Number Theory Test 45

Result

Number Theory Test 45

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Weekly Quiz Competition

Question 1
1 / -0

Mark the correct alternative of the following.
Which of the following are not twin-primes?

A
$$3, 5$$

B
$$5, 7$$

C
$$11, 13$$

D
$$17, 23$$

Solution
Question 2
1 / -0

Mark the correct alternative of the following.
Which of the following numbers are twin primes?

A
$$3, 5$$

B
$$5, 11$$

C
$$3, 11$$

D
$$13, 17$$

Solution
Question 3
1 / -0

The total number of even prime numbers is?

A
$$0$$

B
$$1$$

C
$$2$$

D
Unlimited

Solution

The number $$2$$ is the only prime number, which is even. So, the number of even primes is equal to $$1$$.
Question 4
1 / -0

Mark the correct alternative of the following.
Which of the following is a prime number?

A
$$263$$

B
$$361$$

C
$$323$$

D
$$324$$

Solution

The option (a) is correct answer.
We know that 263 = 1 x 263 = 263 has two factors, 1 and 263
Therefore, it is a prime number
Question 5
1 / -0

Mark the correct alternative of the following.
The smallest number which is neither prime nor composite is?

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

All primes have two positive divisors. There are only two integers that are neither composite or prime and they are 1 and 0.

The number 1 has only 1 positive divisor, itself. The number 0 has an infinite number of divisors.

1 is neither prime nor composite.
Question 6
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \left ( \dfrac{1}{1-4i}-\dfrac{2}{1+i} \right )\left ( \dfrac{3-4i}{5+i} \right )$$

A
$$ \dfrac{307}{442}+i \dfrac{599}{442}i$$

B
$$ \dfrac{307}{442}-i \dfrac{599}{442}i$$

C
$$ \dfrac{-307}{442}+i \dfrac{599}{442}i$$

D
None of the above

Solution
Question 7
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{\left ( 2+i \right )^{3}}{2+3i}$$

A
$$ \dfrac{37}{13}-\dfrac{16}{13}i$$

B
$$ \dfrac{-37}{13}+\dfrac{16}{13}i$$

C
$$ \dfrac{37}{13}+\dfrac{16}{13}i$$

D
None of the above

Solution
Question 8
1 / -0

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form:
$$1-i$$

A
$$\sqrt{2}(cos\,\pi /4+i\, sin\, \pi /4)$$

B
$$\sqrt{2}(cos\,\pi /3-i\, sin\, \pi /3)$$

C
$$\sqrt{2}(cos\,\pi /4-i\, sin\, \pi /4)$$

D
$$\sqrt{2}(cos\,\pi /3+i\, sin\, \pi /3)$$

Solution

Let $$z=1-i$$. Then, $$\left | z \right |=\sqrt{1^{2}+(-1)^{2}}=\sqrt{2}$$.

Let $$\alpha $$ be the acute angle given by $$tan\, \alpha =\dfrac{\left | Im(z) \right |}{\left | Re(z) \right |}$$.

Then,
$$tan\, \alpha =\dfrac{|-1|}{|1|}=1\Rightarrow \alpha =\dfrac{\pi }{4}$$

Clearly, z lies in the fourth quadrant. Therefore, $$arg(z)= -\alpha =-\dfrac{\pi }{4}$$.
Question 9
1 / -0

Express the following complex numbers in the standard from $$ a+ib$$ :
$$ \dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}$$

A
$$ 1-2\sqrt{2}i$$

B
$$ 1+\sqrt{2}i$$

C
$$ 1+2\sqrt{2}i$$

D
$$ 1-\sqrt{2}i$$

Solution

$$ \dfrac{5+\sqrt{2}i}{1-\sqrt{2}i}$$

Rationalizing the denominator, we get

$$ = \dfrac{5+\sqrt{2}i}{1-\sqrt{2}i} \times \dfrac{1+\sqrt{2}i}{1+\sqrt{2}i}$$

$$ = \dfrac{5+2i^2+(5+1)\sqrt{2}i}{1+2}$$

$$ = \dfrac{3+6\sqrt{2}i}{1+2}$$......................$$(\because i^2=-1)$$

$$ = 1+2\sqrt{2}i$$
Question 10
1 / -0

Express the following complex numbers in the standard form $$ a+ib$$ :
$$ \dfrac{3-4i}{\left ( 4-2i \right )\left ( 1+i \right )}$$

A
$$ \dfrac{1}{4}+\dfrac{3}{4}i$$

B
$$ \dfrac{1}{4}-\dfrac{3}{4}i$$

C
$$ \dfrac{-1}{4}-\dfrac{3}{4}i$$

D
None of the above

Solution