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

The possible values of θ ∈ (0, π) such that sin (θ) + sin (4θ) + sin (7θ) = 0 are ______.

Let a, b, c be such that b(a + c) ≠ 0 if

`|(a, a + 1, a - 1),(-b, b + 1, b - 1),(c, c - 1, c + 1)| + |(a + 1, b + 1, c - 1),(a - 1, b - 1, c + 1),((-1)^(n + 2)a, (-1)^(n + 1)b, (-1)^n c)|` = 0, then the value of n is ______.

The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio ______.

`int (dx)/(e^x + e^(-x))` is equal to ______.

`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.

Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?

If `int_0^1(sqrt(2x) - sqrt(2x - x^2))dx = int_0^1(1 - sqrt(1 - y^2) - y^2/2)dy + int_1^2(2 - y^2/2)dy` + I then I equal.

If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.

Let the functions f: R→R and g: R→R be defined as:

f(x) = `[((x + 2",", x < 0)),((x^2",", x ≥ 0))]` and

g(x) = `{{:(x^3",", x < 1),(3x - 2",", x ≥ 1):}`

Then, the number of points in R where (fog)(x) is NOT differentiable is equal to ______.

The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.

The number of real roots of the equation `tan^-1sqrt(x(x + 1)) + sin^-1sqrt(x^2 + x + 1) = π/4` is ______.

Let a be a positive real number such that `int_0^ae^(x-[x])dx` = 10e – 9 where [x] is the greatest integer less than or equal to x. Then, a is equal to ______.

The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.

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?