JEE Main 2023 · 13 Apr 2023 Shift 1

For the differentiable function f : ℝ − {0} → ℝ, let 3f(x) + 2f(1/x) = 1/x − 10, then |f(3) + f′(1/4)| is equal to

1. A7
2. B29/5
3. C33/5
4. D13

The set of all a ∈ ℝ for which the equation x|x − 1| + |x + 2| + a = 0 has exactly one real root, is

1. A(−∞, −3)
2. B(−6, ∞)
3. C(−∞, ∞)
4. D(−6, −3)

Let B = [[1, 3, α], [1, 2, 3], [α, α, 4]], α > 2 be the adjoint of a matrix A and |A| = 2. Then [α −2α α] B [α; −2α; α] is equal to

1. A0
2. B−16
3. C16
4. D32

For the system of linear equations 2x + 4y + 2az = b, x + 2y + 3z = 4, 2x − 5y + 2z = 8, which of the following is NOT correct?

1. AIt has unique solution if a = b = 6
2. BIt has unique solution if a = b = 8
3. CIt has infinitely many solutions if a = 3, b = 8
4. DIt has infinitely many solutions if a = 3, b = 6

The number of symmetric matrices of order 3, with all the entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, is

1. A10⁹
2. B9¹⁰
3. C10⁶
4. D6¹⁰

Let s₁, s₂, s₃, ..., s₁₀ respectively be the sum to 12 terms of 10 A.P.s whose first terms are 1, 2, 3, ..., 10 and the common differences are 1, 3, 5, ..., 19 respectively. Then ∑ᵢ₌₁¹⁰ sᵢ is equal to

1. A7260
2. B7220
3. C7360
4. D7380

Among (S1) : limₙ→∞ 1/n² (2 + 4 + 6 + ...... + 2n) = 1 (S2) : limₙ→∞ 1/n¹⁶ (1¹⁵ + 2¹⁵ + 3¹⁵ + ...... + n¹⁵) = 1/16

1. AOnly (S1) is true
2. BOnly (S2) is true
3. CBoth (S1) and (S2) are true
4. DBoth (S1) and (S2) are false

Let the equation of plane passing through the line of intersection of the planes x + 2y + az = 2 and x − y + z = 3 be 5x − 11y + bz = 6a − 1. For c ∈ ℤ, if the distance of this plane from the point (a, −c, c) is 2/√a, then (a + b)/c is equal to

1. A−4
2. B−2
3. C2
4. D4