JEE Main entrance exam Question Bank Solutions

`tan(2tan^-1  1/5 + sec^-1  sqrt(5)/2 + 2tan^-1  1/8)` is equal to ______.

The set of all values of k for which (tan^–1 x)³ + (cot^–1 x)³ = kπ³, x ∈ R, is the internal ______.

If `cos^-1(2/(3x)) + cos^-1(3/(4x)) = π/2(x > 3/4)`, then x is equal to ______.

If the solution curve of the differential equation `(dy)/(dx) = (x + y - 2)/(x - y)` passes through the point (2, 1) and (k + 1, 2), k > 0, then ______.

Let `{a_n}_(n = 0)^∞` be a sequence such that a₀ = a₁ = 0 and a_n+2 = 2a_n+1 – a_n + 1 for all n ≥ 0. Then, `sum_(n = 2)^∞ a^n/7^n` is equal to ______.

Let y = y(x) be the solution of the differential equation `(dy)/(dx) + (sqrt(2)y)/(2cos^4x - cos2x) = xe^(tan^-1(sqrt(2)cost2x)), 0 < x < π/2` with `y(π/4) = π^2/32`. If `y(π/3) = π^2/18e^(-tan^-1(α))`, then the value of 3α² is equal to ______.

The sum of the infinite series `1 + 5/6 + 12/6^2 + 22/6^3 + 35/6^4 + 51/6^5 + 70/6^6 + ....` is equal to ______.

If y = y(x) is the solution of the differential equation `(1 + e^(2x))(dy)/(dx) + 2(1 + y^2)e^x` = 0 and y(0) = 0, then `6(y^'(0) + (y(log_esqrt(3))))^2` is equal to ______.

The population P = P(t) at time 't' of a certain species follows the differential equation `("dp")/("dt")` = 0.5P – 450. If P(0) = 850, then the time at which population becomes zero is ______.

If `e^((cos^2x + cos^4x + cos^6x + ...∞)log_e2` satisfies the equation t² – 9t + 8 = 0, then the value of `(2sinx)/(sinx + sqrt(3)cosx)(0 < x ,< π/2)` is ______.

If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p² + q²) is equal to ______.

Let y = y(x) be the solution of the differential equation `xtan(y/x)dy = (ytan(y/x) - x)dx, -1 ≤ x ≤ 1, y(1/2) = π/6`. Then the area of the region bounded by the curves x = 0, x = `1/sqrt(2)` and y = y(x) in the upper half plane is ______.

Let y = y(x) be the solution of the differential equation `e^xsqrt(1 - y^2)dx + (y/x)dy` = 0, y(1) = –1. Then, the value of (y(3))² is equal to ______.

If the sum of an infinite GP a, ar, ar², ar³, ...... . is 15 and the sum of the squares of its each term is 150, then the sum of ar², ar⁴, ar⁶, .... is ______.

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.

Let y = y(x) be the solution of the differential equation, `(2 + sinxdy)/(y + 1) (dy)/(dx)` = –cosx. If y > 0, y(0) = 1. If y(π) = a, and `(dy)/(dx)` at x = π is b, then the ordered pair (a, b) is equal to ______.

Let y = y(x) be the solution of the differential equation, `(x^2 + 1)^2 ("dy")/("d"x) + 2x(x^2 + 1)"y"` = 1, such that y(0) = 0. If `sqrt("ay")(1) = π/32` then the value of  'a' is ______.

The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are ______.

Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.

If z₁, z₂, z₃ are complex numbers such that |z₁| = |z₂| = |z₃| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z₁ + z₂ + z₃| is ______.