IPMAT Indore 2023

Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, then the total time taken, in minutes, to complete the journey, will be

The remainder when $1! + 2! + 3! + ... + 95!$ is divided by $15$ is

Let $a, b, c, d$ be positive integers such that $a + b + c + d = 2023$. If $a: b = 2:5$ and $c:d=5:2$, then the maximum possible value of $a + c$ is

In an election with only two contesting candidates, 15% of the voters did not turn up to vote and 50 voters cast invalid votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more than the other candidate, then the number of voters in the voting list is

If $f(1) = 1$ and $f(n) = 3n - f(n - 1)$ for all integers $n > 1$ , then the value of $f(2023)$ is

In the xy-plane let $A = (- 2, 0), B = (2, 0)$ . Define the set S as the collection of all points C on the circle $x ^ 2 + y ^ 2 = 4$ such that the area of the triangle ABC is an integer. The number of points in the set S is

If three consecutive coefficients in the expansion of $(x+y)^n$ are in the ratio $1:9:63$, then the value of $n$ is

In a chess tournament, there are four groups, each containing an equal number of players. Each player plays

1. against every other player belonging to one's own group exactly once;
2. against each player belonging to one of the remaining three groups exactly twice;
3. against each player belonging to one of the remaining two groups exactly three times; and
4. against each player belonging to the remaining group exactly four times.

If there are more than 1000 matches being played in the tournament, the minimum possible number of players in each group is

The product of the roots of the equation $\log_{2} 2^{(\log_{2}x)^{2}} -5 \log_{2}x+6=0$ is

The total number of positive integer solutions of $21 ≤ a + b + c ≤ 25$ is

The length of the line segment joining the two intersection points of the curves $y = 4970 - |x|$ and $y = x ^ 2$ is

If $f(n)= 1 + 2 + 3 +\cdots+(n+1)$ and $g(n)= \sum_{k=1}^{k=n} \dfrac{1}{f(k)}$, then the least value of $n$ for which $g(n)$ exceeds the value $\dfrac{99}{100}$ is:

The polynomial $4x ^ {10} - x ^ 9 + 3x ^ {11} - 5x ^ 7 + c x ^ 6 + 2x ^ 5 - x ^ 4 + x ^ 3 - 4x ^ 2 + 6x - 2$ when divided by $x - 1$ leaves a remainder $2.$ Then the value of $c + 6$ is

Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each year, the total interest earned by her, in INR, in five years from today will be

Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker to complete the task alone would be