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

The number of real solutions of the equation `(9/10)^x` = –3 + x – x² is ______.

If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.

Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos  (α - β)/2` is ______.

8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do no occupy odd places is ______.

Let a, b and c be the length of sides of a triangle ABC such that `(a + b)/7 = (b + c)/8 = (c + a)/9`. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` is equal to ______.

If in a ΔABC, the altitudes from the vertices A, B, C on opposite sides are in H.P, then sin A, sin B, sin C are in ______

If the coefficients of x⁷ and x⁸ in `2 + x^n/3` are equal, then n is ______.

The number of elements in the set S = {θ∈ [–4π, 4π]: 3cos²2θ + 6cos²θ – 10cos²θ + 5 = 0} is ______.

The number of solutions of the equation `2θ - cos^2θ + sqrt(2)` = 0 in R is equal to ______.

The number of solutions of the equation sinx = cos²x in the interval (0, 10) is ______.

Let the coefficients of x^–1 and x^–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn² = ¹⁵C_r, 2^r, then the value of r is equal to ______.

The number of roots of the equation, `(81)^(sin^2x) + (81)^(cos^2x)` = 30 in the interval [0, π] is equal to ______.

If for `x∈(0, π/2), log_10sinx + log_10cosx` = –1 and `log_10(sinx + cosx) = 1/2(log_10n - 1), n > 0`, then the value of n is equal to ______.

If the normal to the curve y(x) = `int_0^x(2t^2 - 15t + 10)dt` at a point (a, b) is parallel to the line x + 3y = –5, a > 1, then the value of |a + 6b| is equal to ______.

If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC is  ______.

If α = `cos^-1(3/5)`, β = `tan^-1(1/3)`, where 0 < α, β < `π/2`, then α – β is equal to ______.

If the sum of all solutions of equation sinx + 2cosx = `1 + sqrt(3)cosx` in [0, 2π] is `(kπ)/6`. The value of k is ______.

If the number of five-digit numbers with distinct digits and 2 at the 10^th place is 336 k, then k is equal to ______.

`d/(dx)x^(logx)` = ______.

Let y = sin²θ + cos²θ + tan²θ + sec²θ + cosec²θ + cot²θ attains its least value (where θ ∈ [0, 4π]), then number of such possible values of θ is ______.