IPM Indore 2019

Let $$\alpha$$, $$\beta$$ be the roots of $$x^{2} - x + p = 0$$ and $$\gamma$$, $$\delta$$ be the roots of $$x^{2}- 4x + q = 0$$ where p and q are integers. If $$\alpha, \beta, \gamma, \delta$$ are in geometric progression then p + q is

If $$(1 + x - 2x^{2})^{6} = A_{0} + \sum_{r = 1}^{12} A_{r}X^{r}$$, then value of $$A_{2} + A_{4} + A_{6}.... + A_{12}$$ is

The number of terms common to both the arithmetic progressions 2,5,8,11,...., 179 and 3,5,7,9,....., 101 is

From a pack of 52 cards, we draw one by one, without replacement. If f(n) is the probability that an Ace will appear at the $$n^{th}$$ turn, then

A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is

In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2,4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to

The value of $$(\log_{3}30)^{−1}+(\log_{4}900)^{−1}+(\log_{5}30)^{−1}$$ is

The inequality $$\log_{a} f(x)<\log_{a} g(x)$$ implies that

Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 $$cm^{2}$$ Then the possible values of the sum of their volumes are

Determine the greatest number among the following four numbers