The least value of |z| where z is complex number which satisfies the inequality exp(((|z|+3)(|z|-1))

Q: Wait, why do we just drop the modulus bars around $|z|+1$?

Great question! Remember that $|z|$ represents the distance of a complex number from the origin, which is always a non-negative real number. Therefore, $|z|+1$ is at least 1, which is strictly positive. Since the absolute value of any positive number is just the number itself, we can safely remove the outer modulus bars.

Q: Why does the inequality sign stay the same when we remove the base 2?

This is a fundamental property of exponential functions. The function $f(x) = 2^x$ is a strictly increasing function. If $2^A \geq 2^B$, it must be true that $A \geq B$. If the base were between 0 and 1, the inequality sign would flip, but since our base is 2, the direction of the inequality is preserved.

Q: Why do we ignore the $|z|+2$ factor when solving the quadratic inequality?

We don't ignore it; we analyze it! When we factor the quadratic $|z|^2 - |z| - 6 \geq 0$ into $(|z|-3)(|z|+2) \geq 0$, we look at the sign of the product. Since $|z| \geq 0$, the term $|z|+2$ is always at least 2, meaning it is strictly positive. Because it is positive, it does not affect the sign of the inequality, so we only need to ensure that $(|z|-3) \geq 0$.

Question

The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp\left( \frac{(|z|+3)(|z|-1)}{||z|+1|} \log_e 2 \right) \geq \log_{\sqrt{2}} |5\sqrt{7} + 9i|$, $i = \sqrt{-1}$, is equal to :

Options:

(A) 3

(B) $\sqrt{5}$

(C) 2

(D) 8

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler, to the beautiful world of complex numbers. Today, we are going to dismantle a problem that, at first glance, looks like a chaotic storm of exponentials, logarithms, and moduli.

By the end of this journey, you will see it for what it truly is: an elegant, structured dance of algebra.

### Taming the Beast

Let us look at our given inequality:

$$ \exp\left( \frac{(|z|+3)(|z|-1)}{||z|+1|} \log_e 2 \right) \geq \log_{\sqrt{2}} |5\sqrt{7} + 9i| $$

The key to solving complex problems is to simplify them piece by piece. Recall the golden rule of logarithms: $e^{a \ln b} = b^a$.

Applying this, our left-hand side transforms beautifully into:

$$ 2^{\frac{(|z|+3)(|z|-1)}{||z|+1||}} $$

Since $|z|$ is a distance, it is always non-negative. Thus, $|z|+1$ is at least 1, which is strictly positive. We can drop those outer modulus bars without a second thought, simplifying the expression to:

$$ 2^{\frac{(|z|+3)(|z|-1)}{|z|+1}} $$

### The Right-Hand Side Revelation

Now, let's turn our attention to the right-hand side. We need to calculate the magnitude of the complex number $5\sqrt{7} + 9i$.

This is the distance from the origin:

$$ \sqrt{(5\sqrt{7})^2 + 9^2} = \sqrt{175 + 81} = \sqrt{256} = 16 $$

So, the right-hand side is $\log_{\sqrt{2}} 16$. We know $16 = 2^4$ and $\sqrt{2} = 2^{1/2}$.

Using the property $\log_{a^k} b^n = \frac{n}{k} \log_a b$, we get:

$$ \frac{4}{1/2} \log_2 2 = 8 = 2^3 $$

Our inequality is now:

$$ 2^{\frac{(|z|+3)(|z|-1)}{|z|+1}} \geq 2^3 $$

### The Final Algebraic Push

Since the base 2 is greater than 1, the exponential function is strictly increasing. This allows us to compare the exponents directly:

$$ \frac{(|z|+3)(|z|-1)}{|z|+1} \geq 3 $$

We can safely multiply both sides by $|z|+1$ because it is positive. This leads us to:

$$ (|z|+3)(|z|-1) \geq 3(|z|+1) $$

Expanding this, we get $|z|^2 + 2|z| - 3 \geq 3|z| + 3$. Bringing everything to one side, we arrive at the quadratic inequality:

$$ |z|^2 - |z| - 6 \geq 0 $$

Factoring this, we get $(|z|-3)(|z|+2) \geq 0$. Since $|z|+2$ is always positive, the inequality holds if and only if $|z|-3 \geq 0$, which means $|z| \geq 3$.

### Conclusion

Geometrically, this means $z$ lies on or outside a circle of radius 3 centered at the origin. The **least value of $|z|$ is therefore 3**.

You have successfully navigated the storm and found the calm center. Remember, no matter how complex the problem, there is always a path to simplicity if you take it one step at a time.

Animated Solution for Mathematics - Complex Numbers: The least value of $∣ z ∣$ where $z$ is complex number which satisfies the inequality $exp (\frac{( ∣ z ∣ + 3 ) ( ∣ z ∣ - 1 )}{∣∣ z ∣ + 1∣} lo g_{e} 2) \geq lo g_{2} ∣5 7 + 9 i ∣$ , $i = - 1$ , is equal to :

Select Answer:

Visualized Solution

The Complex Inequality

Simplifying the Exponential

Handling the Modulus Denominator

Calculating the RHS Modulus

Simplifying the RHS Logarithm

Comparing the Exponents

Cross-Multiplication

Expanding the Expressions

Rearranging to Standard Form

Factoring the Quadratic

Solving for $∣ z ∣$

The Final Answer

The Sigma Insight: Conjugate and Modulus

Analyzing the Setup

Taming the Beast

The Right-Hand Side Revelation

The Final Algebraic Push

Conclusion

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List-I

List-II

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