Number Theory Test 9

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Number Theory Test 9

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Weekly Quiz Competition

Question 1
1 / -0

If $$i^2$$ $$= -1$$, then find the odd one out of the following expressions.

A
$$-i^2$$

B
$$(-i)^2$$

C
$$i^4$$

D
$$(-i)^4$$

E
$$-i^6$$

Solution

$$i$$ is an imagiary number which has a value of $$i=\sqrt{-1}$$. Then
A. $$-i^2=-1*-1=1$$
B. $${(-i)}^2=i^2=-1$$
C. $$i^4={(i^2)}^2={(-1)}^2=1$$
D. $${(-i)}^4={(i^2)}^2={(-1)}^2=1$$
E. $$-i^6=-{(i^2)}^3=-1\times {(-1)}^3=-1\times (-1)=1$$

All are $$1$$ except option $$B$$ which is $$-1$$.
Question 2
1 / -0

Which of the following is a prime number?

A
$$33$$

B
$$81$$

C
$$93$$

D
$$97$$

Solution

Clearly, $$97$$ is a prime number.
Question 3
1 / -0

What is the sum of the least prime number and the greatest negative even integer?

A
$$10$$

B
$$0$$

C
$$1$$

D
$$2$$

E
$$3$$

Solution

The least prime number is $$2$$, and the greatest negative even integer is $$-2$$, so the answer is $$0$$.
Question 4
1 / -0

What is the sum of the smallest prime and the largest prime less than $$10$$?

A
$$7$$

B
$$9$$

C
$$10$$

D
$$11$$

E
$$12$$

Solution

Smallest prime number less than $$10$$ is $$2$$ and largest prime number less than $$10$$ is $$7$$ and their sum is $$2+7=9$$
Question 5
1 / -0

For $$i=\sqrt{-1}$$, what is the sum $$\left(7+3i\right) + \left(-8+9i\right)$$?

A
$$-1+12i$$

B
$$-1-6i$$

C
$$15+12i$$

D
$$15-6i$$

Solution

Given: $$(7+3i)+(-8+9i)$$
$$= (7+(-8))+(3i+9i)$$
$$=(7-8)+12i$$
$$=-1+12i$$
Option A is correct.
Question 6
1 / -0

Perform the indicated operations:
$$(8-2i)-(-2-6i)$$

A
$$6+4i$$

B
$$10+4i$$

C
$$10+8i$$

D
$$10-8i$$

Solution

The value of $$(8-2i)-(-2-6i) $$
$$= 8+2+i(6-2) $$
$$= 10+4i$$
Question 7
1 / -0

What is the least positive integer that is the product of $$3$$ different prime numbers greater than $$2$$?

A
$$27$$

B
$$45$$

C
$$63$$

D
$$75$$

E
$$105$$

Solution

Three different prime numbers which are greater than $$2$$ are $$3,5,7$$.
$$\therefore 3,5,7$$ are three least prime numbers above $$2$$.
Therefore, the product of all these is $$3 \times 5 \times 7 = 105$$.
Question 8
1 / -0

Which one of the following is not a prime number?

A
$$31$$

B
$$61$$

C
$$71$$

D
$$91$$

Solution

$$91$$ is divisible by $$7$$ and $$13$$. So, it is not a prime number.
Rest all the given numbers are prime numbers since they don't have any factors other than $$1$$ and themselves.
Question 9
1 / -0

The complex number $$e^{i\theta }$$ can be expressed in vector form by

A
$$\sin \theta + i\cos \theta $$

B
$$\cos \theta + i\sin \theta $$

C
both $$(a)$$ and $$(b)$$

D
none of these

Solution

Let $$f(x)=\cos x + i \sin x$$
Taking derivative we get,
$$f'(x)=-\sin x + i\cos x=i\times f(x)$$
Any other function $$g(x)$$ has this property if
$$g'(x)=ig(x)$$ i.e.
$$\dfrac{dg}{dx}=i g$$
$$\implies\dfrac{1}{g} dg=i\ dx$$
Taking integration on both sides we get,
$$log(g)=ix+C$$
$$\implies g=e^{(ix+C)}$$
$$\implies g=e^{ix} e^C$$
$$\implies g(x)=e^{ix}c_{1}$$
Put $$x=0$$ we get $$c_{1}=g(0)$$
This function is $$1$$ when $$c_{1}=1$$
Hence, $$\cos x + i\sin x=e^{ix}$$
Question 10
1 / -0

How many prime numbers are less than $$50$$?

A
$$16$$

B
$$15$$

C
$$14$$

D
$$18$$

Solution

Prime numbers less than $$50$$ are:
$$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$$
Their number is $$15$$.

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Re-Attempt Weekly Quiz Competition

Number Theory Test 9

Result

Number Theory Test 9

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SHARING IS CARING

Question 1

$$-i^2$$

$$(-i)^2$$

$$i^4$$

$$(-i)^4$$

$$-i^6$$

Solution

Question 2

$$33$$

$$81$$

$$93$$

$$97$$

Solution

Question 3

$$10$$

$$0$$

$$1$$

$$2$$

$$3$$

Solution

Question 4

$$7$$

$$9$$

$$10$$

$$11$$

$$12$$

Solution

Question 5

$$-1+12i$$

$$-1-6i$$

$$15+12i$$

$$15-6i$$

Solution

Question 6

$$6+4i$$

$$10+4i$$

$$10+8i$$

$$10-8i$$

Solution

Question 7

$$27$$

$$45$$

$$63$$

$$75$$

$$105$$

Solution

Question 8

$$31$$

$$61$$

$$71$$

$$91$$

Solution

Question 9

$$\sin \theta + i\cos \theta $$

$$\cos \theta + i\sin \theta $$

both $$(a)$$ and $$(b)$$

none of these

Solution

Question 10

$$16$$

$$15$$

$$14$$

$$18$$

Solution

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