$$1$$

Solution

We need to find value of $$-i^{48}$$
The power rise to the $$i$$ is even, then the value of $$i$$ is $$-1$$.
The power rise to the $$i$$ is odd, then the value of $$i$$ is $$-i$$.
So, the value of $$-i^{48}=-1$$
Question 9
1 / -0

Select the right statement.

A
Zero is not even and not positive

B
Zero is even and positive

C
The largest factor of $$28$$ is $$14$$

D
The sum of the smallest prime and the greatest negative even integer is zero

E
None

Solution

We know that the smallest prime is $$2$$ and the greatest negative even integer is $$-2$$, so answer D is correct$$.(2-2=0)$$

For A & B. Zero is an even integer, since it can be divided by two without any remainder. Also, zero is neither positive nor negative: it is the integer that divides the number line into negative on the left and positive on the right.

For C. The largest factor of $$28$$ is $$28$$ itself.
Question 10
1 / -0

What is the product of the smallest prime number that is greater than $$50$$ and the greatest prime number that is less than $$50$$ ?

A
$$2400$$

B
$$2491$$

C
$$2450$$

D
$$2241$$

Solution

We know that:
Smallest prime number greater than $$50$$ is $$'53'$$
Greatest prime number less than $$50$$ is $$'47'$$
Product of the above two numbers $$=$$ $$53$$ $$\times$$ $$47$$ $$=$$ $$2491$$
Therefore, product of smallest prime number greater than $$'50'$$ and the greatest prime number less than $$50$$ is $$'2491'$$.

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Number Theory Test 8

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Number Theory Test 8

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SHARING IS CARING

Question 1

$$-\dfrac {2}{7}$$

$$\dfrac {28}{3}$$

$$-\dfrac {2}{7} + i\sqrt {3}$$

$$1 + i \sqrt {3}$$

Solution

Question 2

$$1 + 5i$$

$$1 + 9i$$

$$7 + 5i$$

$$7 + 9i$$

Solution

Question 3

decimal number

fraction

irrational number

integer

Solution

Question 4

Three

Four

Five

Six

Seven

Solution

Question 5

$$-7 + 24i$$

$$9 + 8i$$

$$9 + 16i$$

$$25 - 24i$$

Solution

Question 6

composite

prime

natural

irrational

Solution

Question 7

gives a prime number

gives a composite number

may give a prime number or composite number

none of these

Solution

Question 8

$$-i$$

$$i$$

$$-1$$

$$1$$

Solution

Question 9

Zero is not even and not positive

Zero is even and positive

The largest factor of $$28$$ is $$14$$

The sum of the smallest prime and the greatest negative even integer is zero

None

Solution

Question 10

$$2400$$

$$2491$$

$$2450$$

$$2241$$

Solution

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