Practice IPMAT Indore 2022 Question Paper with Solutions

If \(\log _{\left(x^{2}\right)} y+\log _{\left(y^{2}\right)} x\) \({}= 1\) and \(y=x^{2}-30\), then the value of \(x^{2}+y^{2}\) is ___________.

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

When Geeta increases her speed from \(12\) km/hr to \(20\) km/hr, she takes one hour less than the usual time to cover the distance between her home and office. The distance between her home and office is ___________ km.

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 _________.

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 _________.

Given that \(f(x) = |x| +2|x -1|\) \({}+|x -2| +|x -4|\) \({}+|x -6| +2|x -10|,\) \(x \in(-\infty, \infty)\) the minimum value of \(f(x)\) is _________.

If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\), then the absolute value of the determinant of \(\left(A^{9}+A^{6}+A^{3}+A\right)\) is __________.

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

A new sequence is obtained from the sequence of positive integers \((1,2,3, \ldots)\) by deleting all the perfect squares. Then the \(2022^{\text {nd }}\) term of the new sequence 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 _________.