If a² + b² + c² = -2 and f(x) = |1+a²x (1+b²)x (1+c²)x; (1+a²)x 1+b²x (1+c²)x; (1+a²)x (1+b²)x 1+c²x

Q: Why can't I just expand the determinant directly? It seems straightforward.

Expanding a $3 \times 3$ determinant with these terms will lead to a massive algebraic expression. You'll likely end up with a page full of terms, and the probability of a sign error is nearly 100%. In JEE, we look for elegance, not brute force.

Q: How do I know which row operations to choose?

Look for patterns. If rows look similar, subtraction is your best friend. The goal is to create zeros. Zeros are the 'magic keys' that make expansion trivial.

Q: What if the condition $a^2 + b^2 + c^2 = -2$ was different? Would the degree change?

Absolutely! That condition is the 'hidden key' that collapses the polynomial. If that sum were different, the $x$ term wouldn't vanish, and you'd likely end up with a cubic polynomial (degree 3).

Question

If $a^2 + b^2 + c^2 = -2$ and $f(x) = \begin{vmatrix} 1+a^2x & (1+b^2)x & (1+c^2)x \ (1+a^2)x & 1+b^2x & (1+c^2)x \ (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$, then $f(x)$ is a polynomial of degree

Options:

(A) $1$

(B) $0$

(C) $3$

(D) $2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The determinant $f(x)$ is designed to look like an intimidating fortress. With terms like $1+a^2x$ and $(1+b^2)x$ scattered across the rows, a direct expansion is a trap intended to waste time.

In JEE Advanced, examiners reward the strategic mind over brute force. We are given the critical condition:
$$a^2 + b^2 + c^2 = -2$$
This constraint is the master key to unlocking the entire problem.

### The Surgical Strike: Row Operations

Instead of expanding, we utilize the power of row operations to create zeros. Zeros are the "magic keys" that make expansion trivial.

We apply the following transformations to the determinant:
$$R_2 	o R_2 - R_1$$
$$R_3 	o R_3 - R_1$$

Let us calculate the new elements for row two:
The first element becomes $(1+a^2)x - (1+a^2x) = x + a^2x - 1 - a^2x = x - 1$.
The second element becomes $(1+b^2)x - (1+b^2x) = x + b^2x - 1 - b^2x = x - 1$.
The third element becomes $(1+c^2)x - (1+c^2x) = x - 1$.

By repeating similar logic for row three, we transform the chaos into a structured matrix where rows are filled with simple terms like $(x-1)$.

### The 'Aha!' Moment

Now, our determinant $f(x)$ is significantly simplified. When we expand this along the rows, we find a common factor of $(x-1)^2$ hiding in the expression.

After factoring out $(x-1)^2$, we are left with a bracketed expression:
$$1 + a^2x + (1+b^2)x + (1+c^2)x$$

Expanding the terms inside the bracket:
$$1 + a^2x + x + b^2x + x + c^2x = 1 + x(a^2 + b^2 + c^2 + 2)$$

### The Final Victory

Recall the condition provided at the start: $a^2 + b^2 + c^2 = -2$. Substituting this into our expression, the term inside the bracket becomes:
$$1 + x(-2 + 2) = 1 + x(0) = 1$$

The entire $x$ term vanishes, leaving us with:
$$f(x) = (x-1)^2$$

Expanding this, we get $f(x) = x^2 - 2x + 1$. The highest power of $x$ is $2$.

We have successfully navigated the trap and arrived at the solution with precision. The **degree of the polynomial is 2**.

Animated Solution for Mathematics - Matrices and Determinants: If $a^{2} + b^{2} + c^{2} = - 2$ and $f (x) = 1 + a^{2} x (1 + a^{2}) x (1 + a^{2}) x (1 + b^{2}) x 1 + b^{2} x (1 + b^{2}) x (1 + c^{2}) x (1 + c^{2}) x 1 + c^{2} x$ , then $f (x)$ is a polynomial of degree

Select Answer:

Visualized Solution

$f (x)$ and the Given Condition

Simplifying the Determinant

Applying $R_{2} \to R_{2} - R_{1}$

Applying $R_{3} \to R_{3} - R_{1}$

The Simplified Determinant

Expanding the Determinant

Factoring out $(1 - x)^{2}$

Grouping the Terms

Using the Given Condition

Final Polynomial and Degree

The Sigma Insight: Properties of Determinants

Analyzing the Setup

The Surgical Strike

The 'Aha!' Moment

The Final Victory

Similar Questions

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x) and the Given Condition

Simplifying the Determinant

Applying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R2​→R2​−R1​

Applying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } R3​→R3​−R1​

The Simplified Determinant

Expanding the Determinant

Factoring out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (1−x)2

Grouping the Terms

Using the Given Condition

Final Polynomial and Degree

The Sigma Insight: Properties of Determinants

Analyzing the Setup

The Surgical Strike

The 'Aha!' Moment

The Final Victory

Similar Questions

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Factoring out $(1 - x)^{2}$

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