Q1:
Number System>Unit Digit
Easy
Let $p$ be a positive integer such that the unit digit of $p ^ 3$ is $4$. What are the possible unit digits of $(p + 3) ^ 3$
Correct Answer
Option 1
Correct Answer
IPMAT Indore 2023 PYQsMultiple Choice (Quants). Free, no login required.
Q2:
Number System>Integral Solutions
Medium
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
Correct Answer
Option 1
Correct Answer
Q3:
Algebra>Inequalities
Easy
The set of all real values of x satisfying the inequality $\dfrac{x^2(x+1)}{(x-1)(2x+1)^3} > 0$ is
Correct Answer
Option 2
Correct Answer
Q4:
Arithmetic>Profit & Loss
Easy
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
Correct Answer
Option 2
Correct Answer
Q5:
Geometry>Circles
Medium
Which of the following straight lines are both tangent to the circle $x ^ 2 + y ^ 2 - 6x + 4y - 12 = 0$?
Correct Answer
Option 1
Correct Answer
Q6:
Modern Math>Matrices & Determinants
Medium
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
Correct Answer
Option 2
Correct Answer
Q7:
Arithmetic>Time, Speed & Distance
Easy
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
Correct Answer
Option 3
Correct Answer
Q8:
Arithmetic>Ratio, Proportion & Variation
Easy
$\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)} $ which of the following statements is always true?
Correct Answer
Option 2
Correct Answer
Q9:
Algebra>Polynomials
Medium
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
Correct Answer
Option 4
Correct Answer
Q10:
Logical Reasoning>Tournaments
Medium
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?
Correct Answer
Option 4
Correct Answer
Q11:
Arithmetic>Simple & Compound Interest
Easy
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
Correct Answer
Option 1
Correct Answer
Q12:
Algebra>Progression & Series
Medium
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?
Correct Answer
Option 4
Correct Answer
Q13:
Geometry>Triangles
Hard
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
Correct Answer
Option 1
Correct Answer
Q14:
Modern Math>Logarithms
Medium
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?
Correct Answer
Option 4
Correct Answer
Q15:
Modern Math>Permutation & Combination
Easy
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
Correct Answer
Option 4
Correct Answer
Q16:
Geometry>Conic Sections
Easy
The equation $x^2 + y^2 - 2x - 4y +5 = 0$ represents
Correct Answer
Option 4
Correct Answer
Q17:
Algebra>Progression & Series
Hard
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?
Correct Answer
Option 4
Correct Answer
Q18:
Modern Math>Logarithms
Hard
If $\log_{(cos x)}(sin x) + \log_{(sin x)}(cos x) = 2,$ then the value of $x$ is
Correct Answer
Option 2
Correct Answer
Q19:
Modern Math>Permutation & Combination
Medium
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
Correct Answer
Option 2
Correct Answer
Q20:
Modern Math>Probability
Medium
The probability that a randomly chosen positive divisor of $10 ^ {2023}$ is an integer multiple of $10 ^ {2001}$ is
Correct Answer
Option 2
Correct Answer
Q21:
Modern Math>Set Theory
Easy
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
Correct Answer
Option 3
Correct Answer
Q22:
Number System>Divisibility Rules
Easy
If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4?
Correct Answer
Option 2
Correct Answer
Q23:
Modern Math>Probability
Easy
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
Correct Answer
Option 3
Correct Answer
Q24:
Geometry>Trigonometry
Medium
If $cos \alpha$ + $cos \beta$ = 1 then the maximum value of $sin \alpha - sin \beta$ is
Correct Answer
Option 3
Correct Answer
Q25:
Number System>Remainder
Medium
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
Correct Answer
Option 1
Correct Answer
Q26:
Logical Reasoning>Arrangements
Easy
Correct Answer
Option 4
Correct Answer
Q27:
Logical Reasoning>Arrangements
Easy
Correct Answer
Option 1
Correct Answer
Q28:
Logical Reasoning>Arrangements
Easy
Correct Answer
Option 4
Correct Answer
Q29:
Logical Reasoning>Arrangements
Easy
Correct Answer
Option 4
Correct Answer
Q30:
Logical Reasoning>Arrangements
Easy
Correct Answer
Option 1
Correct Answer
IPMAT Indore 2023 Multiple Choice (Quants) Past Year Question Paper
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