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

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 ______.

Let f: `R - {1/2}→R - {1/2}, f(x) = (x - 2)/(2x - 1)` be a function such that x = m is the solution of f(x) + 2f^–1(x) + 2 = f(f(x)), then m is equal to ______.

Range of 'a' for which x³ – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.

Let f(x) be a function which satisfies f(x³)f’(x) = f’(x)f’(x³) + f”(x²). Given that f(1) = 1 and f”’(1) = `1/4`, then value of 4(f’(1) + f'”(1)) is ______.

If f : R `rightarrow` R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7, then `sum_(r = 1)^nf(r)` is ______.

Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.

Let f: R → R be a function defined by:

f(x) = `{{:(max_(t ≤ x){t^3 - 3t},;, x ≤ 2),(x^2 + 2x - 6,;, 2 < x ≤ 3),([x - 3] + 9,;, 3 < x ≤ 5),(2x + 1,;, x > 5):}`

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and I = `int_-2^2f(x)dx`. Then the ordered pair (m, I) is equal to ______.

Let A = {n ∈ N: n is a 3-digit number}

B = {9k + 2: k ∈ N}

and C = {9k + ℓ: k ∈ N} for some ℓ(0 < ℓ < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then ℓ is equal to ______.