Number Theory Test 27

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Number Theory Test 27

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Weekly Quiz Competition

Question 1
1 / -0

In the complex numbers, where $$i = \sqrt {-1}$$, the conjugate of any value $$a + bi$$ is $$a -ib$$. What is the result when you multiply $$2 + 7i$$ by its conjugate?

A
$$45$$

B
$$-45$$

C
$$45i$$

D
$$53$$

E
$$53i$$

Solution

Conjugate of $$ 2 + 7i $$ is $$ 2-7i$$
$$ (2 + 7i) \times (2-7i) = 2^2 - (7i)^2$$
$$ = 4 -49(-1) $$
$$= 4 + 49 $$
$$= 53 $$
Question 2
1 / -0

The imaginary number $$i$$ is defined such that $$i^2=-1$$. What is the value of $$(1 - i \sqrt {5}) ( 1 + i\sqrt {5})$$?

A
$$\sqrt5$$

B
$$5$$

C
$$6$$

D
$$\sqrt6$$

Solution

Imaginary number is $$i=\sqrt {-1}$$

Value of $$(1-i\sqrt{5})(1+i\sqrt{5})$$

It is in the form of $$(a+b)(a-b)=a^2-b^2$$

So, $$(1-i\sqrt{5})(1+i\sqrt{5})={(1)}^2-{(i\sqrt {5})}^2$$

$$\Rightarrow 1-(i^2\times 5)$$

$$\Rightarrow 1-(-1\times 5)$$

$$\Rightarrow 6$$

$$(1-i\sqrt{5})(1+i\sqrt{5})=6$$
Question 3
1 / -0

In the complex numbers, where $$i^{2} = -1$$, what is the value of $$5 + 6i$$ multiplied by $$3- 2i$$?

A
$$27$$

B
$$27i$$

C
$$27 + 8i$$

D
$$15 + 8i$$

E
$$15 - 18i$$

Solution

We know, $$i^2=-1$$
The required product is $$ (5 + 6i)(3-2i) $$
$$= 15-10i+18i+12 $$
$$= 27 + 8i $$
Question 4
1 / -0

$$\left( \frac{1+cos\dfrac{\pi}{8}-i sin \dfrac{\pi}{8}}{1+cos \dfrac{\pi}{8}+i sin \dfrac{\pi}{8}}\right)^8 =$$

A
$$1$$

B
$$-1$$

C
$$2$$

D
$$\frac{1}{2}$$

Solution

$$\left( \frac{1+cos\dfrac{\pi}{8}-i sin \dfrac{\pi}{8}}{1+cos \dfrac{\pi}{8}+i sin \dfrac{\pi}{8}}\right)^8 = -1$$
Question 5
1 / -0

How many of the prime factors of $$30$$ are greater than $$2$$?

A
One

B
Two

C
Three

D
Four

E
Five

Solution

$$30=2\times 3\times 5$$
So, only $$3$$ and $$5$$ are the two prime factors that are greater than $$2$$.
Question 6
1 / -0

If $$i = \sqrt {-1}$$, find the values of $$n$$ such that $$i^{n} + (i)^{n}$$ have a positive value.

A
$$23$$

B
$$24$$

C
$$25$$

D
$$26$$

E
$$27$$

Solution

$${ i }^{ n }+{ i }^{ n }=2{ i }^{ n }$$
We know that $${ i }^{ 2 }=-1$$ and $${ i }^{ 4 }=1$$, so $$n$$ should be multiple of $$4$$.
Among given options $$24$$ will be divisible by $$4$$.
Therefore the answer is $$24$$.
Question 7
1 / -0

Find $$(5 + 2i)(5 - 2i)$$

A
$$25 - 4i$$

B
$$25 - 20i$$

C
$$21$$

D
$$29$$

E
$$0$$

Solution

Since $$i^2=-1$$ and $$(a+b)(a-b)=a^2-b^2$$
$$\Rightarrow (5+2i)(5-2i)$$ $$=(5)^2-(2i)^2$$
$$=25-4i^2$$
$$=25-4(-1)$$
$$=25+4$$
$$=29$$
Question 8
1 / -0

Find the number of prime numbers between 301 and 320?

A
6

B
5

C
4

D
3

Solution

Prime numbers between 301 and 320 are 307, 311, 313, 317. So there are total $$4$$ prime numbers between 301 and 320
So correct answer will be option C
Question 9
1 / -0

Express $$\dfrac {(-5-i)(-7+8i)}{(2-4i)}$$ in the form of a complex number $$a+bi$$.

A
$$\dfrac{109}{10}-\dfrac{53}{10}i$$

B
$$-\dfrac{109}{10}-\dfrac{53}{10}i$$

C
$$\dfrac{109}{10}+\dfrac{53}{10}i$$

D
$$-\dfrac{109}{10}+\dfrac{53}{10}i$$

Solution

We need to express $$\dfrac {(-5-i)(-7+8i)}{(2-4i)}$$ in the form of $$a+bi$$
On rationalising the given expression as follows:
$$\dfrac {(-5-i)(-7+8i)(2+4i)}{(2-4i)(2+4i)}$$
$$=\dfrac {(35-40i+7i-8i^2)(2+4i)}{(4-16i^2)}$$
$$=\dfrac {(43-33i)(2+4i)}{(4+16)}$$
$$=\dfrac {(86+172i-66i-132i^2)}{20}$$
$$=\dfrac {(218+106i)}{20}$$
$$=\dfrac {109}{10}+\dfrac {53}{10}i$$
Question 10
1 / -0

Every composite number has _____________.

A
no prime divisor

B
one and only one prime divisor

C
atleast one prime divisor

D
atleast two prime divisor

Solution

A composite number is that has at least one prime divisor other than $$1$$ and itself. For example : $$10$$ is composite number that has $$2, 5$$ as its prime divisors.

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Re-Attempt Weekly Quiz Competition

Number Theory Test 27

Result

Number Theory Test 27

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SHARING IS CARING

Question 1

$$45$$

$$-45$$

$$45i$$

$$53$$

$$53i$$

Solution

Question 2

$$\sqrt5$$

$$5$$

$$6$$

$$\sqrt6$$

Solution

Question 3

$$27$$

$$27i$$

$$27 + 8i$$

$$15 + 8i$$

$$15 - 18i$$

Solution

Question 4

$$1$$

$$-1$$

$$2$$

$$\frac{1}{2}$$

Solution

Question 5

One

Two

Three

Four

Five

Solution

Question 6

$$23$$

$$24$$

$$25$$

$$26$$

$$27$$

Solution

Question 7

$$25 - 4i$$

$$25 - 20i$$

$$21$$

$$29$$

$$0$$

Solution

Question 8

6

5

4

3

Solution

Question 9

$$\dfrac{109}{10}-\dfrac{53}{10}i$$

$$-\dfrac{109}{10}-\dfrac{53}{10}i$$

$$\dfrac{109}{10}+\dfrac{53}{10}i$$

$$-\dfrac{109}{10}+\dfrac{53}{10}i$$

Solution

Question 10

no prime divisor

one and only one prime divisor

atleast one prime divisor

atleast two prime divisor

Solution

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