Number Theory Test 21

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Number Theory Test 21

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Weekly Quiz Competition

Question 1
1 / -0

if $$\displaystyle\ z=1+i\ \tan \alpha $$, where $$\displaystyle\ \pi < \alpha < \frac{3\pi }{2}$$ is $$|z|$$ is equal to

A
$$\displaystyle\ \sec \alpha$$

B
$$\displaystyle\ -\sec \alpha $$

C
$$\displaystyle\ cosec\alpha $$

D
none of these

Solution

$$z=1+i\tan\alpha$$
$$\Rightarrow |z| =\sqrt{1+\tan^2\alpha}=\sqrt{\sec^2\alpha}=|\sec\alpha|$$
Now we know that, modulus of any complex number is always non-zero
and it is given that, alpha lies in third quadrant in which $$\sec\alpha <0$$
Hence $$|z|=-\sec\alpha>0$$
Hence option 'B' is correct choice.
Question 2
1 / -0

For a complex number z, the minimum value of $$\displaystyle\ |z| + |z-2|$$ is

A
$$\displaystyle\ 1$$

B
$$\displaystyle\ 2$$

C
$$\displaystyle\ 3$$

D
None of these

Solution

The question is infact asking the distance of an arbitrary point z from the origin and a point on the real axis, (2, 0).
Thus, the minimum distance will be when z is also on the real axis between the two points and thus the minimum distance comes out to be 2 units.
Question 3
1 / -0

The value of the sum $$\displaystyle\ \sum _{n=1}^{13}\left ( i^{n}+i^{n+1} \right ) $$, where $$\displaystyle\ i=\sqrt{-1} $$

A
$$ i $$

B
$$ i-1$$

C
$$ -i$$

D
$$ 0$$

Solution

$$\sum (i^n+i^{n+1})=i+2(i^2+i^3+i^4.........i^{13})+i^{14}$$

$$=i+2(0)-1$$.........................($$i^2=-1,i^3=-i,i^4=1$$)

$$=i-1$$
Question 4
1 / -0

The complex number which satisfies the equation $$z+\sqrt { 2 } \left| z+1 \right| +i=0$$ is

A
$$2-i$$

B
$$-2-i$$

C
$$2+i$$

D
$$-2+i$$

Solution

Since $$z+\sqrt { 2 } \left| z+1 \right| +i=0$$

$$\therefore x+i\left( y+1 \right) +\sqrt { 2 } \left| x+iy+1 \right| =0$$

By comparing real and imaginary parts

$$\Rightarrow y+1=0$$ $$(\because \left| x+iy+1 \right| $$ is real $$)$$

$$\Rightarrow y=-1$$

$$ \therefore x+\sqrt { 2 } \left| x-i+1 \right| =0$$

$$ \Rightarrow { x }^{ 2 }=2\left( { \left( x+1 \right) }^{ 2 }+1 \right) =2\left( { x }^{ 2 }+2x+2 \right) $$

$$ \Rightarrow { x }^{ 2 }+4x+4=0\Rightarrow { \left( x+2 \right) }^{ 2 }=0$$

$$\Rightarrow x=-2$$

$$\therefore z=-2-i$$
Question 5
1 / -0

If $$z$$ be a complex number, then the minimum value of $$\left | z-7 \right |+\left | z \right |$$ is

A
$$\displaystyle \frac{\sqrt{3}-1}{2}$$

B
$$-\sqrt{7}$$

C
$$\displaystyle \frac{7+\sqrt{2}}{2}$$

D
$$7$$

Solution

$$\left | z-7 \right |$$ implies distance between $$(x,y)$$ and $$(7,0)$$.
$$\left | z\right |$$ means distance between $$(x,y)$$ and $$(0,0)$$
Minimum distance would be straight line between $$(0,0)$$ and $$(7,0)$$ which is $$7$$.
Question 6
1 / -0

If $$\displaystyle z= 1+i\sqrt{3}$$,then $$\displaystyle z^{6}$$ equals

A
32

B
-32

C
64

D
None of these

Solution

$$\displaystyle z= 1+i\sqrt{3}$$
$$\displaystyle z^{6}= (1+\sqrt{3})^{6}$$ using Binomial Theorem.
$$\displaystyle = ^{6}\:C_{0}+^{6}\:C_{1}\:(i\sqrt{3})+^{6}\:C_{2}(i\sqrt{3})^{2}+^{6}\:C_{3}(i\sqrt{3})^{3}+^{6}\:C_{4}(i\sqrt{3})^{4}+^{6}\:C_{5}(i\sqrt{3})^{5}+^{6}\:C_{6}\:(i\sqrt{3})^{6}\:$$
$$\displaystyle = ^{6}\:C_{0}+^{6}C_{2}(3)+^{6}C_{4}(9)-^{6}C_{6}(27)+i\sqrt{3}(^{6}C_{1}-3^{6}C_{3}+9^{6}C_{5})$$
$$\displaystyle = (1-15\times 3+135-27)+i\sqrt{3}(60-60)$$
$$\displaystyle = (136-72)+i\sqrt{3}(0)= 64$$
Question 7
1 / -0

The locus of $$\displaystyle z= x+iy$$ which satisfying the inequality $$\displaystyle \log_{1/2}\left | z-1 \right |> \log_{1/2}\left | z-i \right |$$ is given by

A
$$\displaystyle x+y< 0$$

B
$$\displaystyle x-y> 0$$

C
$$\displaystyle x-y< 0$$

D
$$\displaystyle x+y> 0$$

Solution

$$log_{0.5}|z-1|>log_{0.5}|z-i|$$
$$-log_{2}|z-1|>-log_{2}|z-i|$$
$$log_{2}|z-1|<log_{2}|z-i|$$
Hence
$$|z-1|<|z-i|$$
Now
Let $$z=x+iy$$
Hence by squaring both sides
$$(x-1)^{2}+y^{2}<x^{2}+(y-1)^{2}$$
$$(2y-1)<(2x-1)$$
$$2(y-x)<0$$
$$y-x<0$$
Or
$$x-y>0$$
Question 8
1 / -0

If$$\displaystyle\ z_{1}\neq-z_{2}$$ and $$\displaystyle\ |z_{1}+z_{2}|=\left | \frac{1}{z_{1}}+\frac{1}{z_{2}} \right |$$ then

A
at least one of $$\displaystyle\ z_{1}, z_{2}$$ is unimodular

B
both$$\displaystyle\ z_{1}, z_{2}$$ are unimodular

C
$$\displaystyle\ z_{1}. z_{2} = 1$$

D
None of these

Solution

Let
$$z_{1}=x_{1}+iy_{1}$$
$$z_{2}=x_{2}+iy_{2}$$
Now
$$z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})$$
$$\dfrac{1}{z_{1}}+\dfrac{1}{z_{2}}$$
$$=\bar{z_{1}}+\bar{z_{2}}$$
$$=(x_{1}+x_{2})-i(y_{1}+y_{2})$$.
Now
$$|z_{1}+z_{2}|$$
$$=\sqrt{(x_{2}+x_{1})^2+(y_{2}+y_1)^{2}}$$
And
$$|\dfrac{1}{z_{1}}+\dfrac{1}{z_{2}}|$$
$$=|(x_{1}+x_{2})-i(y_{1}+y_{2})|$$
$$=\sqrt{(x_{2}+x_{1})^2+(y_{2}+y_1)^{2}}$$
Hence
$$|z_{1}+z_{2}|=|\dfrac{1}{z_{1}}+\dfrac{1}{z_{2}}|$$ for all values of $$z_{1}$$ and $$z_{2}$$.
Question 9
1 / -0

If $$\displaystyle z=1+i\cot\alpha,-\frac{\pi}{2}<\alpha<0,$$ then $$|z|$$ is equal to

A
$$cosec\alpha$$

B
$$-cosec\alpha$$

C
$$cosec\alpha$$ or $$-cosec\alpha$$

D
none of these

Solution

$$z=1+i\cot\alpha$$
$$\Rightarrow |z|=\sqrt{1+\cot^2\alpha}=\sqrt{co\sec^2\alpha}$$
$$=|co\sec\alpha|=-co\sec\alpha,$$ as $$-\pi/2<\alpha<0$$
Question 10
1 / -0

If $$\displaystyle u_{i}=1-\frac{1}{i}$$ then $$\displaystyle u_{2}\cdot u_{3}\cdot ... \cdot u_{n}$$ is equal to

A
$$\displaystyle \frac{1}{n}$$

B
$$\displaystyle \frac{1}{n!}$$

C
$$1$$

D
none of these

Solution

$$u_i = 1-\cfrac{1}{i}=\cfrac{i-1}{i}$$, here $$i$$ is not a complex number
$$\displaystyle \therefore u_2.u_3...........u_{n-1}.u_n = \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}.....\frac{n-2}{n-1}.\frac{n-1}{n} = \frac{1}{n}$$

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Re-Attempt Weekly Quiz Competition

Number Theory Test 21

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Number Theory Test 21

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SHARING IS CARING

Question 1

$$\displaystyle\ \sec \alpha$$

$$\displaystyle\ -\sec \alpha $$

$$\displaystyle\ cosec\alpha $$

none of these

Solution

Question 2

$$\displaystyle\ 1$$

$$\displaystyle\ 2$$

$$\displaystyle\ 3$$

None of these

Solution

Question 3

$$ i $$

$$ i-1$$

$$ -i$$

$$ 0$$

Solution

Question 4

$$2-i$$

$$-2-i$$

$$2+i$$

$$-2+i$$

Solution

Question 5

$$\displaystyle \frac{\sqrt{3}-1}{2}$$

$$-\sqrt{7}$$

$$\displaystyle \frac{7+\sqrt{2}}{2}$$

$$7$$

Solution

Question 6

32

-32

64

None of these

Solution

Question 7

$$\displaystyle x+y< 0$$

$$\displaystyle x-y> 0$$

$$\displaystyle x-y< 0$$

$$\displaystyle x+y> 0$$

Solution

Question 8

at least one of $$\displaystyle\ z_{1}, z_{2}$$ is unimodular

both$$\displaystyle\ z_{1}, z_{2}$$ are unimodular

$$\displaystyle\ z_{1}. z_{2} = 1$$

None of these

Solution

Question 9

$$cosec\alpha$$

$$-cosec\alpha$$

$$cosec\alpha$$ or $$-cosec\alpha$$

none of these

Solution

Question 10

$$\displaystyle \frac{1}{n}$$

$$\displaystyle \frac{1}{n!}$$

$$1$$

none of these

Solution

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