IPMAT 2019 Quant - Makhija Academy

IPMAT 2019

Question Bank

1. The sum of the interior angles of a convex n-sided polygon is less than 2019^∘ . The maximum possible value of n is

2. Suppose that a, b, and c are real numbers greater than 1. Then the value of

\frac{1}{1 + \log_{a^{2} b} \frac{c}{a}} + \frac{1}{1 + \log_{b^{2} c} \frac{a}{b}} + \frac{1}{1 + \log_{c^{2} a} \frac{b}{c}}

is.

3. A real-valued function f satisfies the relation f(x)f(y) = f(2xy + 3) + 3f(x + y) – 3f(y) + 6y, for all real numbers x and y, then the value of f(8) is

4. Let A, B, C be three 4 X 4 matrices such that det A = 5, det B = -3, and detC = 1∕2. Then the det (2AB-1C3BT) is

5. If A is a 3 X 3 non-zero matrix such that A2= 0 then determinant of [ (1 + A)2– 50A] is equal to.

6. Three friends divided some apples in the ratio 3 : 5 : 7 among themselves. After consuming 16 apples they found that the remaining number of apples with them was equal to largest number of apples received by one of them at the beginning. Total number of apples these friends initially had was

7. A shopkeeper reduces the price of a pen by 25% as a result of which the sales quantity increased by 20%. If the revenue made by the shopkeeper decreases by x% then x is

8. For all real values of x,

\frac{3 x^{2} - 6 x + 12}{x^{2} + 2 x + 4}

lies between 1 and k, and does not take any value above k. Then k equals

9. The maximum distance between the point (-5, 0) and a point on the circle = 4 is

10. If x, y, z are positive real numbers such that x¹² = y¹⁶ = z²⁴,and the three quantities 3log_yx, 4log_zy, nlog_xz are in arithmetic progression, then the value of n is

11. The number of pairs (x, y) satisfying the equation sinx + siny = sin(x + y) and |x| + |y| = 1 is

12. The circle x² + y² - 6x - 10y + k = 0 does not touch or intersect the coordinate axes. If the point (1, 4) does not lie outside the circle, and the range of k is (a, b] then a + b is

13. If a 3 X 3 matrix is filled with +1 ‘s and – 1 ‘s such that the sum of each row and column of the matrix is 1, then the absolute value of its determinant is

14. Let the set = {2,3,4,..., 25}. For each k ∈ P, define Q(k)= {x ∈ P such that x > k and k divides x}. Then the number of elements in the set $P - U_{k=2}^{25} \\$ Q(k) is

15. The number of whole metallic tiles that can be produced by melting and recasting a circular metallic plate, if each of the tiles has a shape of a right-angled isosceles triangle and the circular plate has a radius equal in length to the longest side of the tile (Assume that the tiles and plate are of uniform thickness, and there is no loss of material in the melting and recasting process) is

16. If $|x| < 100\\$ and $|y| < 100\\$ , then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is

17. The average of five distinct integers is 110 and the smallest number among them is 100. The maximum possible value of the largest integer is

18. Assume that all positive integers are written down consecutively from left to right as in 1234567891011...... The 6389^th digit in this sequence is

19. The number of pairs of integers whose sums are equal to their products is

20. You have been asked to select a positive integer N which is less than 1000 , such that it is either a multiple of 4, or a multiple of 6, or an odd multiple of 9. The number of such numbers is

21. If the compound interest earned on a certain sum for 2 years is twice the amount of simple interest for 2 years, then the rate of interest per annum is _______ percent

22. The maximum value of the natural number n for which 21ⁿ divides 50! is

23. The remainder when $29^{29^{29}}$

24. Placing which of the following two digits at the right end of 4530 makes the resultant six digit number divisible by 6,7 and 9?

25. In a school 70% of the boys like cricket and 50% like football. If x% like both Cricket and Football, then

26. In a class of 65 students 40 like cricket, 25 like football and 20 like hockey. 10 students like both cricket and football, 8 students like football and hockey and 5 students like all three sports. If all the students like at least one sport, then the number of students who like both cricket and hockey is

27. If x ∈ (a, b) satisfies the inequality $\frac{x - 3}{x^{2} + 3 x + 2} \geq 1,$ then the largest possible value of b - a is

28. If a, b, c are real numbers a² + b² + c² = 1, then the set of values ab + bc + ca can take is:

29. The inequality $\log _{2} \frac{3x - 1}{2 - x} < 1\\$ holds true for

x ∈ ( $\frac{1}{3}\\$ , 1)
x ∈ ( $\frac{1}{3}\\$ , 2)
x ∈ (0, $\frac{1}{3}\\$ ) ∪ (1,2)
x ∈ (-∞, 1)

30. The set of values of x which satisfy the inequality 0.7^{2x² - 3x + 4} < 0.343 is

( $\frac{1}{2}\\$ , 1)
( $\frac{1}{2}\\$ , ∞)
(-∞, $\frac{1}{2}\\$ )
(-∞, $\frac{1}{2}\\$ ) ∪ (1, ∞)

31. A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

(2 + √3)
(1 + √2)
(7 + 4√3)
(97 + 56√3)

32. Points P, Q, R and S are taken on sides AB, BC, CD and DA of square ABCD respectively, so that AP : PB = BQ : QC = CR : RD = DS : SA = 1 : n . Then the ratio of the area of PQRS to the area of ABCD is

1 : (1 + n)
1 : n
1 + n² : (1 + n)²
(1 + n ) : (1 + n²)

33. On a circular path of radius 6 m a boy starts from a point A on the circumference and walks along a chord AB of length 3 m. He then walks along another chord BC of length 2 m to reach point C. The point B lies on the minor arc AC. The distance between point C from point A is

$\frac{\sqrt{15} + \sqrt{35}}{2}\\$ m
8 m
√13 m
6 m

34. The area enclosed by the curve 2|x| + 3|y| = 6 is

35. Two points on a ground are 1 m apart. If a cow moves in the field in such a way that it’s distance from the two points is always in ratio 3: 2 then

36. Given that cos x + cos y = 1, the range of sin x – sin y is

[-1, 1]
[-2, 2]
[0, √3]
[-√3, √3]

37. If $\sin \theta + \cos \theta = m,\\$ then $\sin ^{6} \theta + \cos ^{6} \theta\\$ equals

$\frac{3(m^{2}+1)}{4}\\$
$\frac{3(m^{2}-1)}{4}\\$
$1-\frac{3(m^{2}-1)}{4}\\$
$1-\frac{3(m^{2}-1)^{2}}{4}\\$

38. if inverse of the matrix$$\begin{bmatrix}2&-0.5\\-1&x\end{bmatrix}$$is$$\begin{bmatrix}1&1\\2&4\end{bmatrix}$$, then the value of $$x$$ is

39. The function f(x) = $\frac{x^{3} - 5x^{2} - 8x}{3}\\$ is

positive and monotonically increasing for x $\in (-\infty, \frac{5-\sqrt{57}}{2}\\$ ) and x $\in (\frac{5+\sqrt{57}}{2}, +\infty\\$ )
negative and monotonically decreasing for x $\in (-\infty, \frac{5-\sqrt{57}}{2}\\$ and x $\in (\frac{5 + \sqrt{57}}{2}, + \infty$
negative and monotonically increasing for x $\in (-\infty, \frac{5-\sqrt{57}}{2}\\$ ) and positive and monotonically increasing for x $\in (\frac{5+\sqrt{57}}{2},+\infty\\$ )
positive and monotonically increasing for x $\in (-\infty, \frac{5-\sqrt{57}}{2}\\$ ) and negative and monotonically decreasing for x $\in (\frac{5+\sqrt{57}}{2},+\infty\\$ )

40. For a > b > c > 0, the minimum value of the function f(x) = |x – a| + |x – b| + |x – c| is

41. Let $\alpha, \beta\\$ be the roots of x² - x + p = 0 and $\gamma, \delta\\$ be the roots of x² - 4x + q = 0 where p and q are integers. If $\alpha, \beta, \gamma, \delta\\$ are in geometric progression then p + q is

42. If (1 + x - 2x²)⁶ = $A_{0}+\sum_{r=1}^{12} A_{r} x^{r}\\$ , then value of $A_{2}+A_{4}+A_{6}+\cdots+A_{12}\\$ is

43. The number of terms common to both the arithmetic progressions 2,5,8,11,…., 179 and 3,5,7,9,….., 101 is

44. From a pack of 52 cards, we draw one by one, without replacement. If f(n) is the probability that an Ace will appear at the n^th turn, then

f(2) = $\frac{1}{13}\\$ > f(3)
$\frac{1}{13}\\$ > f(2) > f(3)
f(3) > f(2) = $\frac{1}{13}\\$
f(2) = f(3) = $\frac{1}{13}\\$

45. A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is

$\frac{1}{6}\\$
$\frac{1}{4}\\$
$\frac{1}{5}\\$
$\frac{1}{10}\\$

46. In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2,4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to

47. The value of $\log _{3} 30^{-1} + \log _{4} 900^{-1} + \log _{5} 30^{-1}\\$ is

48. The inequality $\log _{a}{f(x)} < \log _{a}{g(x)}\\$ implies that

49. Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 cm² Then the possible values of the sum of their volumes are

50. Determine the greatest number among the following four numbers

51. The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 31), and (31, 0) is

52. Two small insects, which are x metres apart, take u minutes to pass each other when they are flying towards each other, and v minutes to meet each other when they are flying in the same direction. Then, the ratio of the speed of the slower insect to that of the faster insect is

$\frac{u}{v}\\$
$\frac{u}{v-u}\\$
$\frac{v-u}{v+u}\\$
$\frac{u}{v+u}\\$

The inequality $\log _{a}{f(x)} < \log _{a}{g(x)}\\$ implies that

53. An alloy P has copper and zinc in the proportion of 5: 2 (by weight), while another alloy Q has the same metals in the proportion of 3: 4 (by weight). If these two alloys are mixed in the proportion of a : b (by weight), a new alloy R is formed, which has equal contents of copper and zinc. Then, the proportion of copper and zinc in the alloy S, formed by mixing the two alloys P and Q in the proportion of b : a (by weight) is

54. How many different numbers can be formed by using only the digits 1 and 3 which are smaller than 3000000 ?

55. There are numbners $a_{1}, a_{2}, a_{3}, \ldots, a_{n}\\$ each of them being +1 or -1. If it is known that $a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + \ldots a_{n-1} a_{n} + a_{n} a_{1} = 0\\$ then

Q.(56 – 60)Analyze the given data for exports and imports of rubber in Rs. crores from 2016 to 2017 and answer the questions based on the analysis.

56. Average annual exports for the given period 2006-2017) was approximately

57. The percentage decline in exports during the period 2006-2011 is more than the percentage decline in exports during 2012-2017 by approximately

58. The maximum difference between imports and exports is

59. Balance of trade is defined as imports subtracted from exports ( = exports – imports). Which of the following blocks of three years has witnessed the largest average negative balance of trade?

60. The percentage increase in imports over the previous year is maximum during

IPMAT 2019​

Question Bank

IPMAT 2019