IPMAT Indore 2023 - QA (MCQ)

IPMAT Indore 2023 - QA (MCQ)

Questions: 30 (+4/-1)                                                          Sectional Time:  40 Minutes

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

• (a) 1/4
• (b) 2/3
• (c) 1/2
• (d) 1/3

17. 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) 12000
• (b) 9600
• (c) 48000
• (d) 24000

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

• (a) an+bn=cn
• (b) (an+b)n=ac
• (c) a+b=c
• (d) (b−a)n=(c−b)

19. 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

• (a) 4−√5
• (b) 2
• (c) 4+√5
• (d) 3

20. 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

• (a) 7
• (b) 8
• (c) 7!
• (d) 8!

21. The set of all real values of x satisfying the inequality x2(x+1)/(x−1)(2x+1)3 >0 is

• (a) (−1,−1/2 ) ∪(0,+∞)
• (b) (−∞,−1)∪(−1/2 ,0)∪(1,+∞)
• (c) (−1,0)∪(1,+∞)
• (d) (−1,−1/2 ) ∪(1,+∞)

22. If A=[ 1 2 3 a ] where a is a real number and det (A^3− 3A^2−5A)=0 then one of the values of a can be

• (a) 1
• (b) 6
• (c) 4
• (d) 5

23. 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

• (a) 170000
• (b) 15000
• (c) 150000
• (d) 17000

24. 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

• (a) Exactly 20
• (b) Between 15 and 25
• (c) Between 20 and 25
• (d) 30 or more

25. 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

• (a) 5
• (b) 7
• (c) 3
• (d) 6

26. 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

• (a) x−3
• (b) 3−x
• (c) 3
• (d) 2

27. 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

• (a) n
• (b) n(n+1)/2
• (c) 3/2
• (d) n(n+2)/2

28. If a+b b+c =c+d d+a , which of the following statements is always true?

• (a) a=c
• (b) a=c,and b=d
• (c) a+b+c+d=0
• (d) a=c,or a+b+c+d=0

29. If logcosx(sinx)+logsinx(cosx)=2, then the value of x is

• (a) nπ +π/4 ,n is an interger
• (b) 2nπ +π/4 ,n is an interger
• (c) nπ/4 +π/4 ,n is an interger
• (d) 4π/4 ,n is an interger

30. 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

• (a) 184 kmph
• (b) 200 kmph
• (c) 250 kmph
• (d) 192 kmph

31. 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

• (a) 11:12
• (b) 11:4√10
• (c) 12:11
• (d) 4√10:11

32. 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?

• (a) 3, 2, 2, 2, 1
• (b) 4, 4, 1, 1, 0
• (c) 2, 2, 2, 2, 2
• (d) 3, 3, 2, 1, 1

33. 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

• (a) 34
• (b) 55
• (c) 144
• (d) 89

34. Let a1,a2,a3 be three distinct real nubers in geometric progression. If the eqations a1x2+2a2x+ a3=0 and b1x2+2b2x+b3+= 0 have a common root, then which of the following is necessarily ture?

• (a) b1/a1 , b2/a2 , b3/a3 are in arithmetic progression
• (b) b1,b2,b3 are in arithmetic progression
• (c) b1/a1 , b2/a2 , b3/a3 are in geometric progression
• (d) b1,b2,b3 are in geometric progression

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

• (a) 4x+3y+19=0,4x+3y−31=0.
• (b) 4x+3y+19=0,4x+3y+31=0.
• (c) 4x+3y−19=0,4x+3y−31=0.
• (d) 4x+3y−19=0,4x+3y+31=0.

36. 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?

• (a) 64 /√5 meters
• (b) 64 meters
• (c) 32 meters
• (d) 32 /√5 meters

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

• (a) a point
• (b) a pair of straight lines
• (c) an ellipse
• (d) a circle

38. If cosα+cosβ=1, then the maximum value of sinα−sinβ is

• (a) 1
• (b) 2
• (c) √3
• (d) √2

39. "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?

• (a) 1, 7, 9
• (b) 4, 7
• (c) 1, 3, 7
• (d) 3

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

• (a) 23/2024
• (b) 23^2/2023^2
• (c) 23/2023
• (d) 23^2/2024^2

Directions (Q.41-Q.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:

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

41. Organism R can be treated with

• (a) Only A and E
• (b) A, B and E
• (c) A, B and C
• (d) Only A and B

42. Drug E works on

• (a) Only R
• (b) Q and R
• (c) Only P
• (d) P and R

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

• (a) Only Q
• (b) Only P
• (c) P and Q
• (d) Q and R

44. Drug D works on

• (a) Only Q
• (b) P and Q
• (c) Q and R
• (d) Only P

45. Organism P can be treated with

• (a) B, C and D
• (b) Only C and D
• (c) Only B and D
• (d) A, B, and D