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

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 ______. 

If the expansion in powers of x of the function `1/((1 - ax)(1 - bx))` is a₀ + a₁x + a₂x² + a₃x³ ....... then a_n is ______.

If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l² + m² + cnl = 0 are parallel, then the positive value of c is ______.

A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval ______.

The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.

Let α and β be the roots of the equation x² + (2i – 1) = 0. Then, the value of |α⁸ + β⁸| is equal to ______.

Let arg (z) represent the principal argument of the complex number z. Then, |z| = 3 and arg (z – 1) – arg (z + 1) = `π/4`intersect ______.

If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.

Let three vectors `veca, vecb` and `vecc` be such that `vecc` is coplanar with `veca` and `vecb, vecc,` = 7 and `vecb` is perpendicular to `vecc` where `veca = -hati + hatj + hatk` and `vecb = 2hati + hatk`, then the value of `2|veca + vecb + vecc|^2` is ______.

Let a complex number z, |z| ≠ 1, satisfy `log_(1/sqrt(2))((|z| + 11)/(|z| - 1)^2) ≤ 2`. Then, the largest value of |z| is equal to ______.

Let z and ω be two complex numbers such that ω = `zbarz - 2z + 2,|(z + i)/(z - 3i)|` = 1 and Re(ω) has minimum value. Then, the minimum value of n∈N for which ωⁿ is real, is equal to ______.

If z and ω are two complex numbers such that |zω| = 1 and arg(z) – arg(ω) = `(3π)/2`, then `"arg"((1 - 2barzω)/(1 + 3barzω))` is ______. (Here arg(z) denotes the principal argument of complex number z)

The equation arg `((z - 1)/(z + 1)) = π/4` represents a circle with ______. 

Let `veca = hati + hatj + hatk` and `vecb = hatj - hatk`. If `vecc` is a vector such that `veca.vecc = vecb` and `veca.vecc` = 3, then `veca.(vecb.vecc)` is equal to ______.