Number Theory Test 46

Result

Number Theory Test 46

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Weekly Quiz Competition

Question 1
1 / -0

The real part of $$(i - \sqrt{3})^{13}$$ is

A
$$2^{-10}\sqrt3$$

B
$$-2^{12}\sqrt3$$

C
$$2^{-12}\sqrt3$$

D
$$-2^{-12}\sqrt3$$

E
$$-2^{10}\sqrt3$$

Solution

$$w^2=\dfrac{-1-i\sqrt3}{2}$$
$$\Rightarrow iw^2=\dfrac{-i+\sqrt3}{2}$$
$$\Rightarrow -2iw^2=i-\sqrt3$$
$$\Rightarrow (-2iw^2)^{13}=(i-\sqrt3)^{13}$$
$$\Rightarrow -2^{13}i^{13}w^{26}=(i-\sqrt3)^{13}$$
$$\Rightarrow -2^{13}iw^2=(i-\sqrt3)^{13}$$
$$\Rightarrow -2^{13}\left({\dfrac{-i+\sqrt3}{2}}\right)=(i-\sqrt3)^{13}$$
$$\Rightarrow -2^{12}(-i+\sqrt3)=(i-\sqrt3)^{13}$$
$$\therefore (i-\sqrt3)^{13}=-2^{12}\sqrt3+i2^{12}$$
Question 2
1 / -0

$$(1-\sqrt{-1})(1+\sqrt{-1})(5-\sqrt{-7})(5+\sqrt{-7})=?$$

A
$$(25+7i)$$

B
$$(32+5i)$$

C
$$(29-3i)$$

D
$$none\ of\ these$$

Solution

Given expression
$$=(1-i)(1+i)(5-\sqrt 7 i)(5+\sqrt 7i)$$
$$=(1-i^2)(25-7i^2)\\=(1+1)(25+7)\\=(2\times 32)=64$$.
Question 3
1 / -0

Mark against the correct answer in each of the following .
$$i^{326}=$$?

A
$$1$$

B
$$-1$$

C
$$i$$

D
$$-i$$

Solution

$$i^{326}\\=(i^4)^{81}\times i^2 =(1)^{81}\times (-1)\\=1\times (-1)=-1$$
Question 4
1 / -0

If $$\mid z^2 - 3\mid = 3\mid z\mid$$ then the maximum value of $$\mid z\mid$$ is

A
1

B
$$\dfrac{3+\sqrt {21}}{2}$$

C
$$\dfrac{\sqrt {21} - 3}{2}$$

D
none of these

Solution

$$ \mid z^{2} - 3 \mid \geq \mid z \mid^{2} - 3$$

$$\Rightarrow 3 \mid z \mid \geq \mid z \mid ^{2} - 3$$

$$\Rightarrow \mid z \mid ^{2} - 3 \mid z \mid -3 \leq 0$$

$$\Rightarrow 0 < \mid z \mid \leq \dfrac{3 + \sqrt{21}}{2}$$
Question 5
1 / -0

Mark against the correct answer in each of the following .
$$i^{91}=$$?

A
$$1$$

B
$$-1$$

C
$$i$$

D
$$-i$$

Solution

$$i^{91}\\=(1^4)^{22}\times i^3 \\=1\times i^2 \times i\\=1\times (-1)\times i\\=-i$$
Question 6
1 / -0

$$(2-3i)(-3+4i)=?$$

A
$$(6+17i)$$

B
$$(6-17i)$$

C
$$(-6+17i)$$

D
$$none\ of\ these$$

Solution

$$(2-3i)(-3+4i)\\=(-6+8i+9i-12i^2)\\=(-6+17i+12)\\=(6+17i)$$.
Question 7
1 / -0

Mark against the correct answer in each of the following .
$$i^{273}=$$?

A
$$i$$

B
$$-i$$

C
$$1$$

D
$$-1$$

Solution

$$i^{273}=(i^4)^{68}\times i=(1)^{68}\times i=(1\times i)=i$$
Question 8
1 / -0

Compare List I with List II and choose the correct answer using codes given below:
List I (Complex number) List II (Its modulus)
$$(4-3i)$$ $$10$$
$$(8+6i)$$ $$\dfrac{1}{5}$$
$$\dfrac{1}{(3+4i)}$$ $$1$$
$$\dfrac{(3-4i)}{(3+4i)}$$ $$5$$

A
$$(i)-(p), (ii)-(s), (iii)-(r), (iv)-(q)$$

B
$$(i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)$$

C
$$(i)-(s), (ii)-(p), (iii)-(r), (iv)-(q)$$

D
$$(i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)$$

Solution
Question 9
1 / -0

Which of the following is a composite number?

A
$$23$$

B
$$29$$

C
$$32$$

D
none of these

Solution

(a) $$23$$
Since it cannot be broken into factors
Hence $$23$$ is not a composite number
(b) $$29$$
Since it cannot be broken into factors
Hence $$29$$ is not a composite number
(c) $$32$$
Since it can be broken into factors i.e.$$2 \times 2 \times 2 \times 2 \times 2$$
Hence $$32$$ is a composite number
Option (c) is the correct answer
Question 10
1 / -0

Which of the following is an odd composite number ?

A
$$ 7 $$

B
$$ 9 $$

C
$$ 11 $$

D
$$ 12 $$

Solution

$$ 9 = 3 \times 3 $$ , is an odd composite number.

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Re-Attempt Weekly Quiz Competition

List I (Complex number)	List II (Its modulus)
$$(4-3i)$$	$$10$$
$$(8+6i)$$	$$\dfrac{1}{5}$$
$$\dfrac{1}{(3+4i)}$$	$$1$$
$$\dfrac{(3-4i)}{(3+4i)}$$	$$5$$

Number Theory Test 46

Result

Number Theory Test 46

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SHARING IS CARING

Question 1

$$2^{-10}\sqrt3$$

$$-2^{12}\sqrt3$$

$$2^{-12}\sqrt3$$

$$-2^{-12}\sqrt3$$

$$-2^{10}\sqrt3$$

Solution

Question 2

$$(25+7i)$$

$$(32+5i)$$

$$(29-3i)$$

$$none\ of\ these$$

Solution

Question 3

$$1$$

$$-1$$

$$i$$

$$-i$$

Solution

Question 4

1

$$\dfrac{3+\sqrt {21}}{2}$$

$$\dfrac{\sqrt {21} - 3}{2}$$

none of these

Solution

Question 5

$$1$$

$$-1$$

$$i$$

$$-i$$

Solution

Question 6

$$(6+17i)$$

$$(6-17i)$$

$$(-6+17i)$$

$$none\ of\ these$$

Solution

Question 7

$$i$$

$$-i$$

$$1$$

$$-1$$

Solution

Question 8

$$(i)-(p), (ii)-(s), (iii)-(r), (iv)-(q)$$

$$(i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)$$

$$(i)-(s), (ii)-(p), (iii)-(r), (iv)-(q)$$

$$(i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)$$

Solution

Question 9

$$23$$

$$29$$

$$32$$

none of these

Solution

Question 10

$$ 7 $$

$$ 9 $$

$$ 11 $$

$$ 12 $$

Solution

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