If \(|x+1| + (y+2)^2 = 0\) and \(ax - 3ay = 1\), then the value of a is

The angle of elevation of the top of a pole from a point A on the ground is \(30^\circ\). The angle of elevation changes to \(45^\circ\), after moving 20 metres towards the base of the pole. Then the height of the pole, in metres, is

If \(\log_4 x = a\) and \(\log_{25} x = b\) then \(\log_x 10\) is

If \(A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}\) is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of A is ______

Let ABC be an equilateral triangle, with each side of length k. If a circle is drawn with diameter AB, then the area of the portion of the triangle lying inside the circle is

The number of triangles with integer sides and with perimeter 15 is _______

Let ABC be a triangle with AB = AC and D be a point on BC such that \(\angle BAD = 30^\circ\). If E is a point on AC such that AD = AE, then \(\angle CDE\) equals

Person A borrows Rs. 4000 from another person B for a duration of 4 years. He borrows a portion of it at 3% simple interest per annum, while the rest at 4% simple interest per annum. If B gets Rs. 520 as total interest, then the amount A borrowed at 3% per annum in Rs. is _______

If 5 boys and 3 girls randomly sit around a circular table, the probability that there will be at least one boy sitting between any two girls, is