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

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 ______.

If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.

If y = f(x), f'(0) = f(0) = 1 and if y = f(x) satisfies `(d^2y)/(dx^2) + (dy)/(dx)` = x, then the value of [f(1)] is ______ (where [.] denotes greatest integer function)

The d.c's of a line whose direction ratios are 2, 3, –6, are ______.

For an increasing G.P. a₁, a₂ , a₃ ........., a_n, if a₆ = 4a₄, a₉ – a₇ = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.

The sum of infinite number of terms of a decreasing G.P. is 4 and the sum of the terms to m squares of its terms to infinity is `16/3`, then the G.P. is ______.

If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.

A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.

A complex number z is moving on `arg((z - 1)/(z + 1)) = π/2`. If the probability that `arg((z^3 -1)/(z^3 + 1)) = π/2` is `m/n`, where m, n ∈ prime, then (m + n) is equal to ______.