Number Theory Test 23

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Number Theory Test 23

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Weekly Quiz Competition

Question 1
1 / -0

Find the modulus and amplitude of $$-2 + 2 \sqrt 3i$$

A
$$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {\pi}{3}$$

B
$$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {2\pi}{3}$$

C
$$|z|=4; amp(z)=\dfrac {2\pi}{3}$$

D
$$|z|=4; amp(z)=\dfrac {\pi}{3}$$

Solution

Given $$z=-2+2\sqrt{3}i$$
Hence, $$|z|=2\sqrt{1+3}\ \Rightarrow 2\sqrt{4}$$
$$\Rightarrow 2\times2=4$$
And let $$\alpha$$ be the argument of the complex number.
Then
$$tan\alpha=-\sqrt{3}$$
Now $$Re(z)<0$$ and $$Im(z)>0$$
$$\therefore $$ the number lies in the II quadrant
Hence,
$$\alpha=\pi-\dfrac{\pi}{3}$$

$$=\dfrac{2\pi}{3}$$
Question 2
1 / -0

The inequality $$|z-4| < |z-2|$$ represents the region given by

A
$$Re(z) > 1$$

B
$$Re(z) < 2$$

C
$$Re(z) > 0$$

D
None of these

Solution

Let, $$z=x+iy$$

Putting in the given inequality,

$$|(x-4)-iy|<|(x-2)+iy|$$

$$\Rightarrow \sqrt{(x-4)^2+y^2}<\sqrt{(x-2)^2+y^2}$$

squaring both sides,

$$\Rightarrow (x-4)^2+y^2<(x-2)^2+y^2$$

$$\Rightarrow -8x+16<-4x+4$$

$$\Rightarrow 4x>12\Rightarrow x>3\Rightarrow \text{Re(z)}>3$$
Question 3
1 / -0

Find the modulus and amplitude of $$-\sqrt 3-i$$

A
$$|z|=2; amp(z)=-\dfrac {5\pi}{6}$$

B
$$|z|=4; amp(z)=\dfrac {5\pi}{6}$$

C
$$|z|=4; amp(z)=-\dfrac {\pi}{6}$$

D
none of these

Solution

$$z=-\sqrt{3}-i$$ $$|z|=2$$

$$=2(\dfrac{-\sqrt{3}-i}{2})$$

$$=2e^{i(\frac{\pi}{6}-\pi)}$$

$$=2e^{i\frac{-5\pi}{6}}$$

$$=|z|.e^{iarg(z)}$$

Hence

$$|z|=2$$ and $$arg(z)=\dfrac{-5\pi}{6}$$.
Question 4
1 / -0

(i) Every prime number is odd.
(ii) The product of any two prime numbers is odd.
Which of the above statements(s) is/are correct?

A
Statement I alone.

B
Statement II alone.

C
Both statements I and II.

D
Neither statement I nor statement II.
Question 5
1 / -0

Find the value of $$x^3 + 7x^2 -x + 16$$, where $$x = 1 + 2i$$

A
$$-17 + 24i$$

B
$$24 + 17i$$

C
$$17 - 24i$$

D
$$24 - 17i$$

Solution

$$f\left(x\right)={x}^{3}+7{x}^{2}-x+16$$
$$f(1+2i)=(1+2i)^3+7(1+2i)^2-(1+2i)+16$$
$$=\left[1+8{i}^{3}+6i\left(1+2i\right)\right]+7\left[1+4{i}^{2}+4i\right]-1-2i+16$$
$$=\left[1-8i+6i-12\right]+7\left[1-4+4i\right]-2i+15$$
$$=\left[-11-2i\right]+7\left[4i-3\right]-2i+15$$
$$=-11-2i+28i-21-2i+15$$
$$=\left(-32+15\right)+24i$$
$$=-17+24i$$
Question 6
1 / -0

Consider the following statements:
A. The sum of two prime numbers is a prime number.
B. The product of two prime numbers is a prime number.
Which of these statements is/are correct ?

A
Neither A nor B

B
A alone

C
B alone

D
Both A and B

Solution

Neither the sum nor the product of two prime numbers is a prime number
Eg.$$3+5=8 $$ not a prime number.
$$3\times 5=15 $$ not a prime number.
Question 7
1 / -0

Which one of the following is the largest prime number of three digits?

A
$$997$$

B
$$999$$

C
$$991$$

D
$$993$$

Solution

There are two numbers $$997$$ and $$991$$ are prime numbers out of given numbers.

$$\therefore $$ largest prime no. is $$997$$

Option $$A$$ is correct.
Question 8
1 / -0

$$(1 + i)^8 + (1 -i)^8 =$$

A
$$16$$

B
$$-16$$

C
$$32$$

D
$$-32$$

Solution

$${ \left( 1+i \right) }^{ 8 }+{ \left( 1-i \right) }^{ 8 }$$
$$\Rightarrow { \left( 1+i \right) }^{ 8 }={ \left[ { { \left( 1+i \right) }^{ 2 } } \right] }^{ 4 }={ \left[ { 1+2i+{ i }^{ 2 } } \right] }^{ 4 }={ \left[ { 1+2i-1 } \right] }^{ 4 }$$
                     $$={ \left[ { { \left( 2i \right) }^{ 2 } } \right] }^{ 2 }$$
                     $$={ \left( -4 \right) }^{ 2 }$$
                     $$=16$$

Similarely $${ \left( 1-i \right) }^{ 8 }=16$$

$$\therefore { \left( 1+i \right) }^{ 8 }+{ \left( 1-i \right) }^{ 8 }$$

$$=16+16$$

$$=32.$$

Hence, the answer is $$32.$$
Question 9
1 / -0

A student was asked to find the sum of all the prime numbers between $$10$$ and $$40.$$He found the sum as $$180.$$Which of the following statements is true?

A
He missed one prime number between $$10$$ and $$20$$

B
He missed one prime number between $$20$$ and $$30$$

C
He added one extra prime number between $$10$$ and $$20$$

D
None of these

Solution

Prime numbers between $$10$$ and $$40$$ are $$11,13,17,19,23,29,31,37$$
Sum of these prime numbers $$= 180$$
$$\therefore$$ Option $$D$$ is correct.
Question 10
1 / -0

Find the value of: $$i^2 + i^4 + i^6$$ +..... upto $$(2n +1)$$ terms.

A
$$i$$

B
$$-i$$

C
$$1$$

D
$$-1$$

Solution

$$i^2$$ + $$i^4$$ + $$i^6$$ + ...upto $$(2n + 1 )$$ terms

$$=(-1 +1)+ (-1 +1)+( -1 + $$ .....upto $$(2n +1))$$terms $$\Rightarrow$$ (odd terms)

$$= 0+ 0 + ....-1$$

$$= -1$$

Hence, option D is correct.

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Number Theory Test 23

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Number Theory Test 23

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Question 1

$$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {\pi}{3}$$

$$|z|=2\sqrt [ 2 ]{ 2 } ; amp(z)=\dfrac {2\pi}{3}$$

$$|z|=4; amp(z)=\dfrac {2\pi}{3}$$

$$|z|=4; amp(z)=\dfrac {\pi}{3}$$

Solution

Question 2

$$Re(z) > 1$$

$$Re(z) < 2$$

$$Re(z) > 0$$

None of these

Solution

Question 3

$$|z|=2; amp(z)=-\dfrac {5\pi}{6}$$

$$|z|=4; amp(z)=\dfrac {5\pi}{6}$$

$$|z|=4; amp(z)=-\dfrac {\pi}{6}$$

none of these

Solution

Question 4

Statement I alone.

Statement II alone.

Both statements I and II.

Neither statement I nor statement II.

Question 5

$$-17 + 24i$$

$$24 + 17i$$

$$17 - 24i$$

$$24 - 17i$$

Solution

Question 6

Neither A nor B

A alone

B alone

Both A and B

Solution

Question 7

$$997$$

$$999$$

$$991$$

$$993$$

Solution

Question 8

$$16$$

$$-16$$

$$32$$

$$-32$$

Solution

Question 9

He missed one prime number between $$10$$ and $$20$$

He missed one prime number between $$20$$ and $$30$$

He added one extra prime number between $$10$$ and $$20$$

None of these

Solution

Question 10

$$i$$

$$-i$$

$$1$$

$$-1$$

Solution

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