Multiple Choice

If α is a complex constant such that az2+z+ ˉ α =0 has a real root then

A$$\displaystyle\ \alpha+\bar{\alpha}=1$$
Correct Answer
B$$\displaystyle\ \alpha+\bar{\alpha}=0$$
C$$\displaystyle\ \alpha+\bar{\alpha}=-1$$
Correct Answer
Dthe absolute value of the real root is 1
Correct Answer

Solution

αz2+z+
ˉ
α
=0

Let x be the real root.
⇒x=
ˉ
x

αx2+x+
ˉ
α
=0 ...(1)

Conjugating equation (1) we get,
ˉ
α

ˉ
x
2+
ˉ
x
+α=0
⟹
ˉ
α

ˉ
x
2+
ˉ
x
+α=0 ...(2)

Subtracting eq. (1) & eq. (2) we get,
x2=1⟹x=±1∴|x|=1

Multiplying eq. (1) by
ˉ
α
and eq. (2) by α and then subtracting we get,
x=−(α+
ˉ
α
)=±1

Ans: A,C &D

SIMILAR QUESTIONS

Complex Numbers

The number of solution of the equation $${z}^2=\overline{z}$$ is

Complex Numbers

If $$z_1$$, $$z_2$$ are two complex number such that $$|z_1|=|z_2|$$ and arg $$z_1+ $$arg $$ z_2=0$$ then $$z_1$$, $$z_2$$ and

Complex Numbers

If $$z_1$$, $$z_2$$are conjugate complex numbers and arg $$\left ( z_1z_2 \right )=$$

Complex Numbers

Let $$w(Im w\neq 0)$$ be a complex number. Then the set of all complex number z satisfying the equation $$w-\overline wz=k(1-z)$$, for some real number k, is :

Complex Numbers

The least positive integer $$n$$ for which $$\left(\displaystyle\frac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)^n=1$$, is

Complex Numbers

Let $$z$$ and $$w$$ be two non-zero complex numbers such that $$|z|=|w|$$ and $$arg(z)+arg(w)=\pi $$, then $$z $$ equals

Complex Numbers

Let $$z_1$$ and $$z_2$$ be complex numbers such that $$z_1 \neq z_2$$ and $$\left|z_1\right| = \left|z_2\right|$$. If $$z_1$$ has positive real part and $$z_2$$ has negative imaginary part, then $$(z_1 + z_2)/(z_1 - z_2)$$ may be

Complex Numbers

The complex number $$\sin x + i \cos 2x$$ and $$\cos x - i \sin 2x$$ are conjugate to each other for :`

Complex Numbers

(a) Let $$n=a+i b$$ be a complex number, where $$a$$ and $$b$$ are real (positive or negative) numbers. Show that the product $$n \overline n$$ is always a positive real number. (b) Let $$m=c+i d$$ be another complex number. Show that $$|n m|=|n||m|$$

Complex Numbers

For any two complex numbers $$z_1, z_2$$ and any real numbers a & b; $$|az_1 - bz_2|^2 + |bz_1 + az_2|^2 = k(a^2 + b^2) (|z_1|^2 + |z_2|^2)$$. What is the value of $$2k$$?

Contact Details