If the polynomial \(ax^2 + bx + 5\) leaves a remainder 3 when divided by \(x - 1\), and a remainder 2 when divided by \(x + 1\), then \(2b - 4a\) equals _____________.

English exam and Math exam were conducted separately for a class of 120 students. The number of students who did not appear for the English exam is twice the number of students who did not appear for the Math exam. The number of students who passed the Math exam is twice the numb…

The number of factors of \(3^5 \times 5^8 \times 7^2\) that are perfect squares 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…

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

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…

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 \(\log_3 (x^2 - 1)\), \(\log_3 (2x^2 + 1)\) 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 _______________.