MCQ Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 2

The number which is neither prime nor composite is

0%
$0$
0%
$1$
0%
$2$
0%
$5$

Explanation

$1$ is the number which is neither prime nor composite.
Because

$1$ has only

$1$ divisor which is

$1$ only.

Which prime number which comes just after

$43$ ?

0%
$49$
0%
$45$
0%
$47$
0%
none of these

Explanation

Numbers after

$43$ are

$43, 44, 45, 46, 47, 48...$

$44$ is a composite number as it has more than two factors

$1, 2, 11, 22, 44$
Similarly,

$45, 46$ has more than two factors.

However

$47$ has only two factors

$1, 47$ .
Thus the next prime number after

$43$ is

$47.$

Number of prime numbers between

$50$ to

$60$ is

0%
$1$
0%
$2$
0%
$3$
0%
$4$

Every prime number is

$.......$

0%
a even number
0%
an odd number
0%
a composite number
0%
none of these

Explanation

Prime numbers have only two factors

$1$ and the number itself.

For example,

$2, 3, 5, 7, 11..$

From this, we notice that all prime numbers cannot be put to one category except being non-composite.

A prime number

0%
has exactly one factor.
0%
has exactly two factors.
0%
is not divisible by $2$ .
0%
is an odd number.

Explanation

A prime number has only two factors.

For eg.

$11, 17$ , etc..

Each of these numbers is divisible by

$1$ and itself.

Hence, option B is correct.

Which of the following are prime numbers?

0%
$137$
0%
$143$
0%
$396$
0%
Both A and B

Explanation

$143=11\times 13$

$396=2\times198$

Only

$137$ is a prime number.

Number of prime numbers between

$50$ to

$60$ is

0%
$1$
0%
$2$
0%
$3$
0%
$4$

Explanation

The prime numbers between

$50$ and

$60$ are

$53$ and

$59.$
Hence, number of prime numbers between

$50$ and

$60$ is

$2.$

$\displaystyle \frac{\displaystyle i^{4n + 3} + (-i)^{8n - 3}}{\displaystyle(i)^{12 n- 1} - i^{2 - 16 n}}, n \varepsilon N$ is equal to

0%
1 + i
0%
2i
0%
-2i
0%
-1 - i

Explanation

Given expression

$= \displaystyle \frac{\displaystyle i^3 + (-1) (i)^{-3}}{(-i)^{-1} - (i)^2} = \frac{\displaystyle -i - i}{\displaystyle i + 1}$

$=\displaystyle \frac{\displaystyle -2i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{\displaystyle -2i - 2}{\displaystyle 1 + 1} = - 1 - i$

Which is the least prime number ?

0%
$1$
0%
$0$
0%
$2$
0%
$3$

Explanation

The prime numbers are

$2,3,5.......$

So least prime is

$2.$

$i^2 = - 1$ , then the value of

$\displaystyle \sum^{200}_{n = 1} i^n$ is

0%
50
0%
-50
0%
0
0%
100

Explanation

Given,

$i^2=-1$

$\displaystyle \sum_{n = 1}^{200} i^n = i + i^2 + i^3 + ......... + i^{200} = \displaystyle \frac{i (1 - i^{200})}{1 - i}$ (since G. P.)

$= \displaystyle \frac{\displaystyle i (1 - 1)}{\displaystyle 1 - i} = 0$

$z = \displaystyle \frac{(3 + 4i)(5 - 7 i)}{(7 + 5i)(4 - 3i)}$ then

$|z| = ?$

0%
5
0%
4
0%
1
0%
None of these

Explanation

Given,

$z = \displaystyle \frac{(3 + 4i)(5 - 7 i)}{(7 + 5i)(4 - 3i)}$

$\therefore |z| = \displaystyle \frac{|(3 + 4i) ||(5 - 7i)|}{|(7 + 5i)||(4 - 3i)|}$

$= \displaystyle \frac{\sqrt{9 + 16} \sqrt{25 + 49}}{\sqrt{49 + 25} \sqrt{16 + 9}} = 1$

By Goldbach's conjecture, every even number greater than 4 can be expressed as a sum of

0%
Two even numbers
0%
Two odd prime numbers
0%
Two composite numbers
0%
Two co-prime numbers

Explanation

Goldbach's conjecture says that "Every integer greater than

$2$ can be expressed as the sum of

$2$ prime numbers."

How many prime numbers are there between 10 and 30?

0%
$6$
0%
$5$
0%
$7$
0%
$8$

Explanation

${\because}$ The prime numbers between

$10$ and

$30$ are

$11,13,17,19,23,29.$

Which of the following is not a composite number ?

0%
$4$
0%
$6$
0%
$7$
0%
$10$

If 2009 =

$\displaystyle p^{a}.q^{b}$ where p and q are prime numbers then find the value of p + q

0%
3
0%
48
0%
51
0%
2009

Explanation

Given

$2009 = p^a q^b$ ...(i)
Where p and q are prime numbers
We know that

$2009 = 7\times 7\times 41= 7^2\times 41^1$ ...(ii)
Comparing equation (1) and (2) we get

$p=7, q=41, a=2 and b= 1$

$\therefore p+q = 7+41=48$

$i^2 + i^4 + i^6 + ........+ i^{2n}$ is

0%
Positive
0%
Negative
0%
Zero
0%
Cannot be determined

Explanation

$1+i^2+i^4+........+i^{2n}$

$=1-1+1-1+1-1......$

Hence it can b positive or negative depending on the value of

$2n$

Hence It cannot be determined from the given data.

Which of the following is not a composite number?

0%
5
0%
6
0%
7
0%
8

The number of prime numbers between

$0$ and

$20$ is

0%
$7$
0%
$8$
0%
$6$
0%
$9$

Explanation

We know that, a prime number is a number having exactly two factors,

$1$ and the number itself.

Hence, the prime numbers between

$0$ and

$20$ are

$2, 3, 5, 7, 11, 13, 17$ and

$19$

$\therefore \$ The number of prime numbers between

$0$ and

$20$ is

$8.$

If i =

$\sqrt {-1}, then 1 + i^2 + i^3 -i^6 + i^8$ is equal to -

0%
2- i
0%
1
0%
-3
0%
-1

Explanation

Given,

$i= \sqrt {-1}$

$\therefore 1+i^{ 2 }+i^{ 3 }-i^{ 6 }+i^{ 8 }=1-1-i+1+1$

$=2-i$

(

$i^{10}+1) (i^9 + 1)(i^8 +1).......(i+1)$ equal to

0%
-1
0%
1
0%
0
0%
i

Explanation

$(i^{10}+1) (i^9 + 1)(i^8 +1).......(i+1)$

$=(i^{10}+1) (i^9 + 1)(i^8 +1).......(i^3+1)(i^2+1)(i+1)$

$=(i^{10}+1) (i^9 + 1)(i^8 +1).......(i^3+1)(-1+1)(i+1)[\because i^2=-1]$

$=(i^{10}+1) (i^9 + 1)(i^8 +1).......(i^3+1)(0)(i+1) =0$

Twin primes consist of

0%
composite numbers
0%
prime numbers
0%
even numbers
0%
none of these

Explanation

Twin primes are prime numbers which differ by

$2.$

So, options

$B$ is correct.

Which of the following are twin primes?

0%
$(2, 3)$
0%
$(5, 7)$
0%
$(3, 5)$
0%
$(11, 13)$

Explanation

All the numbers given in the options are prime numbers.

For options

$(a), (b), (c),$ the numbers differ by two. Therefore, they are twin prime.

Identify prime numbers among the following options.

0%
$2$
0%
$5$
0%
$6$
0%
$8$

Explanation

$2$ has two factors only, namely

$2$ and

$1.$

$5$ has two factors only, namely

$5$ and

$1.$

So, options

$A$ and

$B$ are correct.

The two consecutive prime numbers with difference 2 are called

0%
co-primes
0%
twin primes
0%
composite
0%
none

Explanation

Twin prime numbers are two consecutive prime numbers with a difference of

$2$ .

For eg.

$\{3,5\}$ and

$\{5,7\}$ are twin prime numbers.

Hence, option B is correct.

The number which have more than two factors is called _____.

0%
Even
0%
Prime
0%
Odd
0%
None of these

The product of

$2 \times 7 \times 3 \times 5$ is a .........

0%
composite number
0%
prime number
0%
negative number
0%
none of the above

Explanation

Composite numbers are numbers which have more then

$2$ factors. They have at least

$1$ factor more than

$1$ and the number itself.

The given expression has more than

$2$ factors. So, it is a Composite number.

Therefore,

$A$ is the correct answer.

When

$(3-2i)$ is subtracted from

$(4 + 7i)$ , then the result is

0%
$1 + 5i$
0%
$1 + 9i$
0%
$7 + 5i$
0%
$7 + 9i$

Explanation

We need to subtract

$(3-2i)$ from

$(4+7i)$

$\therefore 4 + 7i - (3 - 2i)$

$=4 + 7i - 3 + 2i$

$= 1 + 9i$

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.

0%
decimal number
0%
fraction
0%
irrational number
0%
integer

Explanation

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.

$u = 3 - 5i$ and

$v = -6 + i$ , then the value of

$(u+v)^2$ is

0%
$-7 + 24i$
0%
$9 + 8i$
0%
$9 + 16i$
0%
$25 - 24i$

Explanation

$u=3-5i$

$v=-6+i$

$\therefore (u+v)^2=[3-5i+[-6+i]$

$=[-3-4i]^2$

$=[-3]^2+[-4i]^2+2*[-3]*[-4i]$

$=9+16i^2+24i$

$=9+16[-1]+24i$

$=-7+24i$
Option A is correct.

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

0%
composite
0%
prime
0%
natural
0%
irrational

Explanation

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.

Identify the prime number among the following options.

0%
$11$
0%
$15$
0%
$18$
0%
$20$

Explanation

$11$ is a prime number because it has only two factors that are

$1$ and

$11$ .

So, option A is correct.

Addition of prime number

0%
gives a prime number
0%
gives a composite number
0%
may give a prime number or composite number
0%
none of these

Explanation

$2$ 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.

The value of

$-i^{48}$ is

0%
$-i$
0%
$i$
0%
$-1$
0%
$1$

Explanation

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

$-i^{48}=-1$

For any complex number

$z$ , the minimum value of

$\left| z \right| +\left| z-1 \right|$ is

0%
$0$
0%
$1$
0%
$2$
0%
$-1$

Explanation

Since,

$\left| -z \right| =\left| z \right|$
and

$\left| { z }_{ 1 }+{ z }_{ 2 } \right| \le \left| { z }_{ 1 } \right| +\left| { z }_{ 2 } \right|$
Now,

$\ge \left| z+\left( 1-z \right) \right|$

$=1$
Alternative
Now,

$1=\left| z-\left( z-1 \right) \right|$

$\Rightarrow \left| z \right| +\left| z-1 \right| \ge 1$

The composite numbers _________ .

0%
has atleast one factors other than itself and 1
0%
has no factor other than 1
0%
only A
0%
none

Select the right statement.

0%
Zero is not even and not positive
0%
Zero is even and positive
0%
The largest factor of $28$ is $14$
0%
The sum of the smallest prime and the greatest negative even integer is zero
0%
None

Explanation

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.

What is the product of the smallest prime number that is greater than

$50$ and the greatest prime number that is less than

$50$ ?

0%
$2400$
0%
$2491$
0%
$2450$
0%
$2241$

Explanation

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

$'2491'$ .

The simplest form of the expression

$\dfrac {10 - \sqrt {-12}}{1 - \sqrt {-27}}$ is

0%
$-\dfrac {2}{7}$
0%
$\dfrac {28}{3}$
0%
$-\dfrac {2}{7} + i\sqrt {3}$
0%
$1 + i \sqrt {3}$

Explanation

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

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

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

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

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

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

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

$i^2$

$= -1$ , then find the odd one out of the following expressions.

0%
$-i^2$
0%
$(-i)^2$
0%
$i^4$
0%
$(-i)^4$
0%
$-i^6$

Explanation

$i$ is an imagiary number which has a value of

$i=\sqrt{-1}$ . Then

$-i^2=-1*-1=1$

${(-i)}^2=i^2=-1$

$i^4={(i^2)}^2={(-1)}^2=1$

${(-i)}^4={(i^2)}^2={(-1)}^2=1$

$-i^6=-{(i^2)}^3=-1\times {(-1)}^3=-1\times (-1)=1$

All are

$1$ except option

$B$ which is

$-1$ .

Find the number of prime numbers between

$20$ and

$40$ , both inclusive.

0%
Three
0%
Four
0%
Five
0%
Six
0%
Seven

Explanation

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

$40$ .

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

0%
$10$
0%
$0$
0%
$1$
0%
$2$
0%
$3$

Explanation

The least prime number is

$2$ , and the greatest negative even integer is

$-2$ , so the answer is

$0$ .

What is the least positive integer that is the product of

$3$ different prime numbers greater than

$2$ ?

0%
$27$
0%
$45$
0%
$63$
0%
$75$
0%
$105$

Explanation

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

$3 \times 5 \times 7 = 105$ .

How many prime numbers are less than

$50$ ?

0%
$16$
0%
$15$
0%
$14$
0%
$18$

Explanation

Prime numbers less than

$50$ are:

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47$
Their number is

$15$ .

What is the value of

$x$ for which

$x, x + 1, x + 3$ are all prime numbers?

0%
$0$
0%
$1$
0%
$2$
0%
$101$

Explanation

Option

$A :$ Substitute

$x=0,$

$x,x+1,x+3=0,1,3$ are not prime numbers.

Option

$B :$ Substitute

$x=1,$

$x,x+1,x+3=1,2,4$ are not prime numbers.

Option

$C :$ Substitute

$x=2,$

$x, x + 1, x + 3=2,3,5$ which are all prime numbers.

Hence, option

$C$ is correct.

The smallest 3 digit prime number is:

0%
$101$
0%
$103$
0%
$109$
0%
$113$

Explanation

The smallest 3-digit number is

$100$ , which is divisible by

$2$ .

$\therefore$

$100$ is not a prime number.

$\sqrt{100}< 11$ and

$101$ is not divisible by any of the prime numbers

$2,3,5,7,11$ .

$\therefore$

$101$ is a prime number.
Hence

$101$ is the smallest 3-digit prime number.

Which of the following number is a prime?

0%
$667$
0%
$861$
0%
$481$
0%
$331$

Explanation

$667$ is divisible by

$23$

$861$ is divisible by

$3$

$481$ is divisible by

$13$

$331$ is divisible only by

$1$ and

$331$ .

$\therefore 331$ is a prime number.

The complex number

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

0%
$\sin \theta + i\cos \theta$
0%
$\cos \theta + i\sin \theta$
0%
both $(a)$ and $(b)$
0%
none of these

Explanation

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,

$\cos x + i\sin x=e^{ix}$

Which of the following is a prime number?

0%
$33$
0%
$81$
0%
$93$
0%
$97$

Explanation

Clearly,

$97$ is a prime number.

Consider the following numbers.

$247$

$203$
Which of the above number is/are prime?

0%
$1$ only
0%
$2$ only
0%
Both $1$ and $2$
0%
Neither $1$ nor $2$

Explanation

A number which is divisible by itself and

$1$ are prime numbers.

$247 = 13\times 19$
2.

$203 = 7\times 79$

Here both the numbers have some divisors.
So, none of the two are prime.

Which one of the following is a prime number?

0%
$184$
0%
$171$
0%
$173$
0%
$221$

Explanation

$184 = 23\times 8$

Therefore, cannot be prime

$171 = 19\times 9$

Therefore,cannot be prime

$221 = 17\times 13$

Therefore, cannot be prime
Hence, by elimination we have

$173$ as a prime number.

CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 2 - MCQExams.com

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Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

CBSE Questions for Class 11 Commerce Applied Mathematics Number Theory Quiz 2 - MCQExams.com

0:0:2

Practice Class 11 Commerce Applied Mathematics Quiz Questions and Answers

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