JEE Main entrance exam Question Bank Solutions

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 ______.

The length of the perpendicular from the point (2, –1, 4) on the straight line `(x + 3)/10 = ("y" - 2)/(-7) = "z"/1`, 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 ______.

Consider a triangle ABC whose vertices are A(0, α, α), B(α, 0, α) and C(α, α, 0), α > 0. Let D be a point moving on the line x + z – 3 = 0 = y and G be the centroid of ΔABC. If the minimum length of GD is `sqrt(57/2)`, then α is equal to ______.

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

 ______.