JEE Main entrance exam Question Bank Solutions

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is ______.

Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct?

If A and B are two events such that P(A) = `1/3`, P(B) = `1/5` and P(A ∪ B) = `1/2`, then P(A|B') + P(B|A') is equal to ______.

If the shortest distance between the lines `vecr_1 = αhati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)`, λ∈R, α > 0 `vecr_2 = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`, μ∈R is 9, then α is equal to ______.

The shortest distance between the line y = x and the curve y² = x – 2 is ______.

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.

If the shortest distance between the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/λ` and `(x - 2)/1 = (y - 4)/4 = (z - 5)/5` is `1/sqrt(3)`, then the sum of all possible values of λ is ______.

The shortest distance between the z-axis and the line x + y + 2z – 3 = 0 = 2x + 3y + 4z – 4, is ______.

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 ______.

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

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 ______.

The foot of the perpendicular drawn from the origin, on the line, 3x + y = λ(λ ≠ 0) is P. lf the line meets x-axis at A and y-axis at B, then the ratio BP : PA is ______.

Choose the correct alternative:
If one of the lines given by 6x² – xy – 4cy² = 0 is 3x + 4y = 0, then c equals to ______.

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A'B (where B is the point (2, 3)) subtend angle `π/4` at the origin, is equal to ______.

If `"z"^2/(("z" - 1))` is always real, then z, can lie on ______.

Let the equation of the pair of lines, y = px and y = qx, can be written as (y – px) (y – qx) = 0. Then the equation of the pair of the angle bisectors of the lines x² – 4xy – 5y² = 0 is ______.

The pair of lines represented by 3ax² + 5xy + (a² – 2)y² = 0 are perpendicular to each other for ______.

A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point ______.

If the length of the chord of the circle, x + y² = r²(r > 0) along the line, y – 2x = 3 is r, then r² is equal to ______.

If the line 3x + 4y = m touches the circle x² + y² = 10x, then m is equal to ______.