JEE Main entrance exam Question Bank Solutions

Let the tangent to the circle C₁: x² + y² = 2 at the point M(–1, 1) intersect the circle C₂: (x – 3)² + (y – 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to ______.

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 16. Let T be a plane passing through the point Q and contains the line `vecr = -hatk + λ(hati + hatj + 2hatk)`, λ ∈ R. Then, which of the following points lies on T?

Let P be a plane Ix + my + nz = 0 containing the line, `(1 - x)/1 = ("y" + 4)/2 = ("z" + 2)/3`. If plane P divides the line segment AB joining points A(–3, –6, 1) and B(2, 4, –3) in ratio k:1 then the value of k is equal to ______.

Let ABCD be a square of side of unit length. Let a circle C₁ centered at A with unit radius is drawn. Another circle C₂ which touches C₁ and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle C₂ meet the side AB at E. If the length of EB is `α + sqrt(3)β`, where α, β are integers, then α + β is equal to ______.

Let P be a plane passing through the points (1, 0, 1), (1, –2, 1) and (0, 1, –2). Let a vector `vec"a" = αhat"i" + βhat"j" + γhat"k"` be such that `veca` is parallel to the plane P, perpendicular to `(hat"i"+2hat"j"+3hat"k")`and `vec"a".(hat"i" + hat"j" + 2hat"j")` = 2, then (α – β + γ)² equals ______.

The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² – 2x – 4y + 4 = 0 at two distinct points is ______.

The sum of the squares of the lengths of the chords intercepted on the circle, x² + y² = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers is ______.

The Probability that A speaks truth is `3/4` and that of B is `4/5`. The probability that they contradict each other in stating the same fact is p, then the value of 40p is ______.

Let P denotes the probability of selecting one white and one black square from the chessboard so that they are not in the same row and also not in the same column (an example of this kind of the choice is shown in figure), then (1024)P is ______.

A speaks truth in 75% of the cases and B in 80% of the cases. The percentage of cases they are likely to contradict each other in making the same statement is ______.

If x² + y² + sin y = 4, then the value of `(d^2y)/(dx^2)` at the point (–2, 0) is ______.

Let for i = 1, 2, 3, p_i(x) be a polynomial of degree 2 in x, p'_i(x) and p''_i(x) be the first and second order derivatives of p_i(x) respectively. Let,

A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`

and B(x) = [A(x)]^T A(x). Then determinant of B(x) ______

The probability that A speaks truth is `4/5`, while the probability for B is `3/4`. The probability that they contradict each other when asked to speak on a fact is ______.

If a, b, c, d > 0 such that a + 2b + 3c + 4d = 50, then the maximum value of `((a^2b^4c^3d)/16)^(1/10)` is equal to ______.

If λ ∈ R is such that the sum of the cubes of the roots of the equation, x² + (2 – λ)x + (10 – λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is ______.

If `|z - 4/z|` = 2, then the maximum value of |z| is equal to ______.

lf x is real, the maximum value of `(3x^2 + 9x + 17)/(3x^2 + 9x + 7)` is ______.

Let A = `[(2, 3),(a, 0)]`, a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew-symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to ______.

If ax⁴ + bx³ + cx² + dx + e = `|(2x, x - 1, x + 1),(x + 1, x^2 - x, x - 1),(x - 1, x + 1, 3x)|`, then the value of e is ______.

Let a real valued function f(x) satisfying f(x + y) + f(x – y) = f(x)f(y) {f(0) ≠ 0} ∀ x, y ∈ R, then f(–2) – f(–1) + f(0) + f(1) – f(2) is equal to ______.