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

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 ______.

Let `hata` and `hatb` be two unit vectors such that the angle between them is `π/4`. If θ is the angle between the vectors `(hata + hatb)` and `(hata xx 2hatb + 2(hata xx hatb))`, then the value of 164 cos²θ is equal to ______.

Let `veca, vecb, vecc` be three vectors mutually perpendicular to each other and have same magnitude. If a vector `vecr` satisfies. `veca xx {(vecr - vecb) xx veca} + vecb xx {(vecr - vecc) xx vecb} + vecc xx {(vecr - veca) xx vecc} = vec0`, then `vecr` is equal to ______.

If the vector `vecb = 3hatj + 4hatk` is written as the sum of a vector `vec(b_1)`, parallel to `veca = hati + hatj` and a vector `vec(b_2)`, perpendicular to `veca`, then `vec(b_1) xx vec(b_2)` is equal to ______.

Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals ______.

If for a > 0, the feet of perpendiculars from the points A(a, –2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, –a, –1) and D respectively, then the length of line segment CD is equal to ______.

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 ______.