Complex Numbers and Quadratic Equations Test 43

Result

Complex Numbers and Quadratic Equations Test 43

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Weekly Quiz Competition

Question 1
1 / -0

If $$z$$ is a complex number of unit modulus and argument $$\theta $$, then the real part of
$$\dfrac { z\left( 1-\bar { z } \right) }{ \bar { z } \left( 1+2 \right) } $$ is:

A
$$1+cos\dfrac { \theta }{ 2 } $$

B
$$1-sin{ \dfrac { \theta }{ 2 } }$$

C
$$-2\sin { ^{ 2 } } \dfrac { \theta }{ 2 } $$

D
$$2cos^{ 2 }\dfrac { \theta }{ 2 } $$

Solution
Question 2
1 / -0

If the roots $$a^2x^2 +2bx+c^2 = 0$$ are imaginary then the roots of $$b(x^2+1)+2acx = 0$$ are

A
complex number

B
real and unequal

C
real and equal

D
none

Solution
Question 3
1 / -0

If $$a > b > c$$, $$a\neq 0$$ and the system of equations
$$ax+by+cz=0$$, $$bx+cy+az=0$$, $$cx+ay+bz=0$$ has non-trivial solutions, then the roots of the quadratic equation $$at^2+bt+c=0$$.

A
Are imaginary

B
Are real and equal

C
Are real and distinct

D
May be real of imaginary

Solution
Question 4
1 / -0

If $$\left| z \right| \ge 5$$ then the least value $$\left| {z + \frac{2}{z}} \right|$$ is

A
$$\frac{{23}}{5}$$

B
$$\frac{{24}}{5}$$

C
$$5$$

D
none of these

Solution
Question 5
1 / -0

If $$z_1, z_2, z_3$$ are three points lying on the circle |z| =2, then the minimum value of $$|z_1 + z_2|^2 + | z_2 + z_3|^2 + | z_3 + z_1|^2$$ is equal to

A
$$6$$

B
$$12$$

C
$$15$$

D
$$24$$

Solution
Question 6
1 / -0

If $$z$$ is a complex number such that $$|z-1|=1$$ then $$arg \left(\dfrac{1}{z}-\dfrac{1}{2}\right)$$ may be

A
$$\dfrac{\pi}{6}$$

B
$$\dfrac{\pi}{2}$$

C
$$\dfrac{\pi}{4}$$

D
$$-\dfrac{\pi}{4}$$

Solution
Question 7
1 / -0

Necessary conditions when taken together for equation $$x^{2}+a^{2}x+b^{2}=0, (a,b\epsilon R)$$ to have distinct roots, each of which exceeds 'c' are

A
$$a^{4} > 4b^{2}$$

B
$$c^{2}+a^{2}c + b^{2} > 0$$

C
$$-\frac{a^{2}}{2} > c$$

D
$$-\frac{a^{2}}{2} < c$$
Question 8
1 / -0

If $$z=\dfrac{-2}{1+i\sqrt{3}}$$, then the value of arg(z) is?

A
$$\pi$$

B
$$\dfrac{\pi}{3}$$

C
$$\dfrac{2\pi}{3}$$

D
$$\dfrac{\pi}{4}$$
Question 9
1 / -0

If $$z^4+1=\sqrt{3}$$i then?

A
$$z^3$$ is purely real

B
z represents vertices of a square of side $$2^{\dfrac{1}{4}}$$

C
$$z^9$$ is purely imaginary

D
z represents vertices of a square of side $$2^{\dfrac{3}{4}}$$

Solution
Question 10
1 / -0

If $$\theta$$ real then the modulus of $$\dfrac{1}{1+\cos\theta+i\sin\theta}$$ is

A
$$\dfrac{1}{2}\sec\dfrac{\theta}{2}$$

B
$$\dfrac{1}{2}\cos\dfrac{\theta}{2}$$

C
$$\sec\dfrac{\theta}{2}$$

D
$$\cos\dfrac{\theta}{2}$$

Solution