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

Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P(exactly one of A, B occurs) = `5/9`, is ______.

If y = y(x) is an implicit function of x such that log_e(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.

Four fair dice are thrown simultaneously. If the probability that the highest number obtained is 4 is `(25a)/1296` then 'a' is equal to ______.

If 2^x + 2^y = 2^x+y, then `(dy)/(dx)` is equal to ______.

Let y = y(x) be a function of x satisfying `ysqrt(1 - x^2) = k - xsqrt(1 - y^2)` where k is a constant and `y(1/2) = -1/4`. Then `(dy)/(dx)` at x = `1/2`, is equal to ______.

If the system of linear equations

2x + y – z = 7

x – 3y + 2z = 1

x + 4y + δz = k, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to ______.

Let d be the distance between the foot of perpendiculars of the points P(1, 2, –1) and Q(2, –1, 3) on the plane –x + y + z = 1. Then d² is equal to ______.

Let `(x - 2)/3 = (y + 1)/(-2) = (z + 3)/(-1)` lie on the plane px – qy + z = 5, for p, q ∈ R. The shortest distance of the plane from the origin is ______.

The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, is ______.

The system of linear equations

3x – 2y – kz = 10

2x – 4y – 2z = 6

x + 2y – z = 5m

is inconsistent if ______.

Let A = `[(i, -i),(-i, i)], i = sqrt(-1)`. Then, the system of linear equations `A^8[(x),(y)] = [(8),(64)]` has ______.

Let P = `[(-30, 20, 56),(90, 140, 112),(120, 60, 14)]` and A = `[(2, 7, ω^2),(-1, -ω, 1),(0, -ω, -ω + 1)]` where ω = `(-1 + isqrt(3))/2`, and I₃ be the identity matrix of order 3. If the determinant of the matrix (P^–1AP – I₃)² is αω², then the value of α is equal to ______.

Let `θ∈(0, π/2)`. If the system of linear equations,

(1 + cos²θ)x + sin²θy + 4sin3θz = 0

cos²θx + (1 + sin²θ)y + 4sin3θz = 0

cos²θx + sin²θy + (1 + 4sin3θ)z = 0

has a non-trivial solution, then the value of θ is

 ______.

A plane P contains the line x + 2y + 3z + 1 = 0 = x – y – z – 6, and is perpendicular to the plane –2x + y + z + 8 = 0. Then which of the following points lies on P?

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point ______.

The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.

The equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) is ______.

If x = `a[cosθ + logtan  θ/2]`, y = asinθ then `(dy)/(dx)` = ______.

The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is ______.

If the following equations

x + y – 3 = 0 

(1 + λ)x + (2 + λ)y – 8 = 0

x – (1 + λ)y + (2 + λ) = 0

are consistent then the value of λ can be ______.