JEE Main entrance exam Question Bank Solutions

A circle touches two of the smaller sides of a ΔABC (a < b < c) and has its centre on the greatest side. Then the radius of the circle is ______.

If ^m+nP₂ = 90 and ^m–nP₂ = 30, then (m, n) is given by ______.

Let x₁, x₂, x₃ be the solutions of `tan^-1((2x + 1)/(x + 1)) + tan^-1((2x - 1)/(x - 1))` = 2tan^–1(x + 1) where x₁ < x₂ < x₃ then 2x₁ + x₂ + x₃² is equal to ______.

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio ______.

In a ΔABC, let BC = 3. D is a point on BC such that BD = 2, Then the value of AB² + 2AC² – 3AD² is ______.

In usual notation a ΔABC, if A, A₁, A₂, A₃ be the area of the in-circle and ex-circles, then `1/sqrt(A_1) + 1/sqrt(A_2) + 1/sqrt(A_3)` is equal to ______.

In an equilateral triangle of side `2sqrt(3)` cm, the circum radius is ______.

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 ______.