IPM Question Paper 2024

Solve and Review 2024 Questions

Prepare effectively for IPMAT 2024 by solving Quantitative Aptitude questions from the Indore paper. Focus on accuracy and speed while attempting each problem, then review detailed solutions to learn correct methods, identify mistakes, and refine strategies.

IPM Indore Question Papers

1. The number of factors of 1800 that are multiple of 6 is ______________

2. The number of real solutions of the equation $ (x^2 – 15x + 55)^{x^2 – 5x + 6} = 1 $ is ____.

3. The following table shows the number of employees and their median age in eight companies located in a district.

Company	Number of employees	Median age
A	32	24
B	28	30
C	43	39
D	39	45
E	35	49
F	29	54
G	23	59
H	16	63

It is known that the age of all employees are integers. It is known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C. the age of every employee in G is strictly less than the age of every employee in H. The highest possible age of an employee of company A is ______

4. In a group of 150 students, 52 like tea, 48 like juice and 62 like coffee. If each student in the group likes at least one among tea, juice and coffee, then the maximum number of students that like more than one drink is _____ .

5. Let ABC be a triangle right-angled at B with AB = BC = 18. The area of largest rectangle that can be inscribed in this triangle and has B as one of the vertices is ______

6. A fruit seller has oranges, apples and bananas in the ratio 3:6:7. If the number of oranges is a multiple of both 5 and 6, then the minimum number of fruits the seller has is ______

160

7. The number of pairs $(x, y)$ of integers satisfying the inequality $| x - 5 | + | y - 5 | \leq 6$ is ______.

8. The price of a chocolate is increased by x% and then reduced by x%. The new price is 96.76% of the original price. Then x is _____ .

9. Let $f$ and $g$ be two functions defined by $f (x) = | x + | x | |$ and $g (x) = \frac{1}{x}$ for $x \neq 0$ . If $f (a) + g (f (a)) = \frac{13}{6}$ for some real a, then the maximum possible value of $f (g (a))$ is ________.

10. The following table shows the number of employees and their median age in eight companies located in a district.

Company	Number of employees	Median age
A	32	24
B	28	30
C	43	39
D	39	45
E	35	49
F	29	54
G	23	59
H	16	63

It is known that the age of all employees are integers. It is known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C….. the age of every employee in G is strictly less than the age of every employee in H. The median age of an employee across the eight companies is______.

11. The following table shows the number of employees and their median age in eight companies located in a district.

Company	Number of employees	Median age
A	32	24
B	28	30
C	43	39
D	39	45
E	35	49
F	29	54
G	23	59
H	16	63

It is known that the age of all employees are integers. It is known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C….. the age of every employee in G is strictly less than the age of every employee in H. In company F, the lowest possible sum of the ages of all employees is ______

1510

12. If $4^{\log_2 x}\; \; – \; 4x + 9^{\log_3 y} \; \; \; 16y + 68 = 0$, then y – x equals.

13. Person A borrows Rs. 4000 from another person B for a duration of 4 years. He borrows a portion of it at 3% simple interest per annum, while the rest at 4% simple interest per annum. If B gets Rs. 520 as total interest, then the amount A borrowed at 3% per annum in Rs. is _______

3000

14. The number of triangles with integer sides and with perimeter 15 is _______

15. If $ A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix} $is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of $A$ is.

288

16. The angle of elevation of the top of a pole from a point A on the ground is $30^{\circ}$ . The angle of elevation changes to $45^{\circ}$ , after moving 20 meters towards the base of the pole. Then the height of the pole, in meters, is

A. 30

B. $15(\sqrt{5}+1)$

C. $20(\sqrt{3}+1)$

D. $10(\sqrt{3}+1)$

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

A. $\frac{1}{5}$

B. $\frac{1}{7}$

C. $\frac{1}{2}$

D. 2

18. If $\log_{4} x = a$ and $\log_{25} x = b$, then $\log_{x} 10$ is

A. $\frac{a+b}{2(a-b)}$

B. $\frac{a+b}{2}$

C. $\frac{a+b}{2ab}$

D. $\frac{a-b}{2ab}$

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. $\left(3\sqrt{3}+\pi\right)\left(\frac{k}{26}\right)$

B. $\left(3\sqrt{3}-\pi\right)\left(\frac{k^2}{24}\right)$

C. $\left(3\sqrt{3}+\pi\right)\left(\frac{k^2}{24}\right)$

D.$\left(3\sqrt{3}-\pi\right)\left(\frac{k^2}{6}\right)$

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

A. $15^\circ$

B. $60^\circ$

C. $30^\circ$

D.$10^\circ$

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. $\frac{2}{7}$

B. $\frac{1}{7}$

C. $\frac{1}{4}$

D. $\frac{1}{3}$

22. The side $A B$ of a triangle $A B C$ is $c$ . The median $B D$ is of length $k$ . If $∠ B D A = θ <$ $90^{\circ}$ , then the area of triangle $A B C$ is

A. $\frac{k^2\cos 2\theta}{2}+k\sin\theta\sqrt{c^2-k^2\cos^2\theta}$

B. $\frac{k^2\sin 2\theta}{2}+k\sin\theta\sqrt{c^2-k^2\sin^2\theta}$

C. $\frac{k^2\cos\theta}{4}-k\sin\theta\sqrt{c^2+k^2\sin^2\theta}$

D. $\frac{k^2\cos 2\theta}{4}+k\sin\theta\sqrt{c^2-k^2\sin^2\theta}$

23. Let $a=\frac{(\log_{7}4)(\log_{7}5-\log_{7}2)}{(\log_{7}25)(\log_{7}8-\log_{7}4)}$.
Then the value of $5^a$ is

A. $\frac{7}{2}$

B. $5$

C. $8$

D. $\frac{5}{2}$

24. For some non-zero real values of $a, b$ and $c$, it is given that
$\left|\frac{c}{a}\right|=4$, $\left|\frac{a}{b}\right|=\frac{1}{3}$ and $\frac{b}{c}=-\frac{3}{4}$.
If $ac>0$, then $\left(\frac{b+c}{a}\right)$ 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
$(x^2-5)^4+(x^2-7)^4=16$ is

A. $2\sqrt{5}$

B. $2\sqrt{7}$

C. $\sqrt{7}$

D. $\sqrt{10}$

26. If $\theta$ is the angle between the pair of tangents drawn from the point
$A(0,\tfrac{7}{2})$ to the circle $x^2+y^2-14x+16y+88=0$, then $\tan\theta$ equals

A. $\frac{5}{4}$

B. $\frac{20}{21}$

C. $\frac{2}{5}$

D. $\frac{4}{5}$

27. The numbers $2^{2024}$ and $5^{2024}$ 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

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

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

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

B. S5

C. S3

D. S4

IPM Question Paper 2024 ​

Solve and Review 2024 Questions​

IPM Indore Question Papers

IPM Question Paper 2024

Solve and Review 2024 Questions