JEE Main entrance exam Question Bank Solutions

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 ______.

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 ______.