Aruna purchases a certain number of apples for INR 20 each and a certain number of mangoes for INR 25 each. If she sells all the apples at 10% profit and all the mangoes at 20% loss, overall she makes neither profit nor loss. Instead, if she sells all the apples at 20% loss and all the mangoes at 10% profit, overall she makes a loss of INR 150. Then the number of apples purchased by Aruna is _________.

The sum of the coefficients of all the terms in the expansion of \((5x - 9)^4\) is __________.

If \(\log_{x^2} y + \log_{y^2} x = 1\) and \(y = x^2 - 30\). Then the value of \(x^2 + y^2\) is ___________.

If \(\sin\alpha + \sin\beta = \frac{\sqrt{2}}{\sqrt{3}}\) and \(\cos\alpha + \cos\beta = \frac{1}{\sqrt{3}}\), then the value of \(\left(20\cos\left(\frac{\alpha-\beta}{2}\right)\right)^2\) is.

Let P(X) denote power set of a set X. If A is the null set, then the number of elements in P(P(P(P(A)))) is _____________.

The number of triangles that can be formed by choosing points from 7 points on a line and 5 points on another parallel line 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 ___________.

The area enclosed by \(2|x| + 3|y| \le 6\) is ____________ sq. units.

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 ____________.