Algebra and inequalities

2011 · Q39

A person has only ₹ 1 and ₹ 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is ₹ 75, then the number of ₹ 1 and ₹ 2 coins are, respectively

1. (a)15 and 35
2. (b)35 and 15
3. (c)30 and 20
4. (d)25 and 25UPSC key

2013 · Q10

Four friends, A, B, C and D distribute some money among themselves in such a manner that A gets one less than B, C gets 5 more than D, D gets 3 more than B. Who gets the smallest amount?

1. (a)AUPSC key
2. (b)B
3. (c)C
4. (d)D

2013 · Q62

A sum of ₹ 700 has to be used to give seven cash prizes to the students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, what is the least value of the prize?

1. (a)₹ 30
2. (b)₹ 40UPSC key
3. (c)₹ 60
4. (d)₹ 80

2014 · Q7

A group of 630 children is seated in rows for a group photo session. Each row contains three less children than the row in front of it. Which one of the following number of rows is not possible?

1. (a)3
2. (b)4
3. (c)5
4. (d)6UPSC key

2014 · Q65

Four persons, Alok, Bhupesh, Chander and Dinesh have a total of ₹ 100 among themselves. Alok and Bhupesh between them have as much money as Chander and Dinesh between them, but Alok has more money than Bhupesh; and Chander has only half the money that Dinesh has. Alok has in fact ₹ 5 more than Dinesh has. Who has the maximum amount of money?

1. (a)AlokUPSC key
2. (b)Bhupesh
3. (c)Chander
4. (d)Dinesh

2015 · Q16

A father is nine times as old as his son and the mother is eight times as old as the son. The sum of the father's and the mother's age is 51 years. What is the age of the son?

1. (a)7 years
2. (b)5 years
3. (c)4 years
4. (d)3 yearsUPSC key

2015 · Q30

Each of A, B, C and D has ₹ 100. A pays ₹ 20 to B, who pays ₹ 10 to C, who gets ₹ 30 from D. In this context, which one of the following statements is not correct?

1. (a)C is the richest.
2. (b)D is the poorest.
3. (c)C has more than what A and D have together.UPSC key
4. (d)B is richer than D.

2015 · Q38

In a parking area, the total number of wheels of all the cars (four-wheelers) and scooters/motorbikes (two-wheelers) is 100 more than twice the number of parked vehicles. The number of cars parked is

1. (a)35
2. (b)45
3. (c)50UPSC key
4. (d)55

2015 · Q50

A person ordered 5 pairs of black socks and some pairs of brown socks. The price of a black pair was thrice that of a brown pair. While preparing the bill, the bill clerk interchanged the number of black and brown pairs by mistake which increased the bill by 100%. What was the number of pairs of brown socks in the original order?

1. (a)10
2. (b)15
3. (c)20
4. (d)25UPSC key

2015 · Q77

A cow costs more than 4 goats but less than 5 goats. If a goat costs between ₹ 600 and ₹ 800, which of the following is a most valid conclusion?

1. (a)A cow costs more than ₹ 2,500.
2. (b)A cow costs less than ₹ 3,600.
3. (c)A cow costs between ₹ 2,600 and ₹ 3,800.
4. (d)A cow costs between ₹ 2,400 and ₹ 4,000.UPSC key

2015 · Q79

In a box of marbles, there are three less white marbles than the red ones and five more white marbles than the green ones. If there are a total of 10 white marbles, how many marbles are there in the box?

1. (a)26
2. (b)28UPSC key
3. (c)32
4. (d)36

2016 · Q14

There are some nectar-filled flowers on a tree and some bees are hovering on it. If one bee lands on each flower, one bee will be left out. If two bees land on each flower, one flower will be left out. The number of flowers and bees respectively are:

1. (a)2 and 4
2. (b)3 and 2
3. (c)3 and 4UPSC key
4. (d)4 and 3

2016 · Q26

A class starts at 11:00 am and lasts till 2:27 pm. Four periods of equal duration are held during this interval. After every period, a rest of 5 minutes is given to the students. The exact duration of each period is:

1. (a)48 minutesUPSC key
2. (b)50 minutes
3. (c)51 minutes
4. (d)53 minutes

2016 · Q29

In aid of charity, every student in a class contributes as many rupees as the number of students in that class. With the additional contribution of Rs. 2 by one student only, the total collection is Rs. 443. Then how many students are there in the class?

1. (a)12
2. (b)21UPSC key
3. (c)43
4. (d)45

2017 · Q14

The sum of income of A and B is more than that of C and D taken together. The sum of income of A and C is the same as that of B and D taken together. Moreover, A earns half as much as the sum of the income of B and D. Whose income is the highest?

1. (a)A
2. (b)BUPSC key
3. (c)C
4. (d)D

2017 · Q36

Six boys A, B, C, D, E and F play a game of cards. Each has a pack of 10 cards. F borrows 2 cards from A and gives away 5 to C who in turn gives 3 to B while B gives 6 to D who passes on 1 to E. Then the number of cards possessed by D and E is equal to the number of cards possessed by

1. (a)A, B and C
2. (b)B, C and FUPSC key
3. (c)A, B and F
4. (d)A, C and F

2017 · Q40

In a test, Randhir obtained more marks than the total marks obtained by Kunal and Debu. The total marks obtained by Kunal and Shankar are more than those of Randhir. Sonal obtained more marks than Shankar. Neha obtained more marks than Randhir. Who amongst them obtained highest marks?

1. (a)Randhir
2. (b)Neha
3. (c)Sonal
4. (d)Data are inadequateUPSC key

2018 · Q23

If X is between −3 and −1, and Y is between −1 and 1, then $X^{2}$ − $Y^{2}$ is in between which of the following?

1. (a)−9 and 1
2. (b)−9 and −1
3. (c)0 and 8
4. (d)0 and 9UPSC key

2018 · Q24

X and Y are natural numbers other than 1, and Y is greater than X. Which of the following represents the largest number?

1. (a)XYUPSC key
2. (b)X/Y
3. (c)Y/X
4. (d)(X + Y)/XY

2018 · Q37

If Pen < Pencil, Pencil < Book and Book > Cap, then which one of the following is always true?

1. (a)Pen > Cap
2. (b)Pen < BookUPSC key
3. (c)Pencil = Cap
4. (d)Pencil > Cap

2018 · Q38

A bookseller sold ‘a’ number of Geography textbooks at the rate of ₹ x per book, ‘a + 2’ number of History textbooks at the rate of ₹ (x + 2) per book and ‘a − 2’ number of Mathematics textbooks at the rate of ₹ (x − 2) per book. What is his total sale in ₹?

1. (a)3x + 3a
2. (b)3ax + 8UPSC key
3. (c)9ax
4. (d)$x^{3}$$a^{3}$

2018 · Q77

If x − y = 8, then which of the following must be true?

1.Both x and y must be positive for any value of x and y.

2.If x is positive, y must be negative for any value of x and y.

3.If x is negative, y must be positive for any value of x and y.

Select the correct answer using the code given below.

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2 nor 3UPSC key

2019 · Q18

In 2002, Meenu's age was one-third of the age of Meera, whereas in 2010, Meenu's age was half the age of Meera. What is Meenu's year of birth?

1. (a)1992
2. (b)1994UPSC key
3. (c)1996
4. (d)1998

2019 · Q19

Rakesh and Rajesh together bought 10 balls and 10 rackets. Rakesh spent ₹ 1300 and Rajesh spent ₹ 1500. If each racket costs three times a ball does, then what is the price of a racket?

1. (a)₹ 70
2. (b)₹ 90
3. (c)₹ 210UPSC key
4. (d)₹ 240

2019 · Q37

If x is greater than or equal to 25 and y is less than or equal to 40, then which one of the following is always correct?

1. (a)x is greater than y
2. (b)(y − x) is greater than 15
3. (c)(y − x) is less than or equal to 15UPSC key
4. (d)(x + y) is greater than or equal to 65

2019 · Q38

Ena was born 4 years after her parents' marriage. Her mother is three years younger than her father and 24 years older than Ena, who is 13 years old. At what age did Ena's father get married?

1. (a)22 years
2. (b)23 yearsUPSC key
3. (c)24 years
4. (d)25 years

2019 · Q69

Sunita cuts a sheet of paper into three pieces. Length of first piece is equal to the average of the three single digit odd prime numbers. Length of the second piece is equal to that of the first plus one-third the length of the third. The third piece is as long as the other two pieces together. The length of the original sheet of paper is

1. (a)13 units
2. (b)15 units
3. (c)16 units
4. (d)30 unitsUPSC key

2019 · Q79

Which one of the following statements is correct?

1. (a)Weight of Mohan is greatest
2. (b)Weight of Sohan is greatest
3. (c)Weight of Rohan is greatest
4. (d)'Whose weight is greatest' cannot be determinedUPSC key

2019 · Q80

Which one of the following statements is correct?

1. (a)Weight of Mohan is least
2. (b)Weight of Sohan is leastUPSC key
3. (c)Weight of Rohan is least
4. (d)'Whose weight is least' cannot be determined

2020 · Q9

Let p, q, r and s be natural numbers such that

p − 2016 = q + 2017 = r − 2018 = s + 2019

Which one of the following is the largest natural number?

1. (a)p
2. (b)q
3. (c)rUPSC key
4. (d)s

2020 · Q50

What is the largest number among the following?

1. (a)($\dfrac{1}{2}$)⁻⁶
2. (b)($\dfrac{1}{4}$)⁻³
3. (c)($\dfrac{1}{3}$)⁻⁴UPSC key
4. (d)($\dfrac{1}{6}$)⁻²

2020 · Q79

A vessel full of water weighs 40 kg. If it is one-third filled, its weight becomes 20 kg. What is the weight of the empty vessel?

1. (a)10 kgUPSC key
2. (b)15 kg
3. (c)20 kg
4. (d)25 kg

2021 · Q7

From January 1, 2021, the price of petrol (in Rupees per litre) on mᵗʰ day of the year is 80 + 0·1m, where m = 1, 2, 3, ..., 100 and thereafter remains constant. On the other hand, the price of diesel (in Rupees per litre) on nᵗʰ day of 2021 is 69 + 0·15n for any n. On which date in the year 2021 are the prices of these two fuels equal?

1. (a)21st May
2. (b)20th MayUPSC key
3. (c)19th May
4. (d)18th May

2021 · Q8

A biology class at high school predicted that a local population of animals will double in size every 12 years. The population at the beginning of the year 2021 was estimated to be 50 animals. If P represents the population after n years, then which one of the following equations represents the model of the class for the population?

1. (a)P = 12 + 50n
2. (b)P = 50 + 12n
3. (c)P = 50 $(2)^{12n}$
4. (d)P = 50 $(2)^{n}$/¹²UPSC key

2021 · Q53

Jay and Vijay spent an equal amount of money to buy some pens and special pencils of the same quality from the same store. If Jay bought 3 pens and 5 pencils, and Vijay bought 2 pens and 7 pencils, then which one of the following is correct?

1. (a)A pencil costs more than a pen
2. (b)The price of a pencil is equal to that of a pen
3. (c)The price of a pen is two times the price of a pencilUPSC key
4. (d)The price of a pen is three times the price of a pencil

2021 · Q55

A person P asks one of his three friends X as to how much money he had. X replied, "If Y gives me ₹ 40, then Y will have half of as much as Z, but if Z gives me ₹ 40, then three of us will have equal amount." What is the total amount of money that X, Y and Z have?

1. (a)₹ 420
2. (b)₹ 360UPSC key
3. (c)₹ 300
4. (d)₹ 270

2021 · Q58

In an objective type test of 90 questions, 5 marks are allotted for every correct answer and 2 marks are deducted for every wrong answer. After attempting all the 90 questions, a student got a total of 387 marks. What is the number of incorrect responses?

1. (a)9UPSC key
2. (b)13
3. (c)27
4. (d)43

2021 · Q77

X said to Y, "At the time of your birth I was twice as old as you are at present." If the present age of X is 42 years, then consider the following statements:

1.8 years ago, the age of X was five times the age of Y.

2.After 14 years, the age of X would be two times the age of Y.

Which of the above statements is/are correct?

1. (a)1 only
2. (b)2 onlyUPSC key
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2

2022 · Q8

A bill for ₹1,840 is paid in the denominations of ₹50, ₹20 and ₹10 notes. 50 notes in all are used. Consider the following statements:

1.25 notes of ₹50 are used and the remaining are in the denominations of ₹20 and ₹10.

2.35 notes of ₹20 are used and the remaining are in the denominations of ₹50 and ₹10.

3.20 notes of ₹10 are used and the remaining are in the denominations of ₹50 and ₹20.

Which of the above statements are not correct?

1. (a)1 and 2 only
2. (b)2 and 3 only
3. (c)1 and 3 only
4. (d)1, 2 and 3UPSC key

2022 · Q9

Which number amongst $2^{40}$, $3^{21}$, $4^{18}$ and $8^{12}$ is the smallest?

1. (a)$2^{40}$
2. (b)$3^{21}$UPSC key
3. (c)$4^{18}$
4. (d)$8^{12}$

2022 · Q28

A has some coins. He gives half of the coins and 2 more to B. B gives half of the coins and 2 more to C. C gives half of the coins and 2 more to D. The number of coins D has now, is the smallest two-digit number. How many coins does A have in the beginning?

1. (a)76
2. (b)68
3. (c)60
4. (d)52UPSC key

2022 · Q45

Five friends P, Q, X, Y and Z purchased some notebooks. The relevant information is given below:

1.Z purchased 8 notebooks more than X did.

2.P and Q together purchased 21 notebooks.

3.Q purchased 5 notebooks less than P did.

4.X and Y together purchased 28 notebooks.

5.P purchased 5 notebooks more than X did.

If each notebook is priced ₹40, then what is the total cost of all the notebooks?

1. (a)₹2,600UPSC key
2. (b)₹2,400
3. (c)₹2,360
4. (d)₹2,320

2023 · Q70

Consider the following:

I.A + B means A is neither smaller nor equal to B.

II.A − B means A is not greater than B.

III.A × B means A is not smaller than B.

IV.A ÷ B means A is neither greater nor equal to B.

V.A ± B means A is neither smaller nor greater than B.

Statement: P × Q, P − T, T ÷ R, R ± S

Conclusion-1: Q ± T

Conclusion-2: S + Q

Which one of the following is correct in respect of the above Statement and the Conclusions?

1. (a)Only Conclusion-1 follows from the Statement.
2. (b)Only Conclusion-2 follows from the Statement.UPSC key
3. (c)Both Conclusion-1 and Conclusion-2 follow from the Statement.
4. (d)Neither Conclusion-1 nor Conclusion-2 follows from the Statement.

2024 · Q29

A father said to his son, “n years back I was as old as you are now. My present age is four times your age n years back”. If the sum of the present ages of the father and the son is 130 years, what is the difference of their ages?

1. (a)30 yearsUPSC key
2. (b)32 years
3. (c)34 years
4. (d)36 years

2024 · Q50

The total cost of 4 oranges, 6 mangoes and 8 apples is equal to twice the total cost of 1 orange, 2 mangoes and 5 apples. Consider the following statements:

1.The total cost of 3 oranges, 5 mangoes and 9 apples is equal to the total cost of 4 oranges, 6 mangoes and 8 apples.

2.The total cost of one orange and one mango is equal to the cost of one apple.

Which of the statements given above is/are correct?

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2UPSC key
4. (d)Neither 1 nor 2

2024 · Q76

If P means ‘greater than (>)’; Q means ‘less than (<)’; R means ‘not greater than (≯)’; S means ‘not less than (≮)’ and T means ‘equal to (=)’, then consider the following statements:

1.If 2x(S)3y and 3x(T)4z, then 9y(P)8z.

2.If x(Q)2y and y(R)z, then x(R)z.

Which of the statements given above is/are correct?

1. (a)1 only
2. (b)2 only
3. (c)Both 1 and 2
4. (d)Neither 1 nor 2UPSC key

2024 · Q80

In some code, letters P, Q, R, S, T represent numbers 4, 5, 10, 12, 15. It is not known which letter represents which number. If Q − S = 2S and T = R + S + 3, then what is the value of P + R − T?

1. (a)1
2. (b)2UPSC key
3. (c)3
4. (d)Cannot be determined due to insufficient data

2025 · Q9

Team X scored a total of N runs in 20 overs. Team Y tied the score in 10% less overs. Had team Y’s average run rate (runs per over) been 50% higher, the scores would have been tied in 12 overs. How many runs were scored by team X?

1. (a)72
2. (b)144
3. (c)216
4. (d)Cannot be determinedUPSC key

2025 · Q19

Consider the following statements:

I.If A ≤ B > C < D > E > F ≥ G = H; then B is always greater than E.

II.If P > Q = R ≥ S = T ≤ U = V > W; then S is always less than V.

Which of the statements given above is/are correct?

1. (a)I only
2. (b)II only
3. (c)Both I and II
4. (d)Neither I nor IIUPSC key

2025 · Q26

If 4 ≤ x ≤ 8 and 2 ≤ y ≤ 7, then what is the ratio of maximum value of (x + y) to minimum value of (x − y)?

1. (a)6
2. (b)$\dfrac{15}{2}$
3. (c)−$\dfrac{15}{2}$
4. (d)None of the aboveUPSC key

2025 · Q76

Let p + q = 10, where p, q are integers.

Value-I = Maximum value of p × q when p, q are positive integers.

Value-II = Maximum value of p × q when p ≥ −6, q ≥ −4.

Which one of the following is correct?

1. (a)Value-I < Value-II
2. (b)Value-II < Value-I
3. (c)Value-I = Value-IIUPSC key
4. (d)Cannot be determined due to insufficient data

2025 · Q79

Let x be a real number between 0 and 1. Which of the following statements is/are correct?

I.$x^{2}$ > $x^{3}$.

II.x > $\sqrt{x}$.

Select the correct answer using the code given below:

1. (a)I onlyUPSC key
2. (b)II only
3. (c)Both I and II
4. (d)Neither I nor II

2026 · Q8

The weight of X, in kg, is denoted by X. The weights of A, B, C, D, P, Q, R and S are measured. Given:

A + B + C + D = 17

A + C = 6

P + Q + S + D = 15

P + Q + R + B = 17

P = R and Q = S

Which one of the following statements is correct?

1. (a)B and D together weigh less than the total weight of P and Q.
2. (b)P and Q together weigh more than the total weight of A and C.UPSC key
3. (c)P weighs more than Q.
4. (d)Q weighs more than P.

2026 · Q13

A toy T jumps forward or backward. In each forward jump, it moves 5′ forward whereas in each backward jump, it moves 2′ backward. If in 31 jumps, T moves exactly 15′ forward, then what is the difference of the number of forward and backward jumps?

1. (a)6
2. (b)7
3. (c)8
4. (d)9UPSC key

2026 · Q21

For $\dfrac{1}{3}$ < x < y < 2, which of the following statements is/are always correct?

I.x + $\dfrac{1}{x}$ < y + $\dfrac{1}{y}$

II.$\sqrt{1 + y^{2}}$/y < $\sqrt{1 + x^{2}}$/x

Select the answer using the code given below.

1. (a)I only
2. (b)II onlyUPSC key
3. (c)Both I and II
4. (d)Neither I nor II

2026 · Q38

In an objective type question paper, 5 marks are awarded for a correct answer and 2 marks are deducted for a wrong answer. A student attempted all the questions and got a score of 69. Had he been awarded 4 marks for a correct answer and 1 mark deducted for a wrong answer, he would have scored 84. How many questions were there in the question paper?

1. (a)99
2. (b)81UPSC key
3. (c)84
4. (d)79

2026 · Q68

There are three types of rectangular tiles: 3′ × 3′, 3′ × 7′ and 3′ × 11′. An area of rectangular shape of dimensions 3′ × 100′ is to be covered using these tiles without breaking them. If x and y are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then x − y is

1. (a)20UPSC key
2. (b)12
3. (c)10
4. (d)7

2026 · Q80

A shopkeeper employs a delivery boy and gives him a motorcycle for home delivery. For every delivery, the boy is given ₹5. At the end of the day, he also gets ₹2 for every kilometre of the distance covered in the day. The boy wants to earn more than ₹500 a day, but does not want to travel more than 100 km. Which of the following numbers of deliveries would definitely meet his target?

1. (a)80
2. (b)85
3. (c)90
4. (d)The question cannot be answered due to insufficient dataUPSC key