Number Theory Test 22

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Number Theory Test 22

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Weekly Quiz Competition

Question 1
1 / -0

The number of non-zero integral solution of the equation $$ \left | \sqrt{3}+i \right |^{x}= 2^{x} $$

A
1

B
2

C
Infinitely many

D
None of these

Solution

Let $$z=\sqrt{3}+i$$

Hence $$|z|$$

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

$$=\sqrt{(\sqrt{3})^{2}+1}$$

$$=\sqrt{4}$$

$$=2$$

So, the left-hand side and right-hand side of the equation become equal.
Hence the above equation has infinitely many solutions.
Question 2
1 / -0

If $$z=4+i\sqrt7$$, then value of $$z^3-4z^2-9z+91$$ equals

A
$$0$$

B
$$1$$

C
$$-1$$

D
$$2$$

Solution

We have,
$$z-4 = i\sqrt 7 $$
$$z^2 - 8z = -23 $$

$$z^3 -4z^2-9z+91 $$
$$= z(z^2-8z) + 4(z^2-8z) +23z +91 $$
$$ =-23z -92 +23z+91$$
$$= -1$$
Question 3
1 / -0

Write all the composite numbers between $$10$$ and $$35.$$

A
$$7, 12, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 27, 28, 30,33$$ and $$34$$

B
$$7, 12, 14, 15, 16, 18, 20, 21, 21, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$

C
$$7, 12, 14, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$

D
None of these

Solution

All composite numbers between $$ 10 $$ and $$ 35 $$ are $$ 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33 $$ and $$ 34. $$
Question 4
1 / -0

$$\displaystyle \left [ \left ( \cos \theta +i \sin \theta \right )\left ( \cos \theta -i\sin \theta \right ) \right ]^{-1}$$

A
$$\displaystyle i$$

B
$$\displaystyle 1$$

C
$$\displaystyle -i$$

D
$$\displaystyle -1$$

Solution

$$\displaystyle \left [ \left ( \cos \theta +i \sin \theta \right )\left ( \cos \theta -i\sin \theta \right ) \right ]^{-1}$$
$$\displaystyle =\left ( \cos ^{2}\theta -i^{2}\sin ^{2}\theta \right )^{-1}$$
$$\displaystyle =\left ( \cos ^{2}\theta +\sin ^{2}\theta \right )^{-1}=1.$$
Alt.$$\displaystyle \left ( e^{i\theta } .e^{-i\theta }\right )^{-1}=1$$

Ans: B
Question 5
1 / -0

If $$\displaystyle \log _{ \sqrt { 3 } }{ \left( \frac { { \left| z \right| }^{ 2 }-\left| z \right| +1 }{ 2+\left| z \right| } \right) } <2$$, then the locus of $$z$$ is

A
$$\left| z \right| <5$$

B
$$\left| z \right| =5$$

C
$$\left| z \right| >5$$

D
None of these

Solution

$$\displaystyle \log _{ \sqrt { 3 } }{ \left( \frac { { \left| z \right| }^{ 2 }-\left| z \right| +1 }{ 2+\left| z \right| } \right) } <2$$
$$\displaystyle\Rightarrow\frac { { \left| z \right| }^{ 2 }-\left| z \right| +1 }{ 2+\left| z \right| } <{ \left( \sqrt { 3 } \right) }^{ 2 }$$
$$\Rightarrow { \left| z \right| }^{ 2 }-\left| z \right| +1<6+3\left| z \right| $$
$$\Rightarrow { \left| z \right| }^{ 2 }-4\left| z \right| -5<0$$
$$\Rightarrow \left( \left| z \right| -5 \right) \left( \left| z \right| +1 \right) <0$$
$$\Rightarrow \left| z \right| -5<0,$$
since $$\left| z \right| +1>0\Rightarrow \left| z \right| <5.$$
Question 6
1 / -0

Solve:
$$\displaystyle \left ( x+iy \right )\left ( 2-3i \right )= 4+i$$

A
$$\displaystyle x= \left ( 8/13 \right ), y= -\left ( 14/13 \right ).$$

B
$$\displaystyle x= \left ( 5/13 \right ), y= \left ( 14/13 \right ).$$

C
$$\displaystyle x= -\left ( 14/13 \right ), y= \left ( 5/13 \right ).$$

D
$$\displaystyle x= \left ( 14/13 \right ), y=- \left ( 8/13 \right ).$$

Solution

Given, $$\displaystyle \left ( x+iy \right )\left ( 2-3i \right )= 4+i$$

$$\displaystyle \left ( 2x+3y \right )+i\left ( -3x+2y \right )= 4+i.$$

Equating real and imaginary parts,

$$\displaystyle 2x+3y= 4, -3x+2y= 1.$$

Solving these, we get

$$\displaystyle x= \left ( \dfrac {5}{13} \right ), y= \left (\dfrac {14}{13} \right ).$$

Ans: B
Question 7
1 / -0

Solve : $$(2-\sqrt{-100})(1+\sqrt{-36})$$

A
$$62+2i$$

B
$$52+2i$$

C
$$-52+2i$$

D
$$-88+2i$$

Solution

$$(2-\sqrt{-100})(1+\sqrt{-36})$$

$$=(2-10i)(1+6i)$$

$$=2+12i-10i+60$$

$$=62+2i$$
Question 8
1 / -0

If $$Re(a), \space Re(b) > 0$$, and $$x = |a - b| - |\bar{a} + b|$$, then

A
$$x<0$$

B
$$x>0$$

C
$$x\ge1$$

D
0

Solution

Lets take $$a=a_{1}+ia_{2}, b=b_{1}+ib_{2}$$
$$x=\left | (a_{1}+ia_{2})-(b_{1}+ib_{2})\right |+\left | (a_{1}-ia_{2})+(b_{1}+ib_{2}) \right |$$
So we can see from above equation
$$\left | \overline{a}+b \right |>\left | a-b \right |$$
Hence $$x<0$$
Question 9
1 / -0

If $$z = re^{i\theta}$$, then the value of $$|e^{iz}|$$ is equal to

A
$$e^{rcos\theta}$$

B
$$e^{-rcos\theta}$$

C
$$e^{rsin\theta}$$

D
$$e^{-rsin\theta}$$

Solution

Ginen,
$$ z=r{ e }^{ i\theta }$$
To find$$ \left| { e }^{ iz } \right| $$
solution,
for given,$$z=r{ e }^{ i\theta }\\ z=r(\cos\theta +i\sin\theta )\left\{ { e }^{ i\theta }=(\cos\theta +i\sin\theta ) \right\} \\ iz=ir(\cos\theta +i\sin\theta )\\ iz=r(i\cos\theta +{ i }^{ 2 }\sin\theta )\\ iz=r(-\sin\theta +i\cos\theta )\\ iz=-r\sin\theta +ir\cos\theta \\ { e }^{ iz }={ e }^{ -r\sin\theta +ir\cos\theta }\\ \left| { e }^{ iz } \right| =\left| { e }^{ -r\sin\theta } \right| \left| { e }^{ ir\cos\theta } \right| $$
We know that$$ \left| { e }^{ iz } \right| =1\\ \therefore \left| { e }^{ iz } \right| =\left| { e }^{ -r\sin\theta } \right| \longrightarrow \{ always\quad positive\} \\ { e }^{ iz }={ e }^{ -r\sin\theta }$$
Question 10
1 / -0

Find the modulus and amplitude of $$-2i$$

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

B
$$|z|=2i; amp(z)=\dfrac {\pi}{2}$$

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

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

Solution

$$z=-2i$$ $$|z|=2$$

$$=2(-i)$$

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

$$=|z|(e^{i\arg(z)})$$

Hence

$$|z|=2$$ and $$arg(z)=\dfrac{-\pi}{2}$$

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Number Theory Test 22

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Number Theory Test 22

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Question 1

1

2

Infinitely many

None of these

Solution

Question 2

$$0$$

$$1$$

$$-1$$

$$2$$

Solution

Question 3

$$7, 12, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 27, 28, 30,33$$ and $$34$$

$$7, 12, 14, 15, 16, 18, 20, 21, 21, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$

$$7, 12, 14, 15, 16, 19, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33$$ and $$34$$

None of these

Solution

Question 4

$$\displaystyle i$$

$$\displaystyle 1$$

$$\displaystyle -i$$

$$\displaystyle -1$$

Solution

Question 5

$$\left| z \right| <5$$

$$\left| z \right| =5$$

$$\left| z \right| >5$$

None of these

Solution

Question 6

$$\displaystyle x= \left ( 8/13 \right ), y= -\left ( 14/13 \right ).$$

$$\displaystyle x= \left ( 5/13 \right ), y= \left ( 14/13 \right ).$$

$$\displaystyle x= -\left ( 14/13 \right ), y= \left ( 5/13 \right ).$$

$$\displaystyle x= \left ( 14/13 \right ), y=- \left ( 8/13 \right ).$$

Solution

Question 7

$$62+2i$$

$$52+2i$$

$$-52+2i$$

$$-88+2i$$

Solution

Question 8

$$x<0$$

$$x>0$$

$$x\ge1$$

0

Solution

Question 9

$$e^{rcos\theta}$$

$$e^{-rcos\theta}$$

$$e^{rsin\theta}$$

$$e^{-rsin\theta}$$

Solution

Question 10

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

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

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

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

Solution

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