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

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 ______.

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is ______.

Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct?

If A and B are two events such that P(A) = `1/3`, P(B) = `1/5` and P(A ∪ B) = `1/2`, then P(A|B') + P(B|A') is equal to ______.

If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to ______.

The shortest distance between the line y = x and the curve y² = x – 2 is ______.

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.