Number Theory Test 24

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Number Theory Test 24

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following statement is true?

A
1 is the smallest prime number

B
Every prime number is an odd number

C
The sum of two prime numbers is always a prime number

D
None of these

Solution

$${\because}$$ All the given statements are false.
Question 2
1 / -0

$$i^n + i^{n + 1} + i^{n + 2}+ i^{n + 3} (n \in N) $$ is equal to

A
$$4$$

B
$$1$$

C
$$0$$

D
$$2$$

Solution

$$\textbf{Step-1: Expanding the terms}$$
$${{i}^{n}}+{{i}^{n+1}}+{{i}^{n+2}}+{{i}^{n+3}}$$

$$= {{i}^{n}}+{{i}^{n}}.i+{{i}^{n}}.{{i}^{2}}+{{i}^{n}}.{{i}^{3}}$$
$$\textbf{Step-2: Taking } \mathbf{{i}^{n}} \textbf{ common}$$

$$= {{i}^{n}}(1+i+{{i}^{2}}+{{i}^{3}})$$

$$= {{i}^{n}}(1+i-1-i) $$ $$\mathbf{[\because {{i}^{2}}=-1,{{i}^{3}}=-i]}$$

$$= {{i}^{n}}(0)=0$$
$$\textbf{Hence the correct option is (C) 0}$$
Question 3
1 / -0

The number $$10$$ has four factors: $$1,2, 5$$ and $$10$$. The table below lists the number of factors for some numbers
Numbers Number of factors
$$21$$ $$4$$
$$23$$ $$2$$
$$25$$ $$3$$
$$27$$ $$4$$
$$29$$ $$2$$
From this, we can say that the number of prime numbers between $$20$$ and $$30$$ is:

A
$$0$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution

We know that prime numbers have two factors that is $$1$$ and number itself.
In the given table only $$23$$ and $$29$$ have $$2$$ factors so there will be two prime numbers that is $$23$$ and $$29$$ so correct answer will be option B
Question 4
1 / -0

The modulus of (1 + i) (1 + 2i) (1 + 3i) is equal to

A
$$\sqrt10$$

B
$$\sqrt 5$$

C
5

D
10

Solution

$$|(1 + i) (1 + 2i) (1 + 3i)|$$ = $$|1+i||1+2i||1+3i|$$ ($$\because |Z_{1}Z_{2}|=|Z_{1}||Z_{2}|$$)
$$=\sqrt {2}\cdot\sqrt {5}\cdot\sqrt {10}$$
$$\therefore |(1 + i) (1 + 2i) (1 + 3i)|=10$$
Hence, option D.
Question 5
1 / -0

If $$z_1, z_2, \varepsilon C$$ are such that $$|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$$ then $$\displaystyle \frac{z_1}{z_2}$$ is

A
zero

B
purely real

C
purely imaginary

D
complex

Solution

$$\because |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2$$

$$\Rightarrow |z_1|^2 + |z_2|^2 + 2 R (z_1 \bar z_2) = |z_1|^2 + |z_2|^2$$

$$\Rightarrow R(z_1 \bar z_2) = 0 \Rightarrow z_1 \bar z_2$$ is imaginary

Now $$\displaystyle \frac{z_1}{z_2} = \frac{z_1 \bar z_2}{z_2 \bar z_2} = \frac{\text{imaginary}}{|z_2|^2} = $$ imaginary
Question 6
1 / -0

The value of $$(x - 1) \displaystyle \left ( x + \frac{1}{2} - \frac{\sqrt{3}}{2} i \right )\left ( x + \frac{1}{2} + \frac{\sqrt 3}{2} i \right )$$ is

A
$$x^3 + x^2 + x 1$$

B
$$x^3 -1$$

C
$$x^3 + 1$$

D
$$x^3 - x^2 + x + 1$$

Solution

$$\left( x-1 \right) \left( x+\dfrac { 1 }{ 2 } -\dfrac { \sqrt { 3 } }{ 2 } i \right) \left( x+\dfrac { 1 }{ 2 } +\dfrac { \sqrt { 3 } }{ 2 } i \right) $$
$$=\left( x-1 \right) \left[ \left( x+\dfrac { 1 }{ 2 } \right) -\left( \dfrac { \sqrt { 3 } }{ 2 } i \right) \right] \left[ \left( x+\dfrac { 1 }{ 2 } \right) +\left( \dfrac { \sqrt { 3 } }{ 2 } i \right) \right] $$
$$=\left( x-1 \right) \left[ \left( x+\dfrac { 1 }{ 2 } \right) ^{ 2 }-\left( \dfrac { \sqrt { 3 } }{ 2 } i \right) ^{ 2 } \right] $$
$$=\left( x-1 \right) \left[ { x }^{ 2 }+x+\dfrac { 1 }{ 4 } +\dfrac { 3 }{ 4 } \right] $$
$$=\left( x-1 \right) \left[ { x }^{ 2 }+x+1 \right] $$
$$=x^3+x^2+x-x^2-x-1$$
$$=x^3-1$$
Hence, the answer is $$x^3-1.$$
Question 7
1 / -0

If z = x + iy and |z 1 + 2i | = | z + 1 2i |,then the locus of z is

A
circle

B
parabola

C
straight line

D
None of these

Solution

We know that $$z=x+iy$$
$$\Rightarrow \left| z-1+2i \right| =\left| z+1-2i \right| ,$$ locus of $$x=?$$
$$\Rightarrow \left| x+iy-1+2i \right| =\left| x+iy+1-2i \right| $$
$$\Rightarrow \left| \left( x-1 \right) +i\left( y+2 \right) \right| =\left| \left( x+1 \right) +i\left( y-2 \right) \right| $$
$$\Rightarrow \sqrt { { \left( x-1 \right) }^{ 2 }+{ \left( y+2 \right) }^{ 2 } } =\sqrt { { \left( x+1 \right) }^{ 2 }+{ \left( y-2 \right) }^{ 2 } } $$
$$\Rightarrow x^2-2x+1+y^2+4y+4=x^2+2x+1+y^2-4y+4$$
$$\Rightarrow -2x+4y+5=2x-4y+5$$
$$\Rightarrow 4x=8y$$
$$\Rightarrow x=2y$$
$$\therefore x-2y=0$$
$$\therefore$$ The locus of $$z$$ is a straight line.
Hence, the answer straight line.
Question 8
1 / -0

Modulus of $$\displaystyle \frac{cos \theta- isin\theta }{sin\theta - icos\theta} is$$

A
0

B
2$$\theta$$

C
$$\pi - 2\theta$$

D
none of these

Solution

$$\displaystyle |\frac{\cos \theta- i\sin\theta }{\sin\theta - i\cos\theta}|$$ $$=\displaystyle \frac{|\cos \theta- i\sin\theta| }{|\sin\theta - i\cos\theta|}$$
$$= \displaystyle \frac{\sin^2 \theta +\cos^2 \theta}{\sin^2 \theta +\cos^2 \theta}=1$$
$$\therefore \displaystyle |\frac{\cos \theta- i\sin\theta }{\sin\theta - i\cos\theta}|=1$$
Hence, option D.
Question 9
1 / -0

If $$z_1$$ and $$z_2$$ are any two complex numbers, then $$\displaystyle \frac{z_2 + z_1}{||z_2| - |z_1||}$$ is

A
$$\leq 1$$

B
$$\geq 1$$

C
$$\geq -1$$

D
none of these

Solution
Question 10
1 / -0

Number of complex numbers $$z$$ satisfying $$\left| 2z \right| =\left| 2z-1 \right| =\left| 2z+1 \right| $$ is equal to-

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution

Let the complex number $$z$$ be $$x+iy$$
Given that $$|2z|=|2z-1|$$
$$\Rightarrow 2\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =\sqrt { { (2x-1) }^{ 2 }+{ y }^{ 2 } } $$
$$\Rightarrow 3{y}^{2}+4x=1$$
Given that $$|2z|=|2z+1|$$
$$\Rightarrow 2\sqrt { { x }^{ 2 }+{ y }^{ 2 } } =\sqrt { { (2x+1) }^{ 2 }+{ y }^{ 2 } } $$
$$\Rightarrow 3{y}^{2}-4x=1$$
By solving above two equation , we get $$x=0 , y=\pm\frac{1}{\sqrt3}$$
But this point doesnt satisfy the equation $$|2z-1|=|2z+1|$$
Therefore the number of points which satisfies the equation $$|2z|=|2z+1|=|2z-1|$$ is $$0$$

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Numbers	Number of factors
$$21$$	$$4$$
$$23$$	$$2$$
$$25$$	$$3$$
$$27$$	$$4$$
$$29$$	$$2$$

Number Theory Test 24

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Number Theory Test 24

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SHARING IS CARING

Question 1

1 is the smallest prime number

Every prime number is an odd number

The sum of two prime numbers is always a prime number

None of these

Solution

Question 2

$$4$$

$$1$$

$$0$$

$$2$$

Solution

Question 3

$$0$$

$$2$$

$$3$$

$$4$$

Solution

Question 4

$$\sqrt10$$

$$\sqrt 5$$

5

10

Solution

Question 5

zero

purely real

purely imaginary

complex

Solution

Question 6

$$x^3 + x^2 + x 1$$

$$x^3 -1$$

$$x^3 + 1$$

$$x^3 - x^2 + x + 1$$

Solution

Question 7

circle

parabola

straight line

None of these

Solution

Question 8

0

2$$\theta$$

$$\pi - 2\theta$$

none of these

Solution

Question 9

$$\leq 1$$

$$\geq 1$$

$$\geq -1$$

none of these

Solution

Question 10

$$0$$

$$1$$

$$2$$

$$3$$

Solution

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