Algebra - Mathematics Previous Year Questions

Five persons entered the lift cabin on the ground floor of an eight floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is

If the real numbers \(x, y, z, t\) be in GP then the value of \((x^2+y^2+z^2)(y^2+z^2+t^2)\) is euqal to

If matrix \(A=\left[\begin{array}{ccc}0 & 2 b & -2 \\ 3 & 1 & 3 \\ 3 a & 3 & -1\end{array}\right]\) is given to be symmetric, then the value of \(a b\) is

If $a, b$ and $c$ are distinct positive numbers, not equal to unity. Such that $a b c=1$, then the value of $\log _b a \cdot \log _c a+\log _c b$ $\cdot \log _a b+\log _a c \log _b c$ is

A bag contains 50 tickets numbered \(1,2,3, ..., 50\) of which five are drawn at random and arranged in ascending order of magnitude \(\left(x_1 < x_2 < x_3 < x_4< x_5\right)\), then the probability that $x_3=30$ is

The value of \(\frac{4}{1 !}+\frac{11}{2 !}+\frac{22}{3 !}+\frac{37}{4 !}+\frac{56}{5 !}+\ldots \infty\) is