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

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be `24/5` and `194/25` respectively. If the mean and variance of the first 4 observation are `7/2` and a respectively, then (4a + x₅) is equal to ______.

Let b₁, b₂, b₃, b₄ be a 4-element permutation with b_i ∈ {1, 2, 3, .......,100} for 1 ≤ i ≤ 4 and b_i ≠ b_j for i ≠ j, such that either b₁, b₂, b₃ are consecutive integers or b₂, b₃, b₄ are consecutive integers. Then the number of such permutations b₁, b₂, b₃, b₄ is equal to ______.

The number of values of α ∈ N such that the variance of 3, 7, 12, α, 43 – α. is a natural number is ______.

`lim_(x→0^+)(int_0^(x^2)(sinsqrt("t"))"dt")/x^3` is equal to ______.

If ¹P₁ + 2. ²p₂ + 3. ³p₃ + ....... 15. ¹⁵P₁₅ = ^qP_r – s, 0 ≤ s ≤ 1, then ^q+sC_r–s is equal to ______.

The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______.

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ______.

If `lim_(x -> 1) (x + x^2 + x^3|+ .... + x^n - n)/(x - 1)` = 820, (n ∈ N) then the value of n is equal to ______.

`lim_(x→-1) (x^3 - 2x - 1)/(x^5 - 2x - 1)` = ______.

`lim_(x→∞)((x + 7)/(x + 2))^(x + 4)` is ______.

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Determine the number of words which have at least one letter repeated.

The value of `lim_(x→0)(sin(ℓn e^x))^2/((e^(tan^2x) - 1))` is ______.

The number of real solutions of `x - 1/(x^2 - 4) = 2 - 1/(x^2 - 4)` is ______.

A circle touches two of the smaller sides of a ΔABC (a < b < c) and has its centre on the greatest side. Then the radius of the circle is ______.

If ^m+nP₂ = 90 and ^m–nP₂ = 30, then (m, n) is given by ______.

Let x₁, x₂, x₃ be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan^–1(x + 1) where x₁ < x₂ < x₃ then 2x₁ + x₂ + x₃² is equal to ______.

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio ______.

In a ΔABC, let BC = 3. D is a point on BC such that BD = 2, Then the value of AB² + 2AC² – 3AD² is ______.

In usual notation a ΔABC, if A, A₁, A₂, A₃ be the area of the in-circle and ex-circles, then `1/sqrt(A_1) + 1/sqrt(A_2) + 1/sqrt(A_3)` is equal to ______.

In an equilateral triangle of side `2sqrt(3)` cm, the circum radius is ______.