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

Let α and β be the roots of the equation, 5x² + 6x – 2 = 0. If S_n = αⁿ + βⁿ, n = 1, 2, 3, ....., then ______.

If α and β be the roots of the equation x² – 2x + 2 = 0, then the least value of n for which `(α/β)^n` = 1 is ______.

The area of the region bounded by the curve y = sin x and the x-axis in [–π, π] is ______.

For the roots of the equation a – bx – x² = 0; (a > 0, b > 0), which statement is true?

If α, β are roots of the equation x² + px – q = 0 and γ, δ are roots of x² + px + r = 0, then the value of (α – y)(α – δ) is ______.

Area in first quadrant bounded by y = 4x², x = 0, y = 1 and y = 4 is ______.

The value of 'a' for which the sum of the squares of the roots of 2x² – 2(a – 2)x – a – 1 = 0 is least is ______.

Area bounded by the curves y = `"e"^(x^2)`, the x-axis and the lines x = 1, x = 2 is given to be α square units. If the area bounded by the curve y = `sqrt(ℓ "n"x)`, the x-axis and the lines x = e and x = e⁴ is expressed as (pe⁴ – qe – α), (where p and q are positive integers), then (p + q) is ______.

If b and c are odd integers, then the equation x² + bx + c = 0 has ______.

If area of the region bounded by y ≥ cot( cot^–1|In|e^|x|) and x² + y² – 6 |x| – 6|y| + 9 ≤ 0, is λπ, then λ is ______.

The area bounded by the x-axis and the curve y = 4x – x² – 3 is ______.

Area bounded by y = sec²x, x = `π/6`, x = `π/3` and x-axis is ______.

The number of integral values of m for which the equation (1 + m²)x² – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is ______.

The line segment joining the points (1, 2) and (−2, 1) is divided by the line 3x + 4y = 7 in the ratio ______.

Let the functions f: R→R and g: R→R be defined as:

f(x) = `[((x + 2",", x < 0)),((x^2",", x ≥ 0))]` and

g(x) = `{{:(x^3",", x < 1),(3x - 2",", x ≥ 1):}`

Then, the number of points in R where (fog)(x) is NOT differentiable is equal to ______.

If A = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))]`, B = `[(1, 0),(i, 1)]`, i = `sqrt(-1)` and Q = A^TBA, then the inverse of the matrix A. Q²⁰²¹ A^T is equal to ______.

The magnitude of the projection of the vector `2hati + 3hatj + hatk` on the vector perpendicular to the plane containing the vector `hati + hatj + hatk` and `hati + 2hatj + 3hatk`, is ______.

Let `veca = a_1hati + a_2hatj + a_3hatk  a_i > 0`, i = 1, 2, 3 be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of `veca` on the vector `3hati + 4hatj` be 7. Let `vecb` a vector obtained by rotating `veca` with 90°. If `veca, vecb` and x-axis are coplanar, then projection of a vector `vecb` on `3hati + 4hatj` is equal to ______.

Let `veca = hati + hatj + sqrt(2) hatk, vecb = b_1hati + b_2hatj + sqrt(2)hatk` and `vecc = 5hati + hatj + sqrt(2)hatk` be three vectors such that the projection vector of `vecb` on `veca`. If `veca + vecb` is perpendicular to `vecc`, then `|vecb|` 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 ______.