JEE Main entrance exam Question Bank Solutions

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 ______.

The statement p→(q→p) is equivalent to ______.

If in a geometric progression {a_n}, a₁ = 3, a_n = 96 and S_n = 189, then the value of n is ______.

If log_α8 = γ, log_βα = –1 and log_1/4β = –1 then `(1/α + 1)^(log_sqrt(5)(β^2 + 4γ^2)` is equal to ______.

`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.

The statement p→(q→p) is equivalent to ______.

Let x₁ = 97, x₂ = `2/x_1`, x₃ = `3/x_2`, x₄ = `4/x_3`, ......, x₈ = `8/x_7` then `log_(3sqrt(2))(prod_(i = 1)^8x_i - 60)` = ______.

If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.

The solution of the differential equation `(1 + y^2) + (x - e^(tan^-1y)) (dy)/(dx)` = 0, is ______.

The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.

Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.

If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.

If some three consecutive coefficients in the binomial expansion of (x+ 1)ⁿ in powers of x are in the ratio 2:15:70, then the average of these three coefficients is ______.

Let A₁, A₂, A₃, .... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = `1/1296` and A₂ + A₄ = `7/36`, then the value of A₆ + A₈ + A₁₀ is equal to ______.