JEE Main entrance exam Question Bank Solutions

Let a vector `veca` be coplanar with vectors `vecb = 2hati + hatj + hatk` and `vecc = hati - hatj + hatk`. If `veca` is perpendicular to `vecd = 3hati + 2hatj + 6hatk` and `|veca| = sqrt(10)`. Then a possible value of `[(veca, vecb, vecc)] + [(veca, vecb, vecd)] + [(veca, vecc, vecd)]` is equal to ______.

If the vectors, `vecp = (a + 1)hati + ahatj + ahatk, vecq = ahati + (a + 1)hatj + ahatk` and `vecr = ahati + ahatj + (a + 1)hatk  (a ∈ R)` are coplanar and `3(vecp.vecq)^2 - λ|vecr xx vecq|^2` = 0, then the value of λ is ______.

Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.

The sum of 162^th power of the roots of the equation x³ – 2x² + 2x – 1 = 0 is ______.

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^–1 (x) – 2sin^–1(x) = cos^–1 (2x) is equal to ______.

Given that the inverse trigonometric function take principal values only. Then, the number of real values of x which satisfy `sin^-1((3x)/5) + sin^-1((4x)/5) = sin^-1x` is equal to ______.

Let A = `[(2, -1),(0, 2)]`. If B = I – ⁵C₁(adj A) + ⁵C₂(adj A)⁵, –... –⁵C₅(adj A)⁵, then the sum of all elements of the matrix B is ______.

Let f(x) = `cos(2tan^-1sin(cot^-1sqrt((1 - x)/x))), 0 < x < 1`. Then ______.

If the orthocentre of the triangle formed by (1, 3) (4, –5) and (a, b) is (2, 4), Then the value of 33b + 22a is ______.

If the vertices of a triangle be (0, 0), (6, 0) and (6, 8), then its incentre will be ______.

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

If A = `[(-2, 1),(0, 3)]` Then 2A² – 3A ______.

The ratio in which the segment joining the points (2, 4, 5), (3, 5, –4) is divided by the yz-plane 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 A is a square matrix such that A(adjA) = `[(3, 0, 0),(0, 3, 0),(0, 0, 3)]`, then `(|adj(adjA)|)/|adjA|` is equal to ______.

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 ______.

The largest interval lying in `((-π)/2, π/2)` for which the function, f(x) = `4^(-x^2) + cos^-1(x/2 - 1) + log(cosx)`, is defined, is ______.