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

Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.

The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.

Let `int_0^∞ (t^4dt)/(1 + t^2)^6 = (3π)/(64k)` then k is equal to ______.

`vecA, vecB` and `vecC` are three non coplanar vectors, then `(vecA + vecB + vecC).((vecA + vecB) xx (vecA + vecC))` is equal to ______.

Let f be continuous periodic function with period 3, such that `int_0^3f(x)dx` = 1. Then the value of `int_-4^8f(2x)dx` is ______.

For any three vectors `veca, vecb` and `vecc`, `(veca - vecb).(vecb - vecc) xx (vecc - veca)` = ______.

If f(x) = `{{:(x^2",", "where"  0 ≤ x < 1),(sqrt(x)",", "when"  1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.

The angle between two lines `(x + 1)/2 = (y + 3)/2 = (z - 4)/(-1)` and `(x - 4)/1 = (y + 4)/2 = (z + 1)/2` is ______.

If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.

`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.

A straight line L through the point (3, –2) is inclined at an angle of 60° to the line `sqrt(3)x + y` = 1. If L also intersects the x-axis, then the equation of L is ______.

The domain of the function f(x) = `(cos^-1((x^2  -  5x  +  6)/(x^2  -  9)))/(log_e(x^2 - 3x + 2)` is ______.

The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.

Let a vector `veca` be coplanar with vectors `vecb = 2hati + hatj + hatk` and `vecc = hati - hatj + hatk`. If `veca` is perpendicular to `vecd = 3hati + 2hatj + 6hatk` and `|veca| = sqrt(10)`. Then a possible value of `[(veca, vecb, vecc)] + [(veca, vecb, vecd)] + [(veca, vecc, vecd)]` is equal to ______.

If the vectors, `vecp = (a + 1)hati + ahatj + ahatk, vecq = ahati + (a + 1)hatj + ahatk` and `vecr = ahati + ahatj + (a + 1)hatk  (a ∈ R)` are coplanar and `3(vecp.vecq)^2 - λ|vecr xx vecq|^2` = 0, then the value of λ is ______.

Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.

The sum of 162^th power of the roots of the equation x³ – 2x² + 2x – 1 = 0 is ______.

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^–1 (x) – 2sin^–1(x) = cos^–1 (2x) is equal to ______.

Given that the inverse trigonometric function take principal values only. Then, the number of real values of x which satisfy `sin^-1((3x)/5) + sin^-1((4x)/5) = sin^-1x` is equal to ______.

Let A = `[(2, -1),(0, 2)]`. If B = I – ⁵C₁(adj A) + ⁵C₂(adj A)⁵, –... –⁵C₅(adj A)⁵, then the sum of all elements of the matrix B is ______.