JEE Main entrance exam Question Bank Solutions

Let f: `R - {1/2}→R - {1/2}, f(x) = (x - 2)/(2x - 1)` be a function such that x = m is the solution of f(x) + 2f^–1(x) + 2 = f(f(x)), then m is equal to ______.

Range of 'a' for which x³ – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.

Let f(x) be a function which satisfies f(x³)f’(x) = f’(x)f’(x³) + f”(x²). Given that f(1) = 1 and f”’(1) = `1/4`, then value of 4(f’(1) + f'”(1)) is ______.

If f : R `rightarrow` R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then `sum_(r = 1)^nf(r)` is ______.

Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.

Let f: R → R be a function defined by:

f(x) = `{{:(max_(t ≤ x){t^3 - 3t},;, x ≤ 2),(x^2 + 2x - 6,;, 2 < x ≤ 3),([x - 3] + 9,;, 3 < x ≤ 5),(2x + 1,;, x > 5):}`

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and I = `int_-2^2f(x)dx`. Then the ordered pair (m, I) is equal to ______.

Let A = {n ∈ N: n is a 3-digit number}

B = {9k + 2: k ∈ N}

and C = {9k + ℓ: k ∈ N} for some ℓ(0 < ℓ < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then ℓ is equal to ______.

If α and β are the roots of the equation x² + px + 2 = 0 and `1/α` and `1/β` are the roots of the equation 2x² + 2qx + 1 = 0, then `(α - 1/α)(β - 1/β)(α + 1/β)(β + 1/α)` is equal to ______.

The roots of the equation (b + c)x² – (a + b + c)x + a = 0 (a, b, c ∈ Q, b + c ≠ a) are ______.

A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in the surface area (in cm²/min.) of the balloon when its diameter is 14 cm, is ______.

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.

The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.

Let 'a' be a real number such that the function f(x) = ax² + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax² – 6x + 15, x∈R has a ______.

Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.

If f(x) = x³ + 4x² + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.

Let f(x) be a function such that; f'(x) = log_1/3(log₃(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.

Let f(x) = tan^–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.

The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.

If f(x) = x⁵ – 20x³ + 240x, then f(x) satisfies ______.

If f(x) = x + cosx – a then ______.