If \(\log_3 (x^2 - 1)\), \(\log_3 (2x^2 + 1)\) and \(\log_3 (6x^2 + 3)\) are the first three terms of an arithmetic progression, then the sum of the next three terms of the progression is _______________.

Five teams – A, B, C, D and E – each consisting of 15 members, are going on expeditions to five different locations. Each team includes members from three different skill sets: biologists, geologists, and explorers. However, the number of members from each skill set varies by tea…

If m and n are two positive integers such that 7m + 11n = 200, then the minimum possible value of m + n is _________.

Eight teams take part in a tournament where each team plays against every other team exactly once. In a particular year, one team got suspended after playing 3 matches, due to a disciplinary issue. The organizers decide to proceed, nonetheless, with the remaining matches. The tot…

If a, b, c are three distinct natural number, all less than 100, such that |a - b| + |b - c| = |c - a|, then the maximum possible value of b is______________.

If \(A = \begin{bmatrix} 2 & n \\ 4 & 1 \end{bmatrix}\) such that \(A^3 = 27 \begin{bmatrix} 4 & q \\ p & r \end{bmatrix}\), then \(p + q + r\) equals ______________ .

A circle of radius 13 cm touches the adjacent sides AB and BC of a square ABCD at M and N, respectively. If AB = 18 cm and the circle intersects the other two sides CD and DA at P and Q, respectively, then the area, in sq. cm, of triangle PMD is___________.

Five teams – A, B, C, D and E – each consisting of 15 members, are going on expeditions to five different locations. Each team includes members from three different skill sets: biologists, geologists, and explorers. However, the number of members from each skill set varies by tea…

If the sum of the first 21 terms of the sequence \(\ln\frac{a}{b}\), \(\ln\frac{a}{b\sqrt{b}}\), \(\ln\frac{a}{b^2}\), \(\ln\frac{a}{b^2\sqrt{b}}\), ....... is \(\ln\frac{a^m}{b^n}\), then the value of m + n is _____________.