JEE Main entrance exam Question Bank Solutions

If `lim_(x→0) (int_0^x((cos2t - 1)(cost - e^(-t^2))t^-n)dr)/(cosx - 1)` is a finite non-zero number, Then the integer value for n is ______.

The normal to the ellipse `x^2/a^2 + y^2/b^2` = 1 at a point P(x₁, y₁) on it, meets the x-axis in G. PN is perpendicular to OX, where O is origin. Then value of ℓ(OG)/ℓ(ON) is ______.

The point on the ellipse x² + 2y² = 6 closest to the line x + y = 7 is (a, b). The value of (a + b) will be ______.

The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to ______.

Let the eccentricity of an ellipse `x^2/a^2 + y^2/b^2` = 1, a > b, be `1/4`. If this ellipse passes through the point ```(-4sqrt(2/5), 3)`, then a² + b² is equal to ______.

If P₁ and P₂ are two points on the ellipse `x^2/4 + y^2` = 1 at which the tangents are parallel to the chord joining the points (0, 1) and (2, 0), then the distance between P₁ and P₂ is ______.

If `lim_(x rightarrow 0) (ax - (e^(4x) - 1))/(ax(e^(4x) - 1))` exists and is equal to b, then the value of a – 2b is  ______.

If `lim_(x rightarrow 1) (x^4 - 1)/(x - 1)` = `lim_(x rightarrow k) (x^3 - k^3)/(x^2 - k^2)`, then k is ______. 

`lim_(x rightarrow π/4) (8sqrt(2) - (cosx + sinx)^7)/(sqrt(2) - sqrt(2)sin2x)` is equal to ______.

`lim_(x rightarrow a) ((a + 2x)^(1/3) - (3x)^(1/3))/((3a + x)^(1/3) - (4x)^(1/3)) (a ≠ 0)` is equal to ______.

Let f : R `rightarrow` R be a differentiable function satisfying f'(3) + f'(2) = 0. Then `lim_(x rightarrow 0)((1 + f(3 + x) - f(3))/(1 + f(2 - x) - f(2)))^(1/x)` is equal to ______.

Let p = `lim_(x rightarrow 0^+)(1 + tan^2sqrt(x))^(1/(2x))` then log p is equal to ______.

If f(x) = 3x¹⁰ – 7x⁸ + 5x⁶ – 21x³ + 3x² – 7, then `lim_(α rightarrow 0) (f(1 - α) - f(1))/(α^3 + 3α)` is ______. 

If the value of `lim_(x→0)(2 - cosxsqrt(cos2x))^(((x + 2)/x^2))` is equal to e^a, then a is equal to ______.

`lim_(n→∞)((1^2 + 2^2 + ...... + n^2)(1^4 + 2^4 + ...... + n^4))/((1^7 + 2^7 + ......  n^7)) = (k + 1)/15`, then k is equal to ______.

If x + y = `"t" + 1/"t"` and x² + y² = `"t"^2 + 1/"t"^2` then `150|x^2("dy")/("d"x)|` is ______.

If y = `[x + sqrt(x^2 - 1)]^15 + [x - sqrt(x^2 - 1)]^15`, then `(x^2 - 1)(d^2y)/(dx^2) + x(dy)/(dx)` is equal to ______.

Let B_i(i = 1, 2, 3) be three independent events in a sample space. The probability that only B₁ occur is α, only B₂ occurs is β and only B₃ occurs is γ. Let p be the probability that none of the events B_i occurs and these 4 probabilities satisfy the equations (α – 2β)p = αβ and (β – 3γ) = 2βy (All the probabilities are assumed to lie in the interval (0, 1)). Then `("P"("B"_1))/("P"("B"_3))` is equal to ______.

Let E^C denote the complement of an event E. Let E₁, E₂ and E₃ be any pairwise independent events with P(E₁) > 0 and P(E₁ ∩ E₂ ∩ E₃) = 0. Then `"P"(("E"_2^"C"  ∩ "E"_3^"C")/"E"_1)` is equal to ______.

Given two independent events, if the probability that exactly one of them occurs is `26/49` and the probability that none of them occurs is `15/49`, then the probability of more probable of the two events is ______.