IPMAT 2022 QUANT - Makhija Academy

IPMAT 2022

Actual Question Paper

01. If $\log _{\left(x^{2}\right)} y+\log _{\left(y^{2}\right)} x=1$ and $y=x^{2}-30$ , then the value of $x^{2}+y^{2}$ is ___________.

02. The area enclosed by 2|x|+3|y|≤6 is ____________ sq. units.

03. When Geeta increases her speed from 12km/hr to 20km/hr, she takes one hour less than the usual time to cover the distance between her home and office. The distance between her home and office is ___________ km.

04. The number of triangles that can be formed by choosing points from 7 points on a line and 5 points on another parallel line is _________.

05. Aruna purchases a certain number of apples for INR 20 each and a certain number of mangoes for INR 25 each. If she sells all the apples at 10% profit and all the mangoes at 20% loss, overall she makes neither profit nor loss. Instead, if she sells all the apples at 20% loss and all the mangoes at 10% profit, overall she makes a loss of INR 150 . Then the number of apples purchased by Aruna is _________.

06. Given that the minimum value of is _________.

07. If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ , then the absolute value of the determinant of $\left(A^{9}+A^{6}+A^{3}+A\right)$ is ________.

08. The sum of the coefficients of all the terms in the expansion of $(5 x-9)^{4}$ is __________.

09. A new sequence is obtained from the sequence of positive integers $(1,2,3, \ldots)$ by deleting all the perfect squares. Then the $2022^{\text {nd }}$ term of the new sequence is ________.

10. If $\sin \alpha+\sin \beta=\frac{\sqrt{2}}{\sqrt{3}}$ and $\cos \alpha+\cos \beta=\frac{1}{\sqrt{3}}$ , then the value of $\left(20 \cos\left(\frac{\alpha-\beta}{2}\right)\right)^{2}$ is _________.

11. The $3^{\text {rd }}, 14^{\text {th }}$ and $69^{\text {th }}$ terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the geometric progression is the $n^{\text {th }}$ term of the arithmetic progression, then $n$ equals ________.

12. Let $P(X)$ denote power set of a set $X$ . If $A$ is the null set, then the number of elements in $P(P(P(P(A))))$ is _________.

13. The numbers $-16,2^{x+3}-2^{2 x-1}-16,2^{2 x-1}+16$ are in an arithmetic progression. Then $x$ equals ________.

14. Mrs and Mr Sharma, and Mrs and Mr Ahuja along with four other persons are to be seated at a round table for dinner. If Mrs and Mr Sharma are to be seated next to each other, and Mrs and Mr Ahuja are not to be seated next to each other, then the total number of seating arrangements is _________.

15. Let 50 distinct positive integers be chosen such that the highest among them is 100 , and the average of the largest 25 integers among them exceeds the average of the remaining integers by 50 . Then the maximum possible value of the sum of all the 50 integers is _________.

16. The set of real values of $x$ for which the inequality $\log _{27} 8 \leq \log _{3} x \lt 9^{\frac{1}{\log _{2} 3}}$ holds is

17. The set of all possible values of $f(x)$ for which $(81)^{x}+(81)^{f(x)}=3$ is

18. The value of k for which the following lines
x−y−1=0
2x+3y−12=0
2x−3y+k=0
are concurrent is

19. The lengths of the sides of a triangle are x,21 and 40 , where x is the shortest side. A possible value of x is

20. In a right-angled triangle ABC, the hypotenuse AC is of length 13 cm. A line drawn connecting the midpoints D and E of sides AB and AC is found to be 6 cm in length. The length of BC is

21. If the five-digit number abcde is divisible by 6 , then which of the following numbers is not necessarily divisible by 6 ?

22. In how many ways can the letters of the word MANAGEMENT be arranged such that no two vowels appear together?

23. In a room, there are n persons whose average height is 160cm . If m more persons, whose average height is 172cm , enter the room, then the average height of all persons in the room becomes 164cm . Then m:n is

24. The sum of the first 15 terms in an arithmetic progression is 200 , while the sum of the next 15 terms is 350 . Then the common difference is

25. Suppose $a, b$ and $c$ are integers such that $a>b>c>0$ , and $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$ . Then the value of the determinant of $A$

26. If $A=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 0\end{array}\right]$ then $A^{2022}$ is

None of these
$\left[\begin{array}{cc}1 & 0 \\ 1011 & 0\end{array}\right]$
$\left[\begin{array}{cc}1 & 0 \\ 2022 & 0\end{array}\right]$
$\left[\begin{array}{cc}1 & 0 \\ \frac{1}{2^{2022}} & 0\end{array}\right]$

27. For $0\lt\theta\lt\frac{\pi}{4}$ , let $a=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \cos \theta\right), b=\left((\cos \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$ , $c=\left((\sin \theta)^{\cos \theta}\right)\left(\log _{2} \cos \theta\right)$ and $d=\left((\sin \theta)^{\sin \theta}\right)\left(\log _{2} \sin \theta\right)$ . Then, the median value in the sequence $a, b, c, d$ is

$\frac{a+b}{2}$
$\frac{a+d}{2}$
$\frac{b+c}{2}$
$\frac{c+d}{2}$

28. Let $A=\{1,2,3\}$ and $B=\{a, b\}$ . Assuming all relations from set $A$ to set $B$ are equally likely, what is the probability that a relation from $A$ to $B$ is also a function?

$\frac{1}{8}$
$\frac{1}{2}$
1
$\frac{3^{2}}{2^{6}}$

29. In a 400-metre race, Ashok beats Bipin and Chandan respectively by 15 seconds and 25 seconds. If Ashok beats Bipin by 150 metres, by how many metres does Bipin beat Chandan in the race?

30. In a bowl containing 60 ml orange juice, 40 ml of water is poured. Thereafter, 100 ml of apple juice is poured to make a fruit punch. Madhu drinks 50 ml of this fruit punch and comments that the proportion of orange juice needs to be higher for better taste. How much orange juice should be poured into the fruit punch that remained, in order to bring up the level of orange juice to 50 percentage?

31. The curve represented by the equation $\frac{x^{2}}{\sin \sqrt{2}-\sin \sqrt{3}}+\frac{y^{2}}{\cos \sqrt{2}-\cos \sqrt{3}}=1$ is

32. A set of all possible values the function $f(x)=\frac{x}{|x|}$ , where $x \neq 0$ , takes is

33. When the square of the difference of two natural numbers is subtracted from the square of the sum of the same two numbers and the result is divided by four, we get

34. The cost of a piece of jewellery is proportional to the square of its weight. A piece of jewellery weighing 10 grams is INR 3600. The cost of a piece of jewellery of the same kind weighing 4 grams is

35. The sum of the squares of all the roots of the equation $x^{2}+|x+4|+|x-4|-35=0$ is

36. Let A and B be two sets such that the Cartesian product A × B consists of four elements. If two elements of A × B are (1,4) and (4,1) , then

37. If $f\left(x^{2}+f(y)\right)=x f(x)+y$ for all non-negative integers $x$ and $y$ , then the value of $[f(0)]^{2}+f(0)$ equals _________.

38. If one of the factors of the number $3^{7} 2^{8} 17^{3}$ is randomly chosen, then the probability that the chosen factor will be a perfect square is

$\frac{5}{36}$
$\frac{1}{12}$
$\frac{3}{40}$
$\frac{5}{32}$

39. The number of four-digit integers which are greater than 1000 and divisible by both 2 and 3, but not by 5, is

40. Ayesha is standing atop a vertical tower 200m high and observes a car moving away from the tower on a straight, horizontal road from the foot of the tower. At 11:00 AM, she observes the angle of depression of the car to be 45^∘ . At 11:02 AM, she observes the angle of depression of the car to be 30^∘ . The speed at which the car is moving is approximately

Q.41 – 45 A showroom is open on all seven days of the week throughout the year. There are five employees Alex, Bhabha, Cathy, Dilip and Ethan who work in the showroom. Every day except Sunday, two employees are required while on Sunday three employees need to work. Every employee works for three days in a week. Some additional information is also provided:

– Every employee works on two consecutive days while the third day is not consecutive.
– Alex and Dilip work together on Tuesday and Wednesday while the other working day differs for them.
– Neither Bhabha nor Cathy works with Alex on any day.
– Cathy does not work either on Saturday or on Monday.

41. Number of days Bhabha and Cathy work together in a week is

42. Which among the following employees do not work together on any of the days?

43. One of the days Alex works on is

44. The consecutive days on which Ethan works are

45. Employees who work on Sunday are