IPMAT Indore 2026 Question Paper

The number of values $$a$$ can take such that $$x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4$$ can be expressed as a product of two quadratic polynomials, $$x^2 + px + 2$$ and $$x^2 + qx - 2$$, where $$p$$ and $$q$$ are real, is

Let $$S = \{1, 2, \ldots, 180\}$$. Define $$A$$ as the set of all multiples of 4 in $$S$$, $$B$$ as the set of all multiples of 6 in $$S$$, and $$C$$ as the set of all multiples of 9 in $$S$$. The number of elements in $$S$$ that belong to exactly one of $$A, B, C$$ is

If $$m$$ is a positive integer then the values of $$k$$ for which $$6m + k$$ cannot be a perfect square are

Let ABCD be a rectangle with AB = 72 cm and BC = 30 cm. A circle passing through points A and C cuts the side AB at P such that AP = 56 cm. The radius, in cm, of the circle is

Two locations A and B are at diametrically opposite ends of a circular track. Rekha starts running along the track from location A in the clockwise direction. Sajal starts running simultaneously along the track in the anticlockwise direction from location B. If the length of the circular track is 14 km, and the speeds of Rekha and Sajal are in the ratio 5: 2, then the distance, in km, travelled by Rekha, when they meet at location B for the first time, is

The number of integer solutions $$(x, y)$$ of the inequality $$x^2 + y^2 \leq 10$$ is

The possible values of $$x$$ in the set $$\{1, 5, 13\}$$ for which the mean of eight observations $$5, 8, 3x + 2, 15, 27, 29, 36, 5x - 2$$ equals their median are

Painter A can paint a building in 12 days while Painter B can paint it in 16 days. If A and B work on alternate days, and A starts the work on the first day, then the number of days required to paint the building is

The equation $$2^x - x^2 = 0$$ has

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is