JEE Main entrance exam Question Bank Solutions

If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.

If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.

If `lim_(x→∞) 1/(x + 1) tan((πx + 1)/(2x + 2)) = a/(π - b)(a, b ∈ N)`; then the value of a + b is ______.

If `θ∈[(5π)/2, 3π]` and 2cosθ + sinθ = 1, then the value of 7cosθ + 6sinθ is ______.

If `lim_(n→∞)sum_(k = 2)^ncos^-1(1 + sqrt((k - 1)(k + 2)(k + 1)k)/(k(k + 1))) = π/λ`, then the value of λ is ______.

If sinθ = `1/sqrt(2)` and `π/2 < θ < π`. Then the value of `(sinθ + cosθ)/tanθ` is ______.

The statement `∼ ("p"leftrightarrow∼"q")` is ______.

If cosec θ = `("p" + "q")/("p" - "q")` (p ≠ q ≠ 0), then `|cot(π/4 + θ/2)|` is equal to ______.

In the expansion of `(x/cosθ + 1/sinθ)^16`. If l₁ is the least value of the term independent of x when `π/8 ≤ θ ≤ π/4` and l₂ is the least value of the term independent of x when `π/16 ≤ θ ≤ π/8`, then the ratio l₂ : l₁ is equal to ______.

Let Δ, ∇ ∈ {∧, ∨} be such that p ∇ q ⇒ ((p ∇ q) ∇ r) is a tautology. Then (p ∇ q) Δ r is logically equivalent to ______.

Let A = `[(1, -1),(2, α)]` and B = `[(β, 1),(1, 0)]`, α, β ∈ R. Let α₁ be the value of α which satisfies (A + B)² = `A^2 + [(2, 2),(2, 2)]` and α₂ be the value of α which satisfies (A + B)² = B² . Then |α₁ – α₂| is equal to ______.

The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.

Let the solution curve of the differential equation `x (dy)/(dx) - y = sqrt(y^2 + 16x^2)`, y(1) = 3 be y = y(x). Then y(2) is equal to ______.

If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to ______.

The shortest distance between the line y = x and the curve y² = x – 2 is ______.

If 2y = `(cot^-1((sqrt3cosx + sinx)/(cosx - sqrt3 sinx)))^2`, x ∈ `(0, π/2)` then `dy/dx` is equal to ______.

If D, E, F are the mid points of the sides BC, CA and AB respectively of a triangle ABC and 'O' is any point, then, `|vec(AD) + vec(BE) + vec(CF)|`, is ______.

The 4^th term from the end in the expansion of `(x^3/2 - 2/x^2)^7` is ______.

In the binomial expansion of `(root(3)(2) + 1/root(3)(3))^n`, the ratio of the 7^th term from the beginning to the 7^th term from the end is 1:6; n is ______.

If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)