IPMAT Indore 2026 Question Paper with Answers

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

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then the determinant of $A + A^2 + A^3 + \cdots + A^{13}$ is

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

Let the circle $x^2 + y^2 = 2ax + 2by$ intersect the x-axis at point $A(\alpha, 0)$ and y-axis at point $B(0, \beta)$, where $\alpha\beta \neq 0$. If the point $C(p, q)$ lies on the chord AB, then $\frac{p+\alpha}{a} + \frac{q+\beta}{b}$ equals

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

The number of integers $n$ such that $1 \leq n \leq 10^7$ and $\gcd(n, 10^7) = 10^3$ 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

In a class, 25% of all students read news from the Internet. Moreover, 45% of all students read news from printed newspaper. Further, 20% of all students read news from both the Internet and printed newspaper and they do not play video games. It is also known that 30% of the students who do not read news play video games. The minimum percentage of students who do not play video games is

The approximate value of the expression $2\log_3 3n - \log_3(n^2 + 1)$ for a sufficiently large $n$ is

If $x$ is a real number such that $\max(\min(x, 2 - x), x - 4, 2x - 8) = \pi - 3$, then the number of possible values of $x$ is

Positive reals $x, y$ satisfy $x \neq y$ and $\frac{x^2 + y^2}{xy} = k$. If replacing $x$ by $x + y$ and $y$ by $|x - y|$ leaves the value of $k$ unchanged, then $k$ equals

Gita starts from point A and walks 1000 m east. She then walks 800 m north, followed by 640 m west and 512 m south, reaching point B. After this, she continues moving in the same cyclic order: east, north, west, south, with each successive movement 20% shorter than the previous one. After infinitely many such moves, approximately how far in meters will Gita be from her starting point A?

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

A circle of non-zero radius has origin as its centre. If it passes through the point of intersection of two curves $y^2 = 4ax$ and $x^2 = 4ay$, then its equation is

A person walks one lap along a circle at a speed $v$. Thereafter, he runs one lap along the boundary of the largest square that can be inscribed in the circle at a speed $3v$. The ratio of the time he walks to the time he runs is

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

A certain number of people contributed to a charity. The first person contributed one rupee. The rule for contribution was that the next person would contribute double the amount already raised. If the total money raised for the charity was 2187 rupees, then the number of people who contributed to the charity is

If $a$, $b$, and $c$ are three prime numbers such that $abc = 23(a + b + c)$, then the maximum possible value of $a + b - c$ is

A nutritionist is designing a daily diet plan of a person using five food items: Milk, Spinach, Almonds, Oats, and Rice. The Minimum Daily Requirement (MDR) of the nutrients Calcium, Iron and Protein should be 1100 mg, 15 mg, and 40 g, respectively. The following table gives nutrient content per serving of each food item:

Food item	Calcium (mg)	Iron (mg)	Protein (g)
Milk	300	0	8
Spinach	90	6	3
Almonds	210	4	6
Oats	120	3	5
Rice	50	1	4

Each serving provides a certain percentage of the MDR. For example, one serving of milk satisfies 20% of MDR of Protein. Further, the number of servings of any food item should be a whole number.

If two servings of Milk are part of the daily diet plan, then the minimum number of servings of any other single item that can satisfy the MDR of all three nutrients is

A nutritionist is designing a daily diet plan of a person using five food items: Milk, Spinach, Almonds, Oats, and Rice. The Minimum Daily Requirement (MDR) of the nutrients Calcium, Iron and Protein should be 1100 mg, 15 mg, and 40 g, respectively. The following table gives nutrient content per serving of each food item:

Food item	Calcium (mg)	Iron (mg)	Protein (g)
Milk	300	0	8
Spinach	90	6	3
Almonds	210	4	6
Oats	120	3	5
Rice	50	1	4

Each serving provides a certain percentage of the MDR. For example, one serving of milk satisfies 20% of MDR of Protein. Further, the number of servings of any food item should be a whole number.

The food item(s) that can satisfy at least half of the MDR for all the three nutrients using up to seven servings of that single item alone is