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 ______

The number of triangles with integer sides and with perimeter 15 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

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

If \(4^{\log_2 x} - 4x + 9^{\log_3 y} - 16y + 68 = 0\), then \(y - x\) equals _______

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

The following table shows the number of employees and their median age in eight companies located in a district. Company | Number of employees | Median age A | 32 | 24 B | 28 | 30 C | 43 | 39 D | 39 | 45 E | 35 | 49 F | 29 | 54 G | 23 | 59 H | 16 | 63 It is known that the age of…

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