Let ω be a complex number such that 2ω + 1 = z where z = √-3. If |1 1 1; 1 -ω²-1 ω²; 1 ω² ω⁷| = 3k,

Q: Why can't I just expand the determinant directly? It seems like a lot of work to simplify it first.

That is a classic trap! If you expand the determinant with $\omega^7$ and $-\omega^2 - 1$ inside, you will end up with a massive algebraic expression that is prone to calculation errors. In JEE Advanced, the 'simplification first' strategy is your best friend. It turns a nightmare calculation into an elegant, two-step process.

Q: Wait, how do I know if $\omega$ is $\frac{-1 + \sqrt{3}i}{2}$ or $\frac{-1 - \sqrt{3}i}{2}$? Does it change the answer?

Great question! The problem defines $z = \sqrt{-3}$, which is $\sqrt{3}i$. This fixes the imaginary part as positive. Therefore, $\omega$ must be the root with the positive imaginary part. Even if you chose the other root, the symmetry of the problem often leads to the same result, but here, precision is key to matching the options.

Q: Why do we use row operations instead of column operations?

You can actually use either! The determinant of a matrix is equal to the determinant of its transpose. Row operations are just a convention that many students find more intuitive. As long as you perform valid elementary operations, the value of the determinant remains invariant.

Question

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $\begin{vmatrix} 1 & 1 & 1 \ 1 & -\omega^2 - 1 & \omega^2 \ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$, then k is equal to:

Options:

(A) $-z$

(B) $z$

(C) $-1$

(D) $1$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers! Today, we are going to peel back the layers of a problem that looks like a daunting matrix calculation but is actually a beautiful dance with the complex cube roots of unity.

Imagine you are standing at the threshold of a complex plane, looking at the equation $2\omega + 1 = \sqrt{-3}$. Many students see this and immediately reach for their calculators or start panicking about imaginary numbers.

But you? You are going to pause. You are going to recognize the signature of the complex cube root of unity. When we isolate $\omega$, we get:

$$\omega = \frac{-1 + i\sqrt{3}}{2}$$

This is not just a number; it is the key to the entire problem. It satisfies the magical properties $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.

### Simplifying the Matrix Elements

Now, look at the determinant. It looks intimidating, doesn't it? But remember the golden rule of JEE Advanced: never expand a determinant until you have simplified it to the bone.

We look at the term $-\omega^2 - 1$. Using our identity $1 + \omega + \omega^2 = 0$, we immediately see that:

$$-\omega^2 - 1 = \omega$$

Then we look at $\omega^7$. Since $\omega^3 = 1$, we know:

$$\omega^7 = (\omega^3)^2 \cdot \omega = \omega$$

The matrix is collapsing into a simple, elegant form.

### Executing the Determinant Calculation

Now, we perform row operations. We subtract the first row from the second and third rows. This creates zeros, which are the best friends of any determinant.

Expanding along the first column becomes trivial. We are left with a simple quadratic expression in $\omega$.

Finally, we equate this to $3k$. The algebra simplifies beautifully, the terms cancel out, and we arrive at the final result:

**$k = -z$**

### Final Thoughts

It is a moment of pure mathematical harmony. You have navigated the complexity, simplified the chaos, and emerged with the truth.

Keep this mindset: look for the structure, simplify before you calculate, and trust the elegance of the math.

Animated Solution for Mathematics - Complex Numbers: Let $ω$ be a complex number such that $2 ω + 1 = z$ where $z = - 3$ . If $111 1 - ω^{2} - 1 ω^{2} 1 ω^{2} ω^{7} = 3 k$ , then k is equal to:

Select Answer:

Visualized Solution

Identifying $z$ and $ω$

Solving for $ω$

Properties of Cube Roots of Unity

Simplifying Matrix Elements (Part 1)

Simplifying Matrix Elements (Part 2)

The Simplified Determinant

Applying Row Operations

Expanding the Determinant

Expanding the Algebraic Squares

Simplifying the Expression

Equating to Find $k$

Substituting Values of $ω$ and $ω^{2}$

Final Calculation for $k$

The Sigma Insight: Cube Roots and nth Roots of Unity

Analyzing the Setup

Simplifying the Matrix Elements

Executing the Determinant Calculation

Final Thoughts

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Let $α$ be a solution of $x^{2} + x + 1 = 0$ , and for some $a$ and $b$ in $R$ , $[4 a b] 1 - 1 - 2 16 - 1 - 14 132 - 8 = [000]$ . If $\frac{4}{α ^{4}} + \frac{m}{α ^{a}} + \frac{n}{α ^{b}} = 3$ , then $m + n$ is equal to

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Visualized Solution

Identifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ω

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ω