IPMAT Indore 2024 - QA (MCQ)

IPMAT 2024 Quantitataive Aptitude (MCQ)

Questions: 30 (+4/-1)                               Total Test time: 40 mins

16. The angle of elevation of the top of a pole from a point A on the ground is 30°. The angle of elevation changes to 45°, after moving 20 metres towards the base of the pole. Then the height of the pole, in metres, is

• (a) 30
• (b) 15(√5+1)
• (c) 20(√3+ 1)
• (d) 10(√3+1)

17. If |𝑥+1|+ (𝑦+2)2 = 0 and 𝑎𝑥−3𝑎𝑦 = 1, then the value of a is

• (a) 1/5
• (b) 1/2
• (c) 1/7
• (d) 2

18. If log4x = a and log25x = b then logx10 is

• (a) a+b/2(a−b)
• (b) a+b/2
• (c) a+b/2ab
• (d) a−b/2ab

19. Let ABC be an equilateral triangle, with each side of length k. If a circle is drawn with diameter AB, then the area of the portion of the triangle lying inside the circle is

• (a) (3√3+𝜋)(𝑘2/6 )
• (b) (3√3−𝜋)(𝑘2/24 )
• (c) (3√3+𝜋)(𝑘2/24 )
• (d) (3√3−𝜋)(𝑘2/6 )

20. Let ABC be a triangle with AB = AC and D be a point on BC such that ∠BAD = 30°. If E is a point on AC such that AD AE, then ∠CDE equals

• (a) 15°
• (b) 60°
• (c) 30°
• (d) 10°

21. If 5 boys and 3 girls randomly sit around a circular table, the probability that there will be at least one boy sitting between any two girls, is

• (a) 2/7
• (b) 1/7
• (c) 1/4
• (d) 1/3

22. The side AB of a triangle ABC is c. The median BD is of length k. If ∠BDA = θ < 90°, then the area of triangle ABC is

• (a) k2cos2θ/2 +ksinθ√(c2−k2cos2θ)
• (b) k2sin2θ/2 +ksinθ√(c2−k2sin2θ)
• (c) k2cosθ/4 −ksinθ√(c2+ k2sin2θ)
• (d) k2cos2θ/4 +ksinθ√(c2− k2sin2θ)

23. Let a = (log74) (log75−log72) (log725) (log78−log74) Then the value of 5𝑎 is

• (a) 7/2
• (b) 5
• (c) 8
• (d) 5/2

24. For some non-zero real values of a, b and c, it is given that |𝑐 𝑎 | = 4, |𝑎 𝑏 |=1 3 and 𝑏 𝑐 =−3 4 . If ac > 0, then (𝑏+𝑐 𝑎 ) equals

• (a) 7
• (b) -7
• (c) -1
• (d) 1

25. The difference between the maximum real root and the minimum real root of the equation (𝑥2−5)4+ (𝑥2−7)4=16 is

• (a) 2√5
• (b) 2√7
• (c) √7
• (d) √10

26. If θ is the angle between the pair of tangents drawn from the point A (0, 7 2 ) to the circle 𝑥² + 𝑦²− 14𝑥+ 16𝑦+88 = 0, then tan θ equals

• (a) 5/4
• (b) 20/21
• (c) 2/5
• (d) 4/5

27. The numbers 22024 and 52024 are expanded and their digits are written out consecutively on one page. The total number of digits written on the page is

• (a) 1987
• (b) 2025
• (c) 2065
• (d) 2000

28. A boat goes 96 km upstream in 8 hours and covers the same distance moving downstream in 6 hours. On the next day boat starts from point A, goes downstream for 1 hour, then upstream for 1 hour and repeats this four more time that is, 5 upstream and 5 downstream journeys. Then the boat would be

• (a) 22.5 km downstream of A
• (b) 15 km downstream of A
• (c) 12.5 km downstream of A
• (d) 20 km downstream of A

29. If the shortest distance of a given point to a given circle is 4 cm and the longest distance is 9 cm, then the radius of the circle is

• (a) 2.5 cm or 6.5 cm.
• (b) 6.5 cm
• (c) 5 cm or 13 cm
• (d) 2.5 cm

30. In a survey of 500 people, it was found that 250 owned a 4-wheeler but not a 2-wheeler, 100 owned a 2-wheeler but not a 4-wheeler, and 100 owned neither a 4-wheeler nor a 2-wheeler. Then the number of people who owned both is

• (a) 75
• (b) 60
• (c) 50
• (d) 100

31. The sum of a given infinite geometric progression is 80 and the sum of its first two terms is 35. Then the value of n for which the sum of its first n terms is closest to 100, is

• (a) 6
• (b) 5
• (c) 7
• (d) 4

32. Let n be the number of ways in which 20 identical balloons can be distributed among 5 girls and 3 boys such that everyone gets at least one balloon and no girl gets fewer balloons than a boy does. Then

• (a) 8000 ≤ n < 9000
• (b) 7000 ≤ n < 8000
• (c) 9000 ≤ n < 10000
• (d) 6000 ≤ n < 7000

33. The greatest number among 2300, 3200, 4100, 2100 + 3100 is

• (a) 3200
• (b) 2100 + 3100
• (c) 4100
• (d) 2300

34. The number of values of x for which C3x+1 17−x is defined as an integer is

• (a) 5
• (b) 6
• (c) 2
• (d) 4

35. The number of solutions of the equation 𝑥1+𝑥2+ 𝑥3+𝑥4=50, where x1, x2, x3, x4 are integers with � �1≥1, 𝑥2≥2,𝑥3≥0,𝑥4≥0 is

• (a) 19600
• (b) 19200
• (c) 20200
• (d) 18400

36. Sagarika divides her savings of 10000 rupees to invest across two schemes A and B. Scheme A offers an interest rate of 10% per annum, compounded half yearly, while scheme B offers a simple interest rate of 12% per annum. If at the end of first year, the value of her investment in scheme B exceeds the value of her investment in scheme A by 2310 rupees, then the total interest, in rupees, earned by Sagarika during the first year of investment is

• (a) 1100
• (b) 1130
• (c) 1111
• (d) 1000

37. A fruit seller had a certain number of apples, bananas and oranges at the start of the day. The number of bananas was 10 more than the number of apples, and the total number of bananas and apples was a multiple of 11. She was able to sell 70% of apples, 60% of bananas, and 50% of oranges during the day. If she was able to sell 55% of the fruits she had at the start of the day, then the minimum number of oranges she had at the start of the day was

• (a) 220
• (b) 190
• (c) 210
• (d) 180

38. The terms of a geometric progression are real and positive. If the pth term of the progression is q and the qth term is p, then the logarithm of the first term is

• (a) (1−q)log(p)−(1−p)log (q)/(𝑝−𝑞)
• (b) (1−q)log(p)+(1−p)log (q)/(𝑝−𝑞)
• (c) (1−q)log(q)+(1−p)log (p)/(𝑝−𝑞)
• (d) (1−q)log(q)−(1−p)log (p)/(p−q)

39. The number of real solutions of the equation 𝑥2− 10 |𝑥|−56=0 is

• (a) 3
• (b) 4
• (c) 2
• (d) 1

40. The smallest possible number of students in a class if the girls in the class are less than 50% but more than 48% is

• (a) 25
• (b) 100
• (c) 27
• (d) 200

Direction (Q.41-Q.45): In an election there were five constituencies S1, S2, S3, S4 and S5 with 20 voters each all of whom voted. Three parties A, B and C contested the elections. The party that gets maximum number of votes in a constituency wins that seat. In every constituency there was a clear winner. The following additional information is available:

• Total number of votes obtained by A, B and C across all constituencies are 49, 35 and 16 respectively.
• S2 and S3 were won by C while A won only S1.
• Number of votes obtained by B in S1, S2, S3, S4 and S5 are distinct natural numbers in increasing order.

41. The constituency in which B got lower number of votes compared to A and C is

• (a) S4
• (b) S1
• (c) S3
• (d) S2

42. The number of votes obtained by B in S2 is

• (a) 5
• (b) 7
• (c) 4
• (d) 6

43. Assume that A and C had formed an alliance and any voter who voted for either A or C would have voted for this alliance. Then the number of seats this alliance would have won is

• (a) 3
• (b) 4
• (c) 5
• (d) 2

44. The number of votes obtained by A is S5 is

• (a) 7
• (b) 9
• (c) 6
• (d) 8

45. Comparing the number votes obtained by A across different constituencies, the lowest number of votes were in constituency

• (a) S2
• (c) S3
• (b) S5
• (d) S4