if z1 is a real number, one of the relations below is true (a) n^2-npq+mq^2=0 (b) m^2-npq+nq^2=0

Name: Problem 187, Chapter 1: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-07-14T17:53:41-08:00
Duration: 3 min 54 s
Channel: Uma Kumari
Description: if z1 is a real number, one of the relations below is true (a) n^2-npq+mq^2=0 (b) m^2-npq+nq^2=0 (c) n^2+npq-mq^2=0 (d) m^2+npq+nq^2=0

step 1 In this question, it is given that two complex numbers Z1 and Z2 satisfy the equation Z squared

Question

if $\mathrm{z}_{1}$ is a real number, one of the relations below is true (a) $\mathrm{n}^{2}-\mathrm{npq}+\mathrm{mq}^{2}=0$ (b) $\mathrm{m}^{2}-\mathrm{npq}+\mathrm{nq}^{2}=0$ (c) $\mathrm{n}^{2}+\mathrm{npq}-\mathrm{mq}^{2}=0$ (d) $\mathrm{m}^{2}+\mathrm{npq}+\mathrm{nq}^{2}=0$

   if $\mathrm{z}_{1}$ is a real number, one of the relations below is true
(a) $\mathrm{n}^{2}-\mathrm{npq}+\mathrm{mq}^{2}=0$
(b) $\mathrm{m}^{2}-\mathrm{npq}+\mathrm{nq}^{2}=0$
(c) $\mathrm{n}^{2}+\mathrm{npq}-\mathrm{mq}^{2}=0$
(d) $\mathrm{m}^{2}+\mathrm{npq}+\mathrm{nq}^{2}=0$

IIT JEE Super Course in Mathematics ALGEBRA II

Trishna Knowledge… 2011 Edition

Chapter 1, Problem 187

Instant Answer

Solved by Expert Uma Kumari

07/14/2021

Step 1

We are given that $z_1$ is a real number. Let's denote $z_1$ as $\alpha$. So, we can rewrite the equation as: \[\alpha^2 + (p + iq)\alpha + (m + in) = 0\] Show more…

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if $\mathrm{z}_{1}$ is a real number, one of the relations below is true (a) $\mathrm{n}^{2}-\mathrm{npq}+\mathrm{mq}^{2}=0$ (b) $\mathrm{m}^{2}-\mathrm{npq}+\mathrm{nq}^{2}=0$ (c) $\mathrm{n}^{2}+\mathrm{npq}-\mathrm{mq}^{2}=0$ (d) $\mathrm{m}^{2}+\mathrm{npq}+\mathrm{nq}^{2}=0$

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Key Concepts

Real Numbers and Complex Numbers

A complex number is real when its imaginary part is zero. In many complex equations, requiring the result to be real forces conditions on the parameters, leading to identities or constraints that must be satisfied.

Equating Real and Imaginary Parts

When a complex expression is set to be real, the coefficients of the non?real components (often the imaginary parts) must vanish. This technique leads to specific algebraic relations among the parameters in the equation.

Discriminant Conditions in Quadratic Equations

In quadratic equations, the discriminant determines whether the roots are real or complex. Ensuring that the discriminant is nonnegative is a key method for imposing conditions that guarantee real solutions, which can manifest as specific parameter relations.

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