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JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Indefinite Integrals

Let

f (x) = \int \frac{7 x ^{10} + 9 x ^{8}}{( 1 + x ^{2} + 2 x ^{9} ) ^{2}} d x, x > 0

,

lim_{x \to 0} f (x) = 0

and

f (1) = \frac{1}{4}

.

A = 0 \frac{1}{4} α^{2} 0 f^{'} (1) 4 111

and

B = adj (adj A)

be such that

∣ B ∣ = 81

, then

α^{2}

is equal to

(A)2
(B)3
(C)1
(D)4✓ Correct

Answer: (D)

Solution

The integrand is the derivative of

\frac{x ^{9}}{1 + x ^{2} + 2 x ^{9}}

, so

f (x) = \frac{x ^{9}}{1 + x ^{2} + 2 x ^{9}} + C

; the condition

lim_{x \to 0} f (x) = 0

gives

C = 0

, consistent with

f (1) = \frac{1}{4}

. Differentiating,

f^{'} (1) = \frac{7 + 9}{( 1 + 1 + 2 ) ^{2}} = 1

. The matrix is

A = 0 \frac{1}{4} α^{2} 014111

with

det A = 1 - α^{2}

. Since

∣ B ∣ = adj (adj A) = ∣ A ∣^{4} = 81

,

∣ A ∣ = \pm 3

; the admissible case

1 - α^{2} = - 3

gives

α^{2} = 4

.

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Single correct option

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