JEE Main 2024 · 5 Apr 2024 Shift 1

Let A = {1, 3, 7, 9, 11} and B = {2, 4, 5, 7, 8, 10, 12}. Then the total number of one-one maps f : A → B, such that f(1) + f(3) = 14, is :

1. A120
2. B180
3. C240
4. D480

Consider the following two statements : Statement I : For any two non-zero complex numbers z₁, z₂, (|z₁| + |z₂|) |z₁/|z₁| + z₂/|z₂|| ≤ 2 (|z₁| + |z₂|), and Statement II : If x, y, z are three distinct complex numbers and a, b, c are three positive real numbers such that a/|y − z| = b/|z − x| = c/|x − y|, then a²/(y − z) + b²/(z − x) + c²/(x − y) = 1. Between the above two statements,

1. Aboth Statement I and Statement II are correct.
2. Bboth Statement I and Statement II are incorrect.
3. CStatement I is correct but Statement II is incorrect.
4. DStatement I is incorrect but Statement II is correct.

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |Aᵀ A(adj(2A))⁻¹ (adj(4B)) (adj(AB))⁻¹ AAᵀ| is equal to :

1. A32
2. B64
3. C81
4. D108

If the system of equations 11x + y + λz = −5 2x + 3y + 5z = 3 8x − 19y − 39z = μ has infinitely many solutions, then λ⁴ − μ is equal to :

1. A45
2. B47
3. C49
4. D51

For the function f(x) = sin x + 3x − 2/π (x² + x), where x ∈ [0, π/2], consider the following two statements : (I) f is increasing in (0, π/2). (II) f′ is decreasing in (0, π/2). Between the above two statements,

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

If 1/(√1 + √2) + 1/(√2 + √3) + ... + 1/(√99 + √100) = m and 1/(1·2) + 1/(2·3) + ... + 1/(99·100) = n, then the point (m, n) lies on the line

1. A11x − 100y = 0
2. B11(x − 1) − 100y = 0
3. C11(x − 2) − 100(y − 1) = 0
4. D11(x − 1) − 100(y − 2) = 0

If the function f(x) = (sin 3x + α sin x − β cos 3x)/x³, x ∈ R, is continuous at x = 0, then f(0) is equal to :

1. A2
2. B−2
3. C4
4. D−4

Let f(x) = x⁵ + 2x³ + 3x + 1, x ∈ R, and g(x) be a function such that g(f(x)) = x for all x ∈ R. Then g(7)/g′(7) is equal to :

1. A1
2. B7
3. C14
4. D42