Question

Let \[ f(x)=\int \frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}\,dx \] and $f(1)=\frac14$. Given that 

 

Hint:

For $n\times n$ matrices, remember: $|\operatorname{adj}(\operatorname{adj}A)|=|A|^{(n-1)^2}$.

Answer 1

Correct Answer: 8

Step 1: Evaluate the integral

Given,

f(x) = ∫ (7x¹⁰ + 9x⁸) / (1 + x² + 2x⁹)² dx

Observe that:

d/dx [ x⁹ / (1 + x² + 2x⁹) ] = (7x¹⁰ + 9x⁸) / (1 + x² + 2x⁹)²

Hence,

f(x) = x⁹ / (1 + x² + 2x⁹) + C

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Step 2: Use the given condition f(1) = 1/4

f(1) = 1 / (1 + 1 + 2) + C

1/4 = 1/4 + C

C = 0

Therefore,

f(x) = x⁹ / (1 + x² + 2x⁹)

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Step 3: Write matrix A

A =

| 0   0   1 |
| 4   1/4   1 |
| α²   1/4   1 |

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Step 4: Compute |A|

Expanding along the first row:

|A| = 1 ×

| 4   1/4 |
| α²   1/4 |

|A| = 4(1/4) − (1/4)α²

|A| = 1 − α²/4

|A| = (4 − α²) / 4

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Step 5: Use property of adjoint

For a 3 × 3 matrix:

|adj(adj A)| = |A|⁴

Given:

|B| = 81

⇒ |A|⁴ = 81

⇒ |A| = ±3

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Step 6: Solve for α²

(4 − α²) / 4 = ±3

|4 − α²| = 12

α² = 16 or −8

Since α ∈ ℝ,

α² = 16

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Final Answer:

α² = 16