IPMAT 2021 QUANT - Makhija Academy

IPMAT 2021

Actual Question Paper

01. The number of positive integers that divide (1890).(130).(170) and are not divisible by 45 is ________.

02. The sum up to 10 terms of the series 1.3 + 5.7 + 9.11 + . . is

03. It is given that the sequence {x_n} satisfies x₁ = 0,
x_n+1 = x_n + 1 + 2√(1+x_n) for n = 1,2, . . . . . Then x₃₁ is _______

04. There are 5 parallel lines on the plane. On the same plane, there are ‘n’ other lines that are perpendicular to the 5 parallel lines. If the number of distinct rectangles formed by these lines is 360, what is the value of n?

05. There are two taps, T₁ and T₂, at the bottom of a water tank, either or both of which may be opened to empty the water tank, each at a constant rate. If T₁ is opened keeping T₁ closed, the water tank (initially full) becomes empty in half an hour. If both T₁ and T₂ are kept open, the water tank (initially full) becomes empty in 20 minutes. Then, the time (in minutes) it takes for the water tank (initially full) to become empty if T₂ is opened while T₁ is closed is

06. A class consists of 30 students. Each of them has registered for 5 courses. Each course instructor conducts an exam out of 200 marks. The average percentage marks of all 30 students across all courses they have registered for, is 80%. Two of them apply for revaluation in a course. If none of their marks reduce, and the average of all 30 students across all courses becomes 80.02%, the maximum possible increase in marks for either of the 2 students is

07. What is the minimum number of weights which enable us to weigh any integer number of grams of gold from 1 to 100 on a standard balance with two pans? (Weights can be placed only on the left pan)

08. If one of the lines given by the equation 2𝑥² + axy + 3y² = 0 coincides with one of those given by 2x² + b𝑥𝑦 - 3𝑦² = 0 and the other lines represented by them are perpendicular then 𝑎² + 𝑏² =

09. If a function f(a) = max (a, 0) then the smallest integer value of ‘x’ for which the equation f(x – 3) + 2f(x + 1) = 8 holds true is _______

10. In a class, 60% and 68% of students passed their Physics and Mathematics examinations respectively. Then atleast ________ percentage of students passed both their Physics and Mathematics examinations.

11. Suppose that a real-valued function f(x) of real numbers satisfies f(x + xy) = f(x) + f(xy) for all real x, y, and that f(2020) = 1. Compute f(2021).

12. Suppose that log₂[log₃ (log₄a)] = log₃ [log₄ (log₂b)] = log₄ [log₂ (log₃c)] = 0 then the value of a + b + c is

13. Let S_n be sum of the first n terms of an A.P. {a_n }. If S₅ = S₉ , what is the ratio of a₃ : a₅

14. If A, B and A + B are non singular matrices and AB = BA then 2A – B – A(A + B)^-1A + B(A + B)^-1B equals

15. If the angles A, B, C of a triangle are in arithmetic progression such that sin(2A + B) = 1/2 then sin(B + 2C) is equal to

16. The unit digit in (743)⁸⁵ – (525)³⁷ + (987)⁹⁶ is ________

17. The set of all real value of p for which the equation 3 sin²x + 12 cos x – 3 = p has one solution is

18. ABCD is a quadrilateral whose diagonals AC and BD intersect at O. If triangles AOB and COD have areas 4 and 9 respectively, then the minimum area that ABCD can have is

19. The highest possible value of the ratio of a four-digit number and the sum of its four digits is

20. Consider the polynomials f(x) = ax² + bx + c, where a > 0, b, c are real, g(x) = -2x. If f(x) cuts the x-axis at (-2, 0) and g(x) passes through (a, b), then the minimum value of f(x) + 9a + 1 is

21. In a city, 50% of the population can speak in exactly one language among Hindi, English and Tamil, while 40% of the population can speak in at least two of these three languages. Moreover, the number of people who cannot speak in any of these three languages is twice the number of people who can speak in all these three languages. If 52% of the population can speak in Hindi and 25% of the population can speak exactly in one language among English and Tamil, then the percentage of the population who can speak in Hindi and in exactly one more language among English and Tamil is

22. A train left point A at 12 noon. Two hours later, another train started from point A in the same direction. It overtook the first train at 8 PM. It is known that the sum of the speeds of the two trains is 140 km/hr. Then, at what time would the second train overtake the first train, if instead the second train had started from point A in the same direction 5 hours after the first train? Assume that both the trains travel at constant speeds.

23. The number of 5-digit numbers consisting of distinct digits that can be formed such that only odd digits occur at odd places is

24. There are 10 points in the plane, of which 5 points are collinear and no three among the remaining are collinear. Then the number of distinct straight lines that can be formed out of these 10 points is

25. The x-intercept of the line that passes through the intersection of the lines x + 2y = 4 and 2x + 3y = 6, and is perpendicular to the line 3x – y = 2 is

In a football tournament six teams A, B, C, D, E, and F participated. Every pair of teams had exactly one match among them. For any team, a win fetches 2 points, a draw fetches 1 point, and a loss fetches no points. Both teams E and F ended with less than 5 points. At the end of the tournament points table is as follows (some of the entries are not shown):

Teams	Played	Wins	Losses	Draws	Points
A	5		0		8
B	5		2		6
C	5		2		5
D	5		1		5
E	5		1
F	5

It is known that: (1) team B defeated team C, and (2) team C defeated team D

26. Total number of matches ending in draw is

27. Which team has the highest number of draws

28. Total points Team F scored was

29. Which team was not defeated by team A

30. Team E was defeated by