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

The position vectors of points A, B, C are respectively `vec"a", vec"b", vec"c"`. If L divides AB in 3:4 and M divides BC in 2:1 both externally, then `vec("LM")` is ______.

The mean of a data set consisting of 20 observations is 40. If one observation 53 was wrongly recorded as 33, then the correct mean will be ______.

`int (a^x - b^x)^2/(a^xb^x)dx` equals ______.

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is ______.

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ______.

The mean and standard deviation of 50 observations are 15 and 2 respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is 70. If the correct mean is 16, then the correct variance is equal to ______.

If ω(≠1) is a cube root of unity and (1 + ω)⁷ = A + Bω, then A and B are respectively the numbers ______.

If the constant term in the expansion of `(3x^3 - 2x^2 + 5/x^5)^10` is 2^k. l, where l is an odd integer, then the value of k is equal to ______.

Let c, k ∈ R. If f(x) = (c + 1)x² + (1 – c²)x + 2k and f(x + y) = f(x) + f(y) – xy, for all x, y ∈ R, then the value of |2(f(1) + f(2) + f(3) + ... + f(20))| is equal to ______.

Let α be a root of the equation 1 + x² + x⁴ = 0. Then the value of α¹⁰¹¹ + α²⁰²² – α³⁰³³ is equal to ______.

Let n ≥ be an integer. If 9ⁿ – 8n–1 = 64α and 6ⁿ – 5n = 25β, then α – β is ______.

The distance of the point (1, 1, 9) from the point of intersection of the line `(x - 3)/1 = ("y" - 4)/2 = ("z" - 5)/2` and the plane x + y + z = 17 is ______.

Let f: (0,2)→R be defined as f(x) = `log_2(1 + tan((πx)/4))`. Then, `lim_(n→∞) 2/n(f(1/n) + f(2/n) + ... + f(1))` is equal to ______.

The value of  `lim_(n→∞)1/n sum_(r = 0)^(2n-1) n^2/(n^2 + 4r^2)` is ______.

Let z = `(1 - isqrt(3))/2`, i = `sqrt(-1)`. Then the value of `21 + (z + 1/z)^3 + (z^2 + 1/z^2) + (z^3 + 1/z^3)^3 + ...... + (z^21 + 1/z^21)^3` is ______.

`lim_(n→∞){(1 + 1/n^2)^(2/n^2)(1 + 2^2/n^2)^(4/n^2)(1 + 3^2/n^2)^(6/n^2) ...(1 + n^2/n^2)^((2n)/n^2)}` is equal to ______.

The sum of the series 1.3² + 2.5² + 3.7² + .... upto 20 terms is ______.

If 1, α₁, α₂, ...... α_n–1 are the roots of unity, then (1 + α₁)(1 + α₂) ...... (1 + α_n–1) is equal to (when n is even) ______.

The variance of first 50 even natural numbers is ______.

The value of the expression `(cos^2 12^circ + 2sin12^circ(sin12^circ + 1))/(4cos^2  39^circ - sin^2 78^circ)` is ______.