IPM Indore 2021 Question Paper

The number of positive integers which divide (1890) ∙ (130) ∙ (170) and are not divisible by 45 is ____________.

The sum up to 10 terms of the series 1.3 + 5.7 + 9.11 + ... is ____________

It is given that the sequence $${x_{n}}$$ satisfies $$x_{1} = 0, x_{n+1} = x_{n} + 1 + 2\sqrt{1+ x_{n}}$$ for 𝑛 = 1, 2, ..... Then $$x_{31}$$ is ______________.

There are 5 parallel lines on the plane. On the same plane, there are n other lines which are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of n?

There are two taps, T1 and T2, at the bottom of a water tank, either or both of which may be opened to empty the water tank, each at a constant rate. If T1 is opened keeping T2 closed, the water tank (initially full) becomes empty in half an hour. If both T1 and T2 are kept open, the water tank (initially full) becomes empty in 20 minutes. Then, the time (in minutes) it takes for the water tank (initially full) to become empty if T2 is opened while T1 is closed is ____________.

A class consists of 30 students. Each of them has registered for 5 courses. Each course instructor conducts an exam out of 200 marks. The average percentage marks of all 30 students across all courses they have registered for, is 80%. Two of them apply for revaluation in a course. If none of their marks reduce, and the average of all 30 students across all courses becomes 80.02%, the maximum possible increase in marks for either of the 2 students is

What is the minimum number of weights which enable us to weigh any integer number of grams of gold from 1 to 100 on a standard balance with two pans? (Weights can be placed only on the left pan.)

If one of the lines given by the equation $$2x^{2} + axy + 3y^{2} = 0$$ coincides with one of those given by $$2x^{2} + bxy — 3𝑦^{2} = 0$$, and the other lines represented by them are perpendicular then $$a^{2} + b^{2}$$ is .

If a function f(a) = max (a, 0) then the smallest integer value of x for which the equation f(x — 3) + 2𝑓(x + 1) = 8 holds true is _______________.

In a class, 60% and 68% of students passed their Physics and Mathematics examinations respectively. Then at least ___________percentage of students passed both their Physics and Mathematics examinations.