JEE Main entrance exam Question Bank Solutions for Mathematics (JEE Main)

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 ______.

Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.

y = log x satisfies for x > 1, the inequality ______.

Function f(x) = x¹⁰⁰ + sinx – 1 is increasing for all x ∈ ______.

Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.

Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.

The function f(x) = tan^–1(sin x + cos x) is an increasing function in ______.

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is ______.