AIEEE 2010 PYQ Mock Test

The number of $3\times 3$ non-singular matrices, with four entries as $1$ and all other entries as $0$, is:

Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\dots ,20\}$.Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$.Statement-2: If the four chosen numbers form an AP, then the set of all possible values of common difference is $\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\}$.

Statement-1: The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$.Statement-2: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.

Let $S_1={\sum }_{j=1}^{10}j(j-1)\left(\genfrac{}{}{0ex}{}{10}{j}\right)$, $S_2={\sum }_{j=1}^{10}j\left(\genfrac{}{}{0ex}{}{10}{j}\right)$ and $S_3={\sum }_{j=1}^{10}{j}^2\left(\genfrac{}{}{0ex}{}{10}{j}\right)$.Statement-1: $S_3=55\times 2^9$ Statement-2: $S_1=90\times 2^8$ and $S_2=10\times 2^8$.

Let $A$ be a $2\times 2$ matrix with non-zero entries and let $A^2=I$.Statement-1: $\text{Tr}(A)=0$ Statement-2: $|A|=1$