IPMAT 2023: Past Year Question Paper and Solution | 100 DILR Questions for CAT Preparation PDF Download

 Page 1


                 
                           Page 1 of 12 
  
IPMAT Indore- 2023 
 
QUANTITATIVE ABILITY - (SA) 
 
Questions: 15 (+4/-0)                                           Sectional Time: 40 Minutes 
 
1. Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. 
The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km 
journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, 
if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, 
then the total time taken, in minutes, to complete the journey, will be______.  
 
2. If three consecutive coefficients in the expansion of (x + y)
n
 are in the ratio 1:9:63, then the value of n is _______. 
 
3. The total number of positive integer solutions of 21 = a + b + c = 25 is______. 
 
4. The product of the roots of the equation log
2
2
(log
2
x)
2
- 5log
2
x + 6 = 0  is ________. 
 
5. If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of ƒ (2023) is _________. 
 
6. If f(n)= 1 + 2 + 3 +···+(n+1) and g(n) = ?
1
f(k)
k=n
k=1
then the least value of n for which g(n) exceeds the value 
99
100 
is_________. 
 
7. The polynomial 4x
10
- x
9
+ 3x
8
- 5x
7
+ cx
6
+ 2x
5
- x
4
+ x
3
- 4x
2
+ 6x - 2 when divided by x - 1 leaves a remainder 
2. Then the value of c + 6 is________. 
 
8. The remainder when 1! + 2! + 3! +···+95! is divided by 15 is______. 
 
9. Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum 
possible value of a + c is________. 
 
10. In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y
2
 = 4 such 
that the area of the triangle ABC is an integer. The number of points in the set S is________. 
 
11. Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and 
resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve 
days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker 
to complete the task alone would be ______. 
 
12. In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid 
votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more 
than the other candidate, then the number of voters in the voting list is_________.  
 
13. Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to 
invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase 
by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% 
per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each 
year, the total interest earned by her, in INR, in five years from today will be__________. 
 
14. In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
?? against every other player belonging to one's own group exactly once; 
?? against each player belonging to one of the remaining three groups exactly twice; 
?? against each player belonging to one of the remaining two groups exactly three times; and 
?? against each player belonging to the remaining group exactly four times. 
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in 
each group is_______. 
 
 
15. The length of the line segment joining the two intersection points of the curves y = 4970 - |x| and y = x² is_________. 
 
  
Page 2


                 
                           Page 1 of 12 
  
IPMAT Indore- 2023 
 
QUANTITATIVE ABILITY - (SA) 
 
Questions: 15 (+4/-0)                                           Sectional Time: 40 Minutes 
 
1. Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. 
The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km 
journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, 
if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, 
then the total time taken, in minutes, to complete the journey, will be______.  
 
2. If three consecutive coefficients in the expansion of (x + y)
n
 are in the ratio 1:9:63, then the value of n is _______. 
 
3. The total number of positive integer solutions of 21 = a + b + c = 25 is______. 
 
4. The product of the roots of the equation log
2
2
(log
2
x)
2
- 5log
2
x + 6 = 0  is ________. 
 
5. If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of ƒ (2023) is _________. 
 
6. If f(n)= 1 + 2 + 3 +···+(n+1) and g(n) = ?
1
f(k)
k=n
k=1
then the least value of n for which g(n) exceeds the value 
99
100 
is_________. 
 
7. The polynomial 4x
10
- x
9
+ 3x
8
- 5x
7
+ cx
6
+ 2x
5
- x
4
+ x
3
- 4x
2
+ 6x - 2 when divided by x - 1 leaves a remainder 
2. Then the value of c + 6 is________. 
 
8. The remainder when 1! + 2! + 3! +···+95! is divided by 15 is______. 
 
9. Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum 
possible value of a + c is________. 
 
10. In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y
2
 = 4 such 
that the area of the triangle ABC is an integer. The number of points in the set S is________. 
 
11. Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and 
resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve 
days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker 
to complete the task alone would be ______. 
 
12. In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid 
votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more 
than the other candidate, then the number of voters in the voting list is_________.  
 
13. Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to 
invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase 
by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% 
per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each 
year, the total interest earned by her, in INR, in five years from today will be__________. 
 
14. In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
?? against every other player belonging to one's own group exactly once; 
?? against each player belonging to one of the remaining three groups exactly twice; 
?? against each player belonging to one of the remaining two groups exactly three times; and 
?? against each player belonging to the remaining group exactly four times. 
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in 
each group is_______. 
 
 
15. The length of the line segment joining the two intersection points of the curves y = 4970 - |x| and y = x² is_________. 
 
  
                 
                           Page 2 of 12 
  
QUANTITATIVE ABILITY - (MCQ) 
 
Questions: 30 (+4/-1)                                           Sectional Time: 40 Minutes 
 
16. If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4? 
 (a)  
1
4
 (b)  
2
3
 (c)  
1
2
 (d)  
1
3
 
  
17. A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid 
spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost 
of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made 
a total profit of 20%, then the listed price of each golden bead, in INR, was 
 (a)  12000 (b)  9600 (c)  48000  (d)  24000 
 
18. Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If log
n
 (log
2
 a) = 1, 
log
n
 (log
2
 b) = 2 and log
n
 (log
2
 c) = 3, then which of the following is true?  
 (a) a
n
+ b
n
= c
n
 (b) (a
n
+ b)
n
= ac (c) a + b = c (d) (b - a)
n
= (c - b) 
 
19. If the harmonic mean of the roots of the equation (5 + v2)x² - bx + 8 + 2v5 = 0 is 4 then the value of b is 
 (a) 4 - v5 (b) 2 (c) 4 + v5 (d) 3 
 
20. Consider an 8 × 8 chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in 
the same row and no two are in the same column is 
 (a) 7 (b) 8 (c) 7! (d) 8! 
 
21. The set of all real values of x satisfying the inequality
x
2
(x+1)
(x-1)(2x+1)
3
> 0 is  
 (a) (-1, -
1
2
) ? (0, +8)   (b) (-8, -1) ? (-
1
2
, 0) ? (1, +8) 
 (c) (-1,0) ? (1, +8)   (d) (-1, -
1
2
) ? (1, +8) 
 
22. If A = [
1 2
3 a
] where a is a real number and det (A
3
- 3A
2
- 5A) = 0 then one of the values of a can be 
 (a) 1  (b) 6 (c) 4 (d) 5 
 
23. If the difference between compound interest and simple interest for a certain amount of money invested for 3 years 
at an annual interest rate of 10% is INR 527, then the amount invested in INR is 
 (a) 170000 (b) 15000 (c) 150000 (d) 17000 
 
24. In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is 
known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support 
Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is 
 (a) Exactly 20 (b) Between 15 and 25 (c) Between 20 and 25 (d) 30 or more 
 
25. The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 
0.8 is 
 (a) 5 (b) 7 (c) 3 (d) 6 
 
26. A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The 
remainder when P(x) is divided by (x - 1) (x - 2) is 
 (a) x - 3 (b) 3 - x (c) 3 (d) 2 
 
27. Let [x] denote the greatest integer not exceeding x and {x} = x –[x]. If n is a natural number, then the sum of all values 
of x satisfying the equation 2[x] = x + n{x} is 
 (a) n (b) 
n(n+1)
2
 (c) 
3
2
 (d) 
n(n+2)
2
 
  
Page 3


                 
                           Page 1 of 12 
  
IPMAT Indore- 2023 
 
QUANTITATIVE ABILITY - (SA) 
 
Questions: 15 (+4/-0)                                           Sectional Time: 40 Minutes 
 
1. Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. 
The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km 
journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, 
if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, 
then the total time taken, in minutes, to complete the journey, will be______.  
 
2. If three consecutive coefficients in the expansion of (x + y)
n
 are in the ratio 1:9:63, then the value of n is _______. 
 
3. The total number of positive integer solutions of 21 = a + b + c = 25 is______. 
 
4. The product of the roots of the equation log
2
2
(log
2
x)
2
- 5log
2
x + 6 = 0  is ________. 
 
5. If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of ƒ (2023) is _________. 
 
6. If f(n)= 1 + 2 + 3 +···+(n+1) and g(n) = ?
1
f(k)
k=n
k=1
then the least value of n for which g(n) exceeds the value 
99
100 
is_________. 
 
7. The polynomial 4x
10
- x
9
+ 3x
8
- 5x
7
+ cx
6
+ 2x
5
- x
4
+ x
3
- 4x
2
+ 6x - 2 when divided by x - 1 leaves a remainder 
2. Then the value of c + 6 is________. 
 
8. The remainder when 1! + 2! + 3! +···+95! is divided by 15 is______. 
 
9. Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum 
possible value of a + c is________. 
 
10. In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y
2
 = 4 such 
that the area of the triangle ABC is an integer. The number of points in the set S is________. 
 
11. Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and 
resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve 
days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker 
to complete the task alone would be ______. 
 
12. In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid 
votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more 
than the other candidate, then the number of voters in the voting list is_________.  
 
13. Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to 
invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase 
by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% 
per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each 
year, the total interest earned by her, in INR, in five years from today will be__________. 
 
14. In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
?? against every other player belonging to one's own group exactly once; 
?? against each player belonging to one of the remaining three groups exactly twice; 
?? against each player belonging to one of the remaining two groups exactly three times; and 
?? against each player belonging to the remaining group exactly four times. 
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in 
each group is_______. 
 
 
15. The length of the line segment joining the two intersection points of the curves y = 4970 - |x| and y = x² is_________. 
 
  
                 
                           Page 2 of 12 
  
QUANTITATIVE ABILITY - (MCQ) 
 
Questions: 30 (+4/-1)                                           Sectional Time: 40 Minutes 
 
16. If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4? 
 (a)  
1
4
 (b)  
2
3
 (c)  
1
2
 (d)  
1
3
 
  
17. A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid 
spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost 
of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made 
a total profit of 20%, then the listed price of each golden bead, in INR, was 
 (a)  12000 (b)  9600 (c)  48000  (d)  24000 
 
18. Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If log
n
 (log
2
 a) = 1, 
log
n
 (log
2
 b) = 2 and log
n
 (log
2
 c) = 3, then which of the following is true?  
 (a) a
n
+ b
n
= c
n
 (b) (a
n
+ b)
n
= ac (c) a + b = c (d) (b - a)
n
= (c - b) 
 
19. If the harmonic mean of the roots of the equation (5 + v2)x² - bx + 8 + 2v5 = 0 is 4 then the value of b is 
 (a) 4 - v5 (b) 2 (c) 4 + v5 (d) 3 
 
20. Consider an 8 × 8 chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in 
the same row and no two are in the same column is 
 (a) 7 (b) 8 (c) 7! (d) 8! 
 
21. The set of all real values of x satisfying the inequality
x
2
(x+1)
(x-1)(2x+1)
3
> 0 is  
 (a) (-1, -
1
2
) ? (0, +8)   (b) (-8, -1) ? (-
1
2
, 0) ? (1, +8) 
 (c) (-1,0) ? (1, +8)   (d) (-1, -
1
2
) ? (1, +8) 
 
22. If A = [
1 2
3 a
] where a is a real number and det (A
3
- 3A
2
- 5A) = 0 then one of the values of a can be 
 (a) 1  (b) 6 (c) 4 (d) 5 
 
23. If the difference between compound interest and simple interest for a certain amount of money invested for 3 years 
at an annual interest rate of 10% is INR 527, then the amount invested in INR is 
 (a) 170000 (b) 15000 (c) 150000 (d) 17000 
 
24. In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is 
known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support 
Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is 
 (a) Exactly 20 (b) Between 15 and 25 (c) Between 20 and 25 (d) 30 or more 
 
25. The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 
0.8 is 
 (a) 5 (b) 7 (c) 3 (d) 6 
 
26. A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The 
remainder when P(x) is divided by (x - 1) (x - 2) is 
 (a) x - 3 (b) 3 - x (c) 3 (d) 2 
 
27. Let [x] denote the greatest integer not exceeding x and {x} = x –[x]. If n is a natural number, then the sum of all values 
of x satisfying the equation 2[x] = x + n{x} is 
 (a) n (b) 
n(n+1)
2
 (c) 
3
2
 (d) 
n(n+2)
2
 
  
                 
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28. If 
a+b
b+c
=
c+d
d+a
, which of the following statements is always true? 
 (a) a = c   (b) a = c, and  b = d 
 (c) a + b + c + d = 0   (d) a = c, or a + b + c + d = 0 
 
29. If log
cos x
(sinx) + log
sin x
(cosx) = 2, then the value of x is 
 (a) np +
p 
4
, n is an interger  (b) 2n p +
p 
4
, n is an interger 
 (c) 
np 
4
+
p 
4
, n is an interger  (d) 
4p 
4
, n is an interger 
 
30. A helicopter flies along the sides of a square field of side length 100 kms. The first side is covered at a speed of 100 
kmph, and for each subsequent side the speed is increased by 100 kmph till it covers all the sides. The average speed 
of the helicopter is  
 (a) 184 kmph (b) 200 kmph (c) 250 kmph (d) 192 kmph 
 
31. In a triangle ABC, let D be the mid-point of BC, and AM be the altitude on BC. If the lengths of AB, BC and CA are in 
the ratio of 2:4:3, then the ratio of the lengths of BM and AD would be 
 (a) 11:12 (b) 11: 4v10 (c) 12:11 (d) 4v10: 11 
 
32. In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results 
in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot 
represent the scores of the five players? 
 (a) 3, 2, 2, 2, 1 (b) 4, 4, 1, 1, 0 (c) 2, 2, 2, 2, 2 (d) 3, 3, 2, 1, 1 
 
33. A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either 
one step at a time or two steps at a time. The number of ways it can reach the top is 
 (a) 34 (b) 55 (c) 144 (d) 89 
 
34. Let a
1
, a
2
, a
3
 be three distinct real nubers in geometric progression. If the eqations a
1
x
2
+ 2a
2
x + a
3
= 0 and b
1
x
2
+
2b
2
x + b
3
+= 0 have a common root, then which of the following is necessarily ture?   
 (a) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in arithmetic progre ssion (b) b
1
, b
2
, b
3
 are in arithmetic progression 
 (c) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in geometric progression  (d) b
1
, b
2
, b
3
 are in geometric progression  
 
35. Which of the following straight lines are both tangent to the circle x
2
+ y
2
- 6x + 4y - 12 = 0  
 (a) 4x + 3y + 19 = 0, 4x + 3y - 31 = 0. (b) 4x + 3y + 19 = 0, 4x + 3y + 31 = 0. 
 (c) 4x + 3y - 19 = 0, 4x + 3y - 31 = 0. (d) 4x + 3y - 19 = 0, 4x + 3y + 31 = 0. 
 
36. A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 
16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting 
point after an infinite number of such walks in this pattern? 
 (a) 64 /v5 meters   (b) 64 meters 
 (c) 32 meters   (d) 32 /v5 meters 
 
37. The equation x
2
+ y
2
- 2x - 4y + 5 = 0 represents  
 (a) a point   (b) a pair of straight lines 
 (c) an ellipse   (d) a circle 
 
38. If cos a + cos ß = 1, then the maximum value of sin a - sin ß is  
 (a) 1 (b) 2 (c) v3 (d) v2 
 
39. "Let p be a positive integer such that the unit digit of" p
3
 is 4. What are the possible unit digits of (p+3)
3
? 
 (a) 1,7,9 (b) 4,7 (c) 1,3,7 (d) 3 
 
40. The probability that a randomly chosen positive divisor of 10
2023
 is an integer multiple of 10
2001
 is  
 (a) 
23
2024
 (b) 
23
2
2023
2
 (c) 
23
2023
 (d) 
23
2
2024
2
 
Page 4


                 
                           Page 1 of 12 
  
IPMAT Indore- 2023 
 
QUANTITATIVE ABILITY - (SA) 
 
Questions: 15 (+4/-0)                                           Sectional Time: 40 Minutes 
 
1. Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. 
The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km 
journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, 
if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, 
then the total time taken, in minutes, to complete the journey, will be______.  
 
2. If three consecutive coefficients in the expansion of (x + y)
n
 are in the ratio 1:9:63, then the value of n is _______. 
 
3. The total number of positive integer solutions of 21 = a + b + c = 25 is______. 
 
4. The product of the roots of the equation log
2
2
(log
2
x)
2
- 5log
2
x + 6 = 0  is ________. 
 
5. If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of ƒ (2023) is _________. 
 
6. If f(n)= 1 + 2 + 3 +···+(n+1) and g(n) = ?
1
f(k)
k=n
k=1
then the least value of n for which g(n) exceeds the value 
99
100 
is_________. 
 
7. The polynomial 4x
10
- x
9
+ 3x
8
- 5x
7
+ cx
6
+ 2x
5
- x
4
+ x
3
- 4x
2
+ 6x - 2 when divided by x - 1 leaves a remainder 
2. Then the value of c + 6 is________. 
 
8. The remainder when 1! + 2! + 3! +···+95! is divided by 15 is______. 
 
9. Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum 
possible value of a + c is________. 
 
10. In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y
2
 = 4 such 
that the area of the triangle ABC is an integer. The number of points in the set S is________. 
 
11. Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and 
resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve 
days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker 
to complete the task alone would be ______. 
 
12. In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid 
votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more 
than the other candidate, then the number of voters in the voting list is_________.  
 
13. Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to 
invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase 
by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% 
per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each 
year, the total interest earned by her, in INR, in five years from today will be__________. 
 
14. In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
?? against every other player belonging to one's own group exactly once; 
?? against each player belonging to one of the remaining three groups exactly twice; 
?? against each player belonging to one of the remaining two groups exactly three times; and 
?? against each player belonging to the remaining group exactly four times. 
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in 
each group is_______. 
 
 
15. The length of the line segment joining the two intersection points of the curves y = 4970 - |x| and y = x² is_________. 
 
  
                 
                           Page 2 of 12 
  
QUANTITATIVE ABILITY - (MCQ) 
 
Questions: 30 (+4/-1)                                           Sectional Time: 40 Minutes 
 
16. If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4? 
 (a)  
1
4
 (b)  
2
3
 (c)  
1
2
 (d)  
1
3
 
  
17. A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid 
spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost 
of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made 
a total profit of 20%, then the listed price of each golden bead, in INR, was 
 (a)  12000 (b)  9600 (c)  48000  (d)  24000 
 
18. Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If log
n
 (log
2
 a) = 1, 
log
n
 (log
2
 b) = 2 and log
n
 (log
2
 c) = 3, then which of the following is true?  
 (a) a
n
+ b
n
= c
n
 (b) (a
n
+ b)
n
= ac (c) a + b = c (d) (b - a)
n
= (c - b) 
 
19. If the harmonic mean of the roots of the equation (5 + v2)x² - bx + 8 + 2v5 = 0 is 4 then the value of b is 
 (a) 4 - v5 (b) 2 (c) 4 + v5 (d) 3 
 
20. Consider an 8 × 8 chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in 
the same row and no two are in the same column is 
 (a) 7 (b) 8 (c) 7! (d) 8! 
 
21. The set of all real values of x satisfying the inequality
x
2
(x+1)
(x-1)(2x+1)
3
> 0 is  
 (a) (-1, -
1
2
) ? (0, +8)   (b) (-8, -1) ? (-
1
2
, 0) ? (1, +8) 
 (c) (-1,0) ? (1, +8)   (d) (-1, -
1
2
) ? (1, +8) 
 
22. If A = [
1 2
3 a
] where a is a real number and det (A
3
- 3A
2
- 5A) = 0 then one of the values of a can be 
 (a) 1  (b) 6 (c) 4 (d) 5 
 
23. If the difference between compound interest and simple interest for a certain amount of money invested for 3 years 
at an annual interest rate of 10% is INR 527, then the amount invested in INR is 
 (a) 170000 (b) 15000 (c) 150000 (d) 17000 
 
24. In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is 
known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support 
Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is 
 (a) Exactly 20 (b) Between 15 and 25 (c) Between 20 and 25 (d) 30 or more 
 
25. The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 
0.8 is 
 (a) 5 (b) 7 (c) 3 (d) 6 
 
26. A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The 
remainder when P(x) is divided by (x - 1) (x - 2) is 
 (a) x - 3 (b) 3 - x (c) 3 (d) 2 
 
27. Let [x] denote the greatest integer not exceeding x and {x} = x –[x]. If n is a natural number, then the sum of all values 
of x satisfying the equation 2[x] = x + n{x} is 
 (a) n (b) 
n(n+1)
2
 (c) 
3
2
 (d) 
n(n+2)
2
 
  
                 
©SuperGrads                           Page 3 of 12 
Replication or other unauthorized use of this material is prohibited by the copyright laws of India    
28. If 
a+b
b+c
=
c+d
d+a
, which of the following statements is always true? 
 (a) a = c   (b) a = c, and  b = d 
 (c) a + b + c + d = 0   (d) a = c, or a + b + c + d = 0 
 
29. If log
cos x
(sinx) + log
sin x
(cosx) = 2, then the value of x is 
 (a) np +
p 
4
, n is an interger  (b) 2n p +
p 
4
, n is an interger 
 (c) 
np 
4
+
p 
4
, n is an interger  (d) 
4p 
4
, n is an interger 
 
30. A helicopter flies along the sides of a square field of side length 100 kms. The first side is covered at a speed of 100 
kmph, and for each subsequent side the speed is increased by 100 kmph till it covers all the sides. The average speed 
of the helicopter is  
 (a) 184 kmph (b) 200 kmph (c) 250 kmph (d) 192 kmph 
 
31. In a triangle ABC, let D be the mid-point of BC, and AM be the altitude on BC. If the lengths of AB, BC and CA are in 
the ratio of 2:4:3, then the ratio of the lengths of BM and AD would be 
 (a) 11:12 (b) 11: 4v10 (c) 12:11 (d) 4v10: 11 
 
32. In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results 
in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot 
represent the scores of the five players? 
 (a) 3, 2, 2, 2, 1 (b) 4, 4, 1, 1, 0 (c) 2, 2, 2, 2, 2 (d) 3, 3, 2, 1, 1 
 
33. A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either 
one step at a time or two steps at a time. The number of ways it can reach the top is 
 (a) 34 (b) 55 (c) 144 (d) 89 
 
34. Let a
1
, a
2
, a
3
 be three distinct real nubers in geometric progression. If the eqations a
1
x
2
+ 2a
2
x + a
3
= 0 and b
1
x
2
+
2b
2
x + b
3
+= 0 have a common root, then which of the following is necessarily ture?   
 (a) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in arithmetic progre ssion (b) b
1
, b
2
, b
3
 are in arithmetic progression 
 (c) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in geometric progression  (d) b
1
, b
2
, b
3
 are in geometric progression  
 
35. Which of the following straight lines are both tangent to the circle x
2
+ y
2
- 6x + 4y - 12 = 0  
 (a) 4x + 3y + 19 = 0, 4x + 3y - 31 = 0. (b) 4x + 3y + 19 = 0, 4x + 3y + 31 = 0. 
 (c) 4x + 3y - 19 = 0, 4x + 3y - 31 = 0. (d) 4x + 3y - 19 = 0, 4x + 3y + 31 = 0. 
 
36. A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 
16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting 
point after an infinite number of such walks in this pattern? 
 (a) 64 /v5 meters   (b) 64 meters 
 (c) 32 meters   (d) 32 /v5 meters 
 
37. The equation x
2
+ y
2
- 2x - 4y + 5 = 0 represents  
 (a) a point   (b) a pair of straight lines 
 (c) an ellipse   (d) a circle 
 
38. If cos a + cos ß = 1, then the maximum value of sin a - sin ß is  
 (a) 1 (b) 2 (c) v3 (d) v2 
 
39. "Let p be a positive integer such that the unit digit of" p
3
 is 4. What are the possible unit digits of (p+3)
3
? 
 (a) 1,7,9 (b) 4,7 (c) 1,3,7 (d) 3 
 
40. The probability that a randomly chosen positive divisor of 10
2023
 is an integer multiple of 10
2001
 is  
 (a) 
23
2024
 (b) 
23
2
2023
2
 (c) 
23
2023
 (d) 
23
2
2024
2
 
                 
                           Page 4 of 12 
  
 Directions (Q.41-Q.45): A pharmaceutical company has tested five drugs on three different organisms. The following 
incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B 
and C work on Q. 
 Organism 
P Q R 
Drug 
A   
? 
B  
? 
 
C  
? 
 
D    
E    
 
Following additional information is available: 
?? Each drug works on at least one organism but not more than two organisms. 
?? Each organism can be treated with at least two and at most three of these five drugs. 
?? On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
?? D and E do not work on the same organism. 
 
41. Organism R can be treated with 
 (a) Only A and E (b) A, B and E (c) A, B and C (d) Only A and B 
 
42. Drug E works on 
 (a) Only R (b) Q and R (c) Only P (d) P and R 
 
43. The organism(s) that can be treated with three of these five drugs is(are) 
 (a) Only Q (b) Only P (c) P and Q (d) Q and R 
 
44. Drug D works on 
 (a) Only Q (b) P and Q (c) Q and R (d) Only P 
 
45. Organism P can be treated with 
 (a) B, C and D (b) Only C and D (c) Only B and D (d) A, B, and D 
 
  
Page 5


                 
                           Page 1 of 12 
  
IPMAT Indore- 2023 
 
QUANTITATIVE ABILITY - (SA) 
 
Questions: 15 (+4/-0)                                           Sectional Time: 40 Minutes 
 
1. Vinita drives a car which has four gears. The speed of the car in the fourth gear is five times its speed in the first gear. 
The car takes twice the time to travel a certain distance in the second gear as compared to the third gear. In a 100 km 
journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes to complete the journey. Instead, 
if the distances covered in the first, second, third, and fourth gears are 4 km, 4 km, 32 km, and 60 km, respectively, 
then the total time taken, in minutes, to complete the journey, will be______.  
 
2. If three consecutive coefficients in the expansion of (x + y)
n
 are in the ratio 1:9:63, then the value of n is _______. 
 
3. The total number of positive integer solutions of 21 = a + b + c = 25 is______. 
 
4. The product of the roots of the equation log
2
2
(log
2
x)
2
- 5log
2
x + 6 = 0  is ________. 
 
5. If f(1) = 1 and f(n) = 3n - f(n - 1) for all integers n > 1 , then the value of ƒ (2023) is _________. 
 
6. If f(n)= 1 + 2 + 3 +···+(n+1) and g(n) = ?
1
f(k)
k=n
k=1
then the least value of n for which g(n) exceeds the value 
99
100 
is_________. 
 
7. The polynomial 4x
10
- x
9
+ 3x
8
- 5x
7
+ cx
6
+ 2x
5
- x
4
+ x
3
- 4x
2
+ 6x - 2 when divided by x - 1 leaves a remainder 
2. Then the value of c + 6 is________. 
 
8. The remainder when 1! + 2! + 3! +···+95! is divided by 15 is______. 
 
9. Let a, b, c, d be positive integers such that a + b + c + d = 2023. If a : b = 2 : 5 and c : d = 5 : 2 then the maximum 
possible value of a + c is________. 
 
10. In the xy-plane let A = (-2,0), B = (2,0). Define the set S as the collection of all points C on the circle x² + y
2
 = 4 such 
that the area of the triangle ABC is an integer. The number of points in the set S is________. 
 
11. Amisha can complete a particular task in twenty days. After working for four days she fell sick for four days and 
resumed the work on the ninth day but with half of her original work rate. She completed the task in another twelve 
days with the help of a co-worker who joined her from the ninth day. The number of days required for the co-worker 
to complete the task alone would be ______. 
 
12. In an election with only two contesting candidates, 15% of the voters did not turn up to vote, and 50 voters cast invalid 
votes. It is known that 44% of all the voters in the voting list voted for the winner. If the winner got 200 votes more 
than the other candidate, then the number of voters in the voting list is_________.  
 
13. Assume it is the beginning of the year today. Ankita will earn INR 10,000 at the end of the year, which she plans to 
invest in a bank deposit immediately at a fixed simple interest of 0.5% per annum. Her yearly income will increase 
by INR 10,000 every year, and the fixed simple interest offered by the bank on new deposits will also increase by 0.5% 
per annum every year. If Ankita continues to invest all her yearly income in new bank deposits at the end of each 
year, the total interest earned by her, in INR, in five years from today will be__________. 
 
14. In a chess tournament, there are four groups, each containing an equal number of players. Each player plays 
?? against every other player belonging to one's own group exactly once; 
?? against each player belonging to one of the remaining three groups exactly twice; 
?? against each player belonging to one of the remaining two groups exactly three times; and 
?? against each player belonging to the remaining group exactly four times. 
If there are more than 1000 matches being played in the tournament, the minimum possible number of players in 
each group is_______. 
 
 
15. The length of the line segment joining the two intersection points of the curves y = 4970 - |x| and y = x² is_________. 
 
  
                 
                           Page 2 of 12 
  
QUANTITATIVE ABILITY - (MCQ) 
 
Questions: 30 (+4/-1)                                           Sectional Time: 40 Minutes 
 
16. If a three-digit number is chosen at random, what is the probability that it is divisible neither by 3 nor by 4? 
 (a)  
1
4
 (b)  
2
3
 (c)  
1
2
 (d)  
1
3
 
  
17. A goldsmith bought a large solid golden ball at INR 1,000,000 and melted it to make a certain number of solid 
spherical beads such that the radius of each bead was one-fifth of the radius of the original ball. Assume that the cost 
of making golden beads is negligible. If the goldsmith sold all the beads at 20% discount on the listed price and made 
a total profit of 20%, then the listed price of each golden bead, in INR, was 
 (a)  12000 (b)  9600 (c)  48000  (d)  24000 
 
18. Let a, b, c be real numbers greater than 1, and n be a positive real number not equal to 1. If log
n
 (log
2
 a) = 1, 
log
n
 (log
2
 b) = 2 and log
n
 (log
2
 c) = 3, then which of the following is true?  
 (a) a
n
+ b
n
= c
n
 (b) (a
n
+ b)
n
= ac (c) a + b = c (d) (b - a)
n
= (c - b) 
 
19. If the harmonic mean of the roots of the equation (5 + v2)x² - bx + 8 + 2v5 = 0 is 4 then the value of b is 
 (a) 4 - v5 (b) 2 (c) 4 + v5 (d) 3 
 
20. Consider an 8 × 8 chessboard. The number of ways 8 rooks can be placed on the board such that no two rooks are in 
the same row and no two are in the same column is 
 (a) 7 (b) 8 (c) 7! (d) 8! 
 
21. The set of all real values of x satisfying the inequality
x
2
(x+1)
(x-1)(2x+1)
3
> 0 is  
 (a) (-1, -
1
2
) ? (0, +8)   (b) (-8, -1) ? (-
1
2
, 0) ? (1, +8) 
 (c) (-1,0) ? (1, +8)   (d) (-1, -
1
2
) ? (1, +8) 
 
22. If A = [
1 2
3 a
] where a is a real number and det (A
3
- 3A
2
- 5A) = 0 then one of the values of a can be 
 (a) 1  (b) 6 (c) 4 (d) 5 
 
23. If the difference between compound interest and simple interest for a certain amount of money invested for 3 years 
at an annual interest rate of 10% is INR 527, then the amount invested in INR is 
 (a) 170000 (b) 15000 (c) 150000 (d) 17000 
 
24. In a group of 120 students, 80 students are from the Science stream and the rest are from the Commerce stream. It is 
known that 70 students support Mumbai Indians in the Indian Premier League; all the other students support 
Chennai Super Kings. The number of Science students who are supporters of Mumbai Indians is 
 (a) Exactly 20 (b) Between 15 and 25 (c) Between 20 and 25 (d) 30 or more 
 
25. The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 
0.8 is 
 (a) 5 (b) 7 (c) 3 (d) 6 
 
26. A polynomial P(x) leaves a remainder 2 when divided by (x - 1) and a remainder 1 when divided by (x-2). The 
remainder when P(x) is divided by (x - 1) (x - 2) is 
 (a) x - 3 (b) 3 - x (c) 3 (d) 2 
 
27. Let [x] denote the greatest integer not exceeding x and {x} = x –[x]. If n is a natural number, then the sum of all values 
of x satisfying the equation 2[x] = x + n{x} is 
 (a) n (b) 
n(n+1)
2
 (c) 
3
2
 (d) 
n(n+2)
2
 
  
                 
©SuperGrads                           Page 3 of 12 
Replication or other unauthorized use of this material is prohibited by the copyright laws of India    
28. If 
a+b
b+c
=
c+d
d+a
, which of the following statements is always true? 
 (a) a = c   (b) a = c, and  b = d 
 (c) a + b + c + d = 0   (d) a = c, or a + b + c + d = 0 
 
29. If log
cos x
(sinx) + log
sin x
(cosx) = 2, then the value of x is 
 (a) np +
p 
4
, n is an interger  (b) 2n p +
p 
4
, n is an interger 
 (c) 
np 
4
+
p 
4
, n is an interger  (d) 
4p 
4
, n is an interger 
 
30. A helicopter flies along the sides of a square field of side length 100 kms. The first side is covered at a speed of 100 
kmph, and for each subsequent side the speed is increased by 100 kmph till it covers all the sides. The average speed 
of the helicopter is  
 (a) 184 kmph (b) 200 kmph (c) 250 kmph (d) 192 kmph 
 
31. In a triangle ABC, let D be the mid-point of BC, and AM be the altitude on BC. If the lengths of AB, BC and CA are in 
the ratio of 2:4:3, then the ratio of the lengths of BM and AD would be 
 (a) 11:12 (b) 11: 4v10 (c) 12:11 (d) 4v10: 11 
 
32. In a chess tournament there are 5 contestants. Each player plays against all the others exactly once. No game results 
in a draw. The winner in a game gets one point and the loser gets zero point. Which of the following sequences cannot 
represent the scores of the five players? 
 (a) 3, 2, 2, 2, 1 (b) 4, 4, 1, 1, 0 (c) 2, 2, 2, 2, 2 (d) 3, 3, 2, 1, 1 
 
33. A rabbit is sitting at the base of a staircase which has 10 steps. It proceeds to the top of the staircase by climbing either 
one step at a time or two steps at a time. The number of ways it can reach the top is 
 (a) 34 (b) 55 (c) 144 (d) 89 
 
34. Let a
1
, a
2
, a
3
 be three distinct real nubers in geometric progression. If the eqations a
1
x
2
+ 2a
2
x + a
3
= 0 and b
1
x
2
+
2b
2
x + b
3
+= 0 have a common root, then which of the following is necessarily ture?   
 (a) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in arithmetic progre ssion (b) b
1
, b
2
, b
3
 are in arithmetic progression 
 (c) 
b
1
a
1
,
b
2
a
2
,
b
3
a
3
 are in geometric progression  (d) b
1
, b
2
, b
3
 are in geometric progression  
 
35. Which of the following straight lines are both tangent to the circle x
2
+ y
2
- 6x + 4y - 12 = 0  
 (a) 4x + 3y + 19 = 0, 4x + 3y - 31 = 0. (b) 4x + 3y + 19 = 0, 4x + 3y + 31 = 0. 
 (c) 4x + 3y - 19 = 0, 4x + 3y - 31 = 0. (d) 4x + 3y - 19 = 0, 4x + 3y + 31 = 0. 
 
36. A person standing at the center of an open ground first walks 32 meters towards the east, takes a right turn and walks 
16 meters, takes another right turn and walks 8 meters, and so on. How far will the person be from the original starting 
point after an infinite number of such walks in this pattern? 
 (a) 64 /v5 meters   (b) 64 meters 
 (c) 32 meters   (d) 32 /v5 meters 
 
37. The equation x
2
+ y
2
- 2x - 4y + 5 = 0 represents  
 (a) a point   (b) a pair of straight lines 
 (c) an ellipse   (d) a circle 
 
38. If cos a + cos ß = 1, then the maximum value of sin a - sin ß is  
 (a) 1 (b) 2 (c) v3 (d) v2 
 
39. "Let p be a positive integer such that the unit digit of" p
3
 is 4. What are the possible unit digits of (p+3)
3
? 
 (a) 1,7,9 (b) 4,7 (c) 1,3,7 (d) 3 
 
40. The probability that a randomly chosen positive divisor of 10
2023
 is an integer multiple of 10
2001
 is  
 (a) 
23
2024
 (b) 
23
2
2023
2
 (c) 
23
2023
 (d) 
23
2
2024
2
 
                 
                           Page 4 of 12 
  
 Directions (Q.41-Q.45): A pharmaceutical company has tested five drugs on three different organisms. The following 
incomplete table reports if a drug works on the given organism. For example, drug A works on organism R while B 
and C work on Q. 
 Organism 
P Q R 
Drug 
A   
? 
B  
? 
 
C  
? 
 
D    
E    
 
Following additional information is available: 
?? Each drug works on at least one organism but not more than two organisms. 
?? Each organism can be treated with at least two and at most three of these five drugs. 
?? On whichever organism A works, B also works. Similarly, on whichever organism C works, D also works. 
?? D and E do not work on the same organism. 
 
41. Organism R can be treated with 
 (a) Only A and E (b) A, B and E (c) A, B and C (d) Only A and B 
 
42. Drug E works on 
 (a) Only R (b) Q and R (c) Only P (d) P and R 
 
43. The organism(s) that can be treated with three of these five drugs is(are) 
 (a) Only Q (b) Only P (c) P and Q (d) Q and R 
 
44. Drug D works on 
 (a) Only Q (b) P and Q (c) Q and R (d) Only P 
 
45. Organism P can be treated with 
 (a) B, C and D (b) Only C and D (c) Only B and D (d) A, B, and D 
 
  
IPMA T 2023 Solution
Q u e s 1 . V i n i t a d r i v e s a c a r w h i c h h a s f o u r g e a r s . T h e s p e e d o f t h e c a r
i n t h e f o u r t h g e a r i s f i v e t i m e s i t s s p e e d i n t h e f i r s t g e a r . T h e c a r t a k e s
t w i c e t h e t i m e t o t r a v e l a c e r t a i n d i s t a n c e i n t h e s e c o n d g e a r a s
c o m p a r e d t o t h e t h i r d g e a r . I n a 1 0 0 k m j o u r n e y , i f V i n i t a t r a v e l s e q u a l
d i s t a n c e s i n e a c h o f t h e g e a r s , s h e t a k e s 5 8 5 m i n u t e s t o c o m p l e t e t h e
j o u r n e y . I n s t e a d , i f t h e d i s t a n c e s c o v e r e d i n t h e f i r s t , s e c o n d , t h i r d ,
a n d f o u r t h g e a r s a r e 4 k m , 4 k m , 3 2 k m , a n d 6 0 k m , r e s p e c t i v e l y , t h e n
t h e t o t a l t i m e t a k e n , i n m i n u t e s , t o c o m p l e t e t h e j o u r n e y , w i l l
b e _ _ _ _ _ _ .
S o l u . L e t t h e s p e e d o f t h e c a r i n t h e f i r s t g e a r b e x k m / h r .
S p e e d i n f o u r t h g e a r = 5 x k m / h r .
T i m e t a k e n i n s e c o n d g e a r = 2 * T i m e t a k e n i n t h i r d g e a r ( g i v e n )
W e c a n r e p r e s e n t t i m e w i t h t h e f o l l o w i n g e q u a t i o n ( d i s t a n c e = s p e e d *
t i m e ) :
T 1 + T 2 + T 3 + T 4 = 1 0 0 / x + T 2 + 1 0 0 / ( 2 x ) + 1 0 0 / ( 5 x ) = 5 8 5 / 6 0 ( g i v e n , t o t a l
t i m e = 5 8 5 m i n u t e s )
S i m p l i f y i n g t h e e q u a t i o n :
1 0 0 / x + 2 ( 1 0 0 / ( 2 x ) ) + 1 0 0 / ( 5 x ) = 1 9 . 5
S o l v i n g f o r x :
x = 5 k m / h r ( s p e e d i n f i r s t g e a r )
N o w , l e t ' s c a l c u l a t e t h e t i m e t a k e n i n e a c h g e a r f o r t h e s e c o n d s c e n a r i o
( d i s t a n c e s m e n t i o n e d ) :
T 1 = 4 / 5 ( t i m e i n f i r s t g e a r ) T 2 = 4 / ( 2 * 5 ) ( t i m e i n s e c o n d g e a r , s p e e d i s h a l f
o f f i r s t g e a r ) T 3 = 3 2 / 5 ( t i m e i n t h i r d g e a r ) T 4 = 6 0 / 2 5 ( t i m e i n f o u r t h g e a r ,
s p e e d i s f i v e t i m e s f i r s t g e a r )
T o t a l t i m e ( T ) = T 1 + T 2 + T 3 + T 4
T = ( 4 / 5 ) + ( 2 / 5 ) + ( 3 2 / 5 ) + ( 1 2 / 5 ) = 5 0 / 5
T = 1 0 m i n u t e s ( t o t a l t i m e f o r t h e s e c o n d s c e n a r i o )