JEE Main 2024 · 31 Jan 2024 Shift 1

If f(x) = (4x + 3)/(6x − 4), x ≠ 2/3 and (f∘f)(x) = g(x), where g : ℝ − {2/3} → ℝ − {2/3}, then (g∘g∘g)(4) is equal to

1. A19/20
2. B4
3. C−19/20
4. D−4

Let S be the set of positive integral values of a for which (ax² + 2(a + 1)x + 9a + 4)/(x² − 8x + 32) < 0, ∀ x ∈ ℝ. Then, the number of elements in S is:

1. A1
2. B3
3. C0
4. D∞

If f(x) = | x³ 2x² + 1 1 + 3x; 3x² + 2 2x x³ + 6; x³ − x 4 x² − 2 | for all x ∈ ℝ, then 2 f(0) + f′(0) is equal to

1. A24
2. B18
3. C42
4. D48

If the system of linear equations x − 2y + z = −4, 2x + αy + 3z = 5, 3x − y + βz = 3 has infinitely many solutions, then 12α + 13β is equal to

1. A54
2. B58
3. C60
4. D64

The sum of the series 1/(1 − 3·1² + 1⁴) + 2/(1 − 3·2² + 2⁴) + 3/(1 − 3·3² + 3⁴) + ... up to 10-terms is

1. A45/109
2. B55/109
3. C−55/109
4. D−45/109

limₓ→₀ (e²|sin x| − 2|sin x| − 1)/x²

1. Ais equal to 2
2. Bis equal to 1
3. Cis equal to −1
4. Ddoes not exist

Let a be the sum of all coefficients in the expansion of (1 − 2x + 2x²)²⁰²³ (3 − 4x² + 2x³)²⁰²⁴ and b = limₓ→₀ [(∫₀ˣ log(1 + t)/(t²⁰²⁴ + 1) dt)/x²]. If the equations cx² + dx + e = 0 and 2bx² + ax + 4 = 0 have a common root, where c, d, e ∈ ℝ, then d : c : e equals

1. A2 : 1 : 4
2. B1 : 1 : 4
3. C4 : 1 : 4
4. D1 : 2 : 4

For 0 < c < b < a, let (a + b − 2c)x² + (b + c − 2a)x + (c + a − 2b) = 0 and α ≠ 1 be one of its root. Then, among the two statements (I) If α ∈ (−1, 0), then b cannot be the geometric mean of a and c (II) If α ∈ (0, 1), then b may be the geometric mean of a and c

1. Aonly (I) is true
2. Bonly (II) is true
3. CBoth (I) and (II) are true
4. DNeither (I) nor (II) is true