IPMAT 2023 - Quantitative Aptitude

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

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 ______.

If f(n) = 1 + 2 + 3 + ??? +(n + 1) and then the least value of n for which g(n) exceeds the value 99/100 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_________.

The total number of positive integer solutions of 21 ≤ a + b + c ≤ 25 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 _______.

The polynomial 4x¹⁰ – x⁹ + 3x⁸ – 5x⁷ + cx⁶ + 2x⁵ – x⁴ + x³ – 4x² + 6x – 2  when divided by x – 1 leaves a remainder 2. Then the value of c + 6 is________.

The product of the roots of the equation is ________.

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

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______.

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 the xy-plane let A = (– 2, 0), B = (2, 0). Define the set S as the collection of all points C on the circle x² + y² = 4 such that the area of the triangle ABC is an integer. The number of points in the set S is________.

The length of the line segment joining the two intersection points of the curves y = 4970 – |x| and y = x² 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__________.

In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
i)    Against every other player belonging to one's own group exactly once; 
ii)    Against each player belonging to one of the remaining three groups exactly twice; 
iii)    Against each player belonging to one of the remaining two groups exactly three times; and 
iv)    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_______.

A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting point after an infinite number of such walks in this pattern ?

The equation x² + y² − 2x − 4y + 5 = 0 represents

A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made a total profit of 20%, then the listed price of each golden bead, in INR, was

A helicopter flies along the sides of a square field of side length 100 kms. The first side is covered at a speed of 100 kmph, and for each subsequent side the speed is increased by 100 kmph till it covers all the sides. The average speed of the helicopter is

The probability that a randomly chosen positive divisor of 10²⁰²³ is an integer multiple of 10²⁰⁰¹ is

In a triangle ABC, let D be the mid-point of BC, and AM be the altitude on BC. If the lengths of AB, BC and CA are in the ratio of 2 : 4 : 3, then the ratio of the lengths of BM and AD would be

If where as is a real number and det?(A³ - 3A² - 5A) = 0 then one of the value of a can be

Which of the following straight lines are both tangent to the circle x² + y² − 6x + 4y − 12 = 0

Let a₁, a₂, a₃ be three distinct real numbers in geometric progression. If the equations a₁x² +2a₂x + a₃ = 0 and b₁x² + 2b₂x + b₃ = 0 has a common root, then which of the following is necessarily true ?

A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either one step at a time or two steps at a time. The number of ways it can reach the top is

The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 0.8 is

Consider an 8 × 8 chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in the same row and no two are in the same column is

Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If log_n?(log₂?a ) = 1, log_n?(log₂?b ) = 2 and log_n?(log₂?c ) = 3, then which of the following is true ?

If the difference between compound interest and simple interest for a certain amount of money invested for 3 years at an annual interest rate of 10% is INR 527, then the amount invested in INR is

If the harmonic mean of the roots of the equation (5 + √2)x² - bx + 8 + 2√5 = 0 is 4 then the value of b is

Let [x] denote the greatest integer not exceeding x and {x} = x – [x]. If n is a natural number, then the sum of all values of x satisfying the equation 2[x] = x + n{x} is

If (a + b)/(b + c) = (c + d)/(d + a), which of the following statements is always true ?

The set of all real values of x satisfying the inequality is

In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot represent the scores of the five players ?

f cosα + cosβ = 1,then the maximum value of sinα − sinβ is

A polynomial P(x) leaves a remainder 2 when divided by (x − 1) and a remainder 1 when divided by (x − 2). The remainder when P(x) is divided by (x − 1) (x − 2) is

In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is

Let p be a positive integer such that the unit digit of p³ is 4. What are the possible unit digits of (p + 3)³ ?

If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4 ?

Direction Qs. 41 – 45: A pharmaceutical company has tested five drugs on three different organisms. The following incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B and C work on Q.

Following additional information is available: 

i)    Each drug works on at least one organism but not more than two organisms. 
ii)    Each organism can be treated with at least two and at most three of these five drugs. 
iii)    On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
iv)    D and E do not work on the same organism.

Drug D works on

Direction Qs. 41 – 45: A pharmaceutical company has tested five drugs on three different organisms. The following incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B and C work on Q.

Following additional information is available: 

i)    Each drug works on at least one organism but not more than two organisms. 
ii)    Each organism can be treated with at least two and at most three of these five drugs. 
iii)    On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
iv)    D and E do not work on the same organism.

Organism P can be treated with

Direction Qs. 41 – 45: A pharmaceutical company has tested five drugs on three different organisms. The following incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B and C work on Q.

Following additional information is available: 

i)    Each drug works on at least one organism but not more than two organisms. 
ii)    Each organism can be treated with at least two and at most three of these five drugs. 
iii)    On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
iv)    D and E do not work on the same organism.

Organism R can be treated with

Direction Qs. 41 – 45: A pharmaceutical company has tested five drugs on three different organisms. The following incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B and C work on Q.

Following additional information is available: 

i)    Each drug works on at least one organism but not more than two organisms. 
ii)    Each organism can be treated with at least two and at most three of these five drugs. 
iii)    On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
iv)    D and E do not work on the same organism.

Drug E works on

Direction Qs. 41 – 45: A pharmaceutical company has tested five drugs on three different organisms. The following incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B and C work on Q.

Following additional information is available: 

i)    Each drug works on at least one organism but not more than two organisms. 
ii)    Each organism can be treated with at least two and at most three of these five drugs. 
iii)    On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
iv)    D and E do not work on the same organism.

The organism(s) that can be treated with three of these five drugs is (are)