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

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)

Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x²dy = (2xy + y²)dx, then `f(1/2)` is equal to ______.

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.

The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is ______.

Let [x] denote greatest integer less than or equal to x. If for n ∈ N, (1 – x + x³)ⁿ = `sum_("j" = 0)^(3"n")"a"_"j"x^"j"`, then `sum_("j" = 0)^([(3"n")/2]) "a"_(2"j") + 4sum_("j" = 0)^([(3"n" - 1)/2])"a"_(2"j" + 1)` is equal to ______.

Let ABC be a triangle with A(–3, 1) and ∠ACB = θ, 0 < θ < `π/2`. If the equation of the median through B is 2x + y – 3 = 0 and the equation of angle bisector of C is 7x – 4y – 1 = 0, then tan θ is equal to ______.