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

The orthocentre of the triangle formed by the lines 4x – 7y + 10 = 0, x + y = 5 and 7x + 4y = 15, is ______.

The coordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2) then the coordinates of its centroid are ______.

If 3x + y = 0 is a tangent to the circle with centre at the point (2, –1), then the equation of the other tangent to the circle from the origin is ______.

If H is the orthocentre of the triangle ABC, then AH is equal to ______.

If α, β, γ, δ are roots of x⁴ – 100x³ + 2x² + 4x + 10 = 0, then `1/α + 1/β + 1/γ + 1/δ` is equal to ______.

Let α, β be the roots of the equation `x^2 - sqrt(2)x + sqrt(6)` = 0 and `1/α^2 + 1, 1/β^2 + 1` be the roots of the equation x² + ax + b = 0. Then the roots of the equation x² – (a + b – 2)x + (a + b + 2) = 0 are ______.

Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point ______.

S = `tan^-1(1/(n^2 + n + 1)) + tan^-1(1/(n^2 + 3n + 3)) + ... + tan^-1(1/(1 + (n + 19)(n + 20)))`, then tan S is equal to ______.

The area enclosed between the curve y = log_e(x + e) and the coordinate axes is ______.

The area (in sq.units) of the part of the circle x² + y² = 36, which is outside the parabola y² = 9x, is ______.

The probability of selecting integers a ∈ [–5, 30] such that x² + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is ______.

If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1) is equal to ______.

The sum of all integral values of k(k ≠ 0) for which the equation `2/(x - 1), 1/(x - 2) = 2/k` in x has no real roots, is ______.

If a function f(x) defined by

f(x) = `{{:("ae"^x + "be"^-x",", -1 ≤ x < 1),("c"x^2",", 1 ≤ x ≤ 3),("a"x^2 + 2"c"x",", 3 < x ≤ 4):}`

be continuous for some a, b, c ε R and f'(0) + f'(2) = e, then the value of a is ______.

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 ______.