IPMAT Indore 2023

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

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

The set of all real values of x satisfying the inequality $\dfrac{x^2(x+1)}{(x-1)(2x+1)^3} > 0$ is

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 or each golden bead, in INR, was

Which of the following straight lines are both tangent to the circle $x ^ 2 + y ^ 2 - 6x + 4y - 12 = 0$?

If $A = \begin{bmatrix} 1 & 2 \newline 3 & a \end{bmatrix}$ where $a$ is a real number and det $(A ^ 3 - 3A ^ 2 - 5A) = 0$ then one of the values of $a$ can be

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

$\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)}$ which of the following statements is always true?

If the harmonic mean of the roots of the equation $(5 + \sqrt{2}) x ^ 2 - bx + 8 + 2\sqrt{5} = 0$ is $4$ then the value of $b$ 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 points. Which of the following sequences cannot represent the scores of the five players?

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

Let $a_{1}, a_{2}, a_{3}$ be three distinct real numbers in geometric progression. If the equations $a_{1} x ^ 2 + 2a_{2}x + a_{3} = 0$ and $b_{1} x ^ 2 + 2b_{2}x + b_{3} = 0$ have a common root, then which of the following is necessarily true?

In a triangle ABC, let D be the midpoint 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

Let $a, b, c$ be real numbers greater than 1, and $n$ be a positive real number not equal to 1. If $log_n(log_2a) = 1; log_n(log_2b) = 2$ and $log_n(log_2c) = 3$ then which of the following is true?

Consider an $8 \times 8$ 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

The equation $x^2 + y^2 - 2x - 4y +5 = 0$ represents

A person standing at the centre 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?

If $\log_{(cos x)}(sin x) + \log_{(sin x)}(cos x) = 2,$ then the value of $x$ is

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 probability that a randomly chosen positive divisor of $10 ^ {2023}$ is an integer multiple of $10 ^ {2001}$ 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

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

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

If $cos \alpha$ + $cos \beta$ = 1 then the maximum value of $sin \alpha - sin \beta$ 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

Drug E works on

Drug D works on:

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

Organism R can be treated with

Organism P can be treated with: