When Geeta increases her speed from 12 km/hr to 20 km/hr, she takes one hour less than the usual time to cover the distance between her home and office. The distance between her home and office is ________ km.

Given that \(f(x) = |x| + 2|x - 1| + |x - 2| + |x - 4| + |x - 6| + 2|x - 10|\), \(x \in (-\infty, \infty)\), the minimum value of \(f(x)\) is _________.

Let 50 distinct positive integers be chosen such that the highest among them is 100, and the average of the largest 25 integers among them exceeds the average of the remaining integers by 50. Then the maximum possible value of the sum of all the 50 integers is _________.

The numbers \(-16,\ 2^{x+3}-2^{2x-1}-16,\ 2^{2x-1}+16\) are in an arithmetic progression. Then x equals _________.

Mrs and Mr Sharma and Mrs and Mr Ahuja along with four other persons are to be seated at a round table for dinner. If Mrs and Mr Sharma are to be seated next to each other, and Mrs and Mr Ahuja are not to be seated next to each other, then the total number of seating arrangements…

If \(A=\begin{bmatrix}1&0&0\\0&0&1\\0&1&0\end{bmatrix}\), then the absolute value of the determinant of \((A^0 + A^6 + A^3 + A)\) is ____________.

The sum of the squares of all the roots of the equation \(x^2 + |x + 4| + |x - 4| - 35 = 0\) is

A new sequence is obtained from the sequence of positive integers (1, 2, 3, ...) by deleting all the perfect squares. Then the 2022nd term of the new sequence is ___________.

Let \(A = \{1, 2, 3\}\) and \(B = \{a, b\}\). Assuming all relations from set A to set B are equally likely, what is the probability that a relation from A to B is also a function?