JEE Main entrance exam Question Bank Solutions

The value of `((1 + sin  (2π)/9 + icos  (2π)/9)/(1 + sin  (2π)/9 - icos  (2pi)/9))^3` is ______.

The integral `int_0^2||x - 1| -x|dx` is equal to ______.

If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = log_ex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.

`int_0^1|3x - 1|dx` equals ______.

`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.

If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.

For three vectors `vecu, vecv, vecw` which of the following expressions is not equal to any of the remaining three?

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 ______.