Complex Numbers and Quadratic Equations Test 25

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Complex Numbers and Quadratic Equations Test 25

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Weekly Quiz Competition

Question 1
1 / -0

If $$z + \sqrt {2}|z + 1| + i = 0$$ and $$z = x + iy$$, then

A
$$x = -2$$

B
$$x = 2$$

C
$$y = -2$$

D
$$y = 1$$

Solution

On putting $$z=x+iy$$, We get

=> $$x+iy+\sqrt{2(x+1)^2+2y^2} +i=0$$

=> $$x+\sqrt{2(x+1)^2+2y^2} +i(y+1)=0$$

=> On comparing real part and imaginary part, We get

=> $$x+\sqrt{2(x+1)^2+2y^2} )=0$$ and $$y+1=0$$

So, $$y=-1$$

and , $$\sqrt{2(x+1)^2+2y^2} )=-x$$

=> $${2(x+1)^2+2} =x^2$$

=> $$x=-2$$
Question 2
1 / -0

If $$(a, 0)$$ is a point on a diameter segment of the circle $$x^2+y^2=4$$, then $$x^2-4x-a^2=0$$ has.

A
exactly one real root in $$(-1, 0]$$

B
Exactly one real root in $$[2, 5]$$

C
distinct roots greater than $$-1$$

D
Distinct roots less than $$5$$

Solution

Given $$x^2-4x-a^2=0$$
$$D=16-4.1.(-a^2)$$
$$D=16+4a^2$$
$$\alpha+\beta=4$$
$$\alpha \cdot \beta=-a^2$$
That mean one root is positive and other is negative
$$-2\leq a\leq2$$
So, answer is $$A$$
Question 3
1 / -0

Let $$P(e^{i\theta_1})$$, $$Q(e^{i\theta_2})$$ and $$R(e^{i\theta_3})$$ be the vertices of a triangle PQR in the Argand Plane. The orthocenter of the triangle PQR is

A
$$2e^(\theta_1+\theta_2+\theta_3)$$

B
$$\frac{2}{3}e^({\theta_1+\theta_2+\theta_3})$$

C
$$e^{\theta_1}$$+$$e^{\theta_2}$$+$$e^{\theta_3}$$

D
None of these

Solution

$$P,Q,R$$ lie on the unit circle and hence origin is the cirumcenter , The centrid is easily found as $$\dfrac{(p+q+r)}{3}$$,Using euler line theorem that centroid divides the line joining the line joining orthocenter and circumcenter in the ratio of $$2:1$$
Let orthocenter be $$x,y$$

$$Using \quad section \quad formula;$$

$$Re(\dfrac{(p+q+r)}{3})=\dfrac{(1.x+2.0)}{3}$$
$$Im(\dfrac{(p+q+r)}{3})=\dfrac{(1.y+2.0)}{3}$$

we get orthocenter as
$$p+q+r\rightarrow$$ $$e^{\theta_1}+$$$$e^{\theta_2}+$$$$e^{\theta_3}$$
Question 4
1 / -0

Let $$z,\omega$$ be complex numbers such that $$\vec{z}+i\vec{\omega}=0$$ and $$Arg(z\omega)=\pi$$ then $$Arg(z)=$$.

A
$$\displaystyle\frac{\pi}{4}$$

B
$$\displaystyle\frac{5\pi}{4}$$

C
$$\displaystyle\frac{3\pi}{4}$$

D
$$\displaystyle\frac{\pi}{2}$$

Solution

Since $$z+iw=0$$
$$\Rightarrow z=-iw$$
$$\Rightarrow w=-iz$$
Also $$arg(zw)=\pi$$
$$\Rightarrow arg(-iz^2)=\pi$$
$$\Rightarrow arg(-i)+2arg(z)=\pi$$
$$\Rightarrow \dfrac{-\pi}{2}+2arg(z)=\pi$$ ....... $$\left[\because arg(-i)=\dfrac{-\pi}{2}\right]$$
$$\Rightarrow 2arg(z)=\dfrac{3\pi}{2}$$
$$\Rightarrow arg(z)=\dfrac{3\pi}{4}$$
Hence, the answer is $$\dfrac{3\pi}{4}$$.
Question 5
1 / -0

Study the statements carefully.
Statement I: Both the roots of the equation $$x^2-x+1=0$$ are real.
Statement II: The roots of the equation $$ax^2+bx+c=0$$ are real if and only if $$b^2-4ac \geq 0$$.
Which of the following options hold?

A
Both Statement I and Statement II are true

B
Both Statement I and Statement II are false

C
Statement I is true and Statement II is false

D
Statement I is false and Statement II is true

Solution

As we know condition for the roots to be real is $$ b^2-4ac \geq 0 $$
so as doing for the statement 1 we are getting $$ (-1)^2-4*1*1=(-3) $$ which is less than $$ 0$$ so the roots will be unequal so statement 1 is false and statement 2 gives the right condition so its true.
Hence option d is correct.
Question 6
1 / -0

If '$$\omega$$' is a complex cube root of unity,then $$\omega ^{ \begin{pmatrix} \frac { 1 }{ 3 } & +\frac { 2 }{ 9 } +\frac { 4 }{ 27 } ...\infty \end{pmatrix} }+\omega^{ \begin{pmatrix} \frac { 1 }{ 2 } & +\frac { 3 }{ 8 } +\frac { 9 }{ 32 } ...\infty \end{pmatrix} }=$$

A
$$1$$

B
$$-1$$

C
$$\omega$$

D
$$i$$

Solution

$$\dfrac{1}{3}+\dfrac{2}{9}+\dfrac{4}{27}+......+\infty$$ (infinte G.P.)

$$S_\omega = \dfrac{a}{1-r} =1$$

$$\dfrac{1}{2}+\dfrac{3}{8}+\dfrac{9}{32}+......+\infty$$ (infinte G.P.)

$$S_\omega = \dfrac{a}{1-r} =2$$

$$\therefore\, \omega^1+\omega^2=-1$$
Question 7
1 / -0

If $${z}_{1}=-3+5i;{z}_{2}=-5-3i$$ and $$z$$ is a complex number lying on the line segment joining $${z}_{1}$$ and $${z}_{2}$$, then $$arg(z)$$ can be:

A
$$-\cfrac { 3\pi }{ 4 } $$

B
$$-\cfrac { \pi }{ 4 } $$

C
$$\cfrac { \pi }{ 6 } $$

D
$$\cfrac { 5\pi }{ 6 } $$
Question 8
1 / -0

If $$\dfrac {lz_{2}}{mz_{1}}$$ is purely imaginary number, then $$\left |\dfrac {\lambda z_{1} + \mu z_{2}}{\lambda z_{1} - \mu z_{2}}\right |$$ is equal to

A
$$\dfrac {l}{m}$$

B
$$\dfrac {\lambda}{\mu}$$

C
$$\dfrac {-\lambda}{\mu}$$

D
$$1$$

Solution

$$\cfrac { { lz }_{ 2 } }{ m{ z }_{ 1 } } =ki\\ \cfrac { { z }_{ 2 } }{ { z }_{ 1 } } =\cfrac { mki }{ l } \\ \implies |\cfrac { { \lambda z }_{ 1 }+\mu { z }_{ 2 } }{ { \lambda z }_{ 1 }-\mu { z }_{ 2 } } |\\ \implies |\cfrac { \lambda +\mu { z }_{ 2 }/{ z }_{ 1 } }{ \lambda -\mu { z }_{ 2 }/{ z }_{ 1 } } |\\ \implies |\cfrac { \lambda +\mu \cfrac { mki }{ l } }{ \lambda -\mu \cfrac { mki }{ l } } |\\ \implies|\cfrac { \lambda l+\mu mki }{ \lambda l-\mu mki } |$$
$$\implies|\cfrac{re^{i\theta}}{re^{-i\theta}}|$$
$$\implies |e^{2i\theta}|$$
$$\implies 1$$
Question 9
1 / -0

The complex numbers $$z=x+iy$$ which satisfy the equation
$$\left| \cfrac { z-5i }{ z+5i } \right| =1$$ lie on:

A
the x-axis

B
straight line $$y=5$$

C
a circle through the origin

D
none of these

Solution

$$|z-5i|$$$$=|z+5i|$$
$$|x+iy-5i|$$$$=|x+iy+5i|$$
$$\sqrt{x^2+(y-5)^2}$$$$=\sqrt{x^2+(y+5)^2}$$
$$\implies x^2+y^2+25-10y$$$$=x^2+y^2+25+10y$$
$$\implies y=0$$
Hence $$x$$ axis.
Question 10
1 / -0

Find a complex number z satisfying the equation $$z+\sqrt{2}|z+1|+i=0$$

A
$$2-i$$

B
$$-2-i$$

C
$$\sqrt{2}-i$$

D
None of these

Solution

Let $$z=x+iy$$. Then,

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

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

$$x+\sqrt{2}\sqrt{(x+1)^2+y^2}+(y+1)i=0$$

$$x+\sqrt{2(x+1)^2+2y^2}+(y+1)i=0$$

$$x+\sqrt{2(x+1)^2+2}=0$$ and $$y=-1$$

$$\sqrt{2(x+1)^2+2}=-x$$ and $$y=-1$$

$$2(x+1)^2+2=x^2$$ and $$y=-1$$

$$x^2+4x+4=0$$ and $$y=-1$$

$$(x+2)^2=0$$ and $$y=-1$$

$$x=-2$$ and $$y=-1$$

Hence, $$z=x+iy=-2-i$$

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Re-Attempt Weekly Quiz Competition

Complex Numbers and Quadratic Equations Test 25

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Question 1

$$x = -2$$

$$x = 2$$

$$y = -2$$

$$y = 1$$

Solution

Question 2

exactly one real root in $$(-1, 0]$$

Exactly one real root in $$[2, 5]$$

distinct roots greater than $$-1$$

Distinct roots less than $$5$$

Solution

Question 3

$$2e^(\theta_1+\theta_2+\theta_3)$$

$$\frac{2}{3}e^({\theta_1+\theta_2+\theta_3})$$

$$e^{\theta_1}$$+$$e^{\theta_2}$$+$$e^{\theta_3}$$

None of these

Solution

Question 4

$$\displaystyle\frac{\pi}{4}$$

$$\displaystyle\frac{5\pi}{4}$$

$$\displaystyle\frac{3\pi}{4}$$

$$\displaystyle\frac{\pi}{2}$$

Solution

Question 5

Both Statement I and Statement II are true

Both Statement I and Statement II are false

Statement I is true and Statement II is false

Statement I is false and Statement II is true

Solution

Question 6

$$1$$

$$-1$$

$$\omega$$

$$i$$

Solution

Question 7

$$-\cfrac { 3\pi }{ 4 } $$

$$-\cfrac { \pi }{ 4 } $$

$$\cfrac { \pi }{ 6 } $$

$$\cfrac { 5\pi }{ 6 } $$

Question 8

$$\dfrac {l}{m}$$

$$\dfrac {\lambda}{\mu}$$

$$\dfrac {-\lambda}{\mu}$$

$$1$$

Solution

Question 9

the x-axis

straight line $$y=5$$

a circle through the origin

none of these

Solution

Question 10

$$2-i$$

$$-2-i$$

$$\sqrt{2}-i$$

None of these

Solution

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