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Algebra & Equations — UPSC CSAT Paper II Quantitative Aptitude previous year questions
20 free solved previous year questions on Algebra & Equations, papers from 2021 to 2026, drawn from UPSC CSAT, covering questions of the kind UPSC asks in CSAT Paper II. Every answer key has been checked again against published sources and solved independently, and each explanation is written fresh.
Read all 20 questions with answers
- #1UPSC CSAT 2026The 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 = 17A + C = 6P + Q + S + D = 15P + Q + R + B = 17P = R and Q = SWhich one of the following statements is correct?
- AB and D together weigh less than the total weight of P and Q.
- BP and Q together weigh more than the total weight of A and C.
- CP weighs more than Q.
- DQ weighs more than P.
Answer: (B) P and Q together weigh more than the total weight of A and C.Explanation
From A + B + C + D = 17 and A + C = 6, we get B + D = 11.
Put R = P and S = Q in the two P-Q equations: P + 2Q + D = 15 and 2P + Q + B = 17.
Add them: 3P + 3Q + (B + D) = 32.
So 3(P + Q) = 32 – 11 = 21, and P + Q = 7.
Now A + C = 6, so P + Q (7) is more than A + C (6).
B + D = 11 is more than 7, so the first option is wrong. Nothing fixes P against Q.
Answer: option (b).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #2UPSC CSAT 2026A 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?
- A6
- B7
- C8
- D9
Answer: (D) 9Explanation
Let f be the forward jumps and b the backward jumps.
Total jumps: f + b = 31.
Net movement: 5f – 2b = 15.
Put b = 31 – f: 5f – 62 + 2f = 15, so 7f = 77 and f = 11.
Then b = 31 – 11 = 20.
Check: 55 – 40 = 15. Correct.
Difference = 20 – 11 = 9, option (d).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #3UPSC CSAT 2026For 1/3 < x < y < 2, which of the following statements is/are always correct?I. x + 1/x < y + 1/yII. √(1 + y²)/y < √(1 + x²)/xSelect the answer using the code given below.
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Answer: (B) II onlyExplanation
Statement I: the function t + 1/t falls until t = 1 and rises after that. The interval 1/3 to 2 contains 1.
Test x = 0.5, y = 1.5: x + 1/x = 2.5, y + 1/y = 2.17. Here the left side is larger, so I is not always true.
Statement II: √(1 + t²)/t = √(1/t² + 1).
As t grows, 1/t² shrinks, so this value always falls.
Since x < y, the value at y is smaller than at x. II is always true.
Only II holds. Answer: (b).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #4UPSC CSAT 2026In 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?
- A99
- B81
- C84
- D79
Answer: (B) 81Explanation
Let c be correct answers and w wrong answers.
First scheme: 5c – 2w = 69.
Second scheme: 4c – w = 84, so w = 4c – 84.
Put this in the first: 5c – 8c + 168 = 69, so 3c = 99 and c = 33.
Then w = 132 – 84 = 48.
Check: 165 – 96 = 69 and 132 – 48 = 84.
Total questions = 33 + 48 = 81, option (b).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #5UPSC CSAT 2026A 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?
- A80
- B85
- C90
- DThe question cannot be answered due to insufficient data
Answer: (D) The question cannot be answered due to insufficient dataExplanation
Earnings = ₹5 × (number of deliveries) + ₹2 × (km travelled).
For 80 deliveries: 400 + 2d. This exceeds 500 only if d is above 50 km.
For 85 deliveries: 425 + 2d. This needs d above 37.5 km.
For 90 deliveries: 450 + 2d. This needs d above 25 km.
The distance is not fixed. It may be small, so none of the numbers guarantees more than ₹500.
The data are insufficient. Answer: option (d).Source: UPSC CSAT 2026 Paper II (Set A); answer key checked by independent computation and against UPSC's provisional key
- #6UPSC CSAT 2025How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?
- ANone
- BOne
- CThree
- DMore than three
Answer: (C) ThreeExplanation
Solving 1/p + 1/q + 1/r = 1 in natural numbers gives only the triples (3,3,3), (2,4,4) and (2,3,6) in some order.
Their sums are 9, 10 and 11, all different.
So there are three possible values.
Answer: option (c).Source: UPSC CSAT 2025 question; answer key checked by independent computation
- #7UPSC CSAT 2025Team 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?
- A72
- B144
- C216
- DCannot be determined
Answer: (D) Cannot be determinedExplanation
Team Y took 18 overs, so its rate is N/18. A 50% higher rate is 1.5N/18 = N/12 per over.
In 12 overs this gives exactly N runs, which is true for every N.
The second condition adds no new information, so N cannot be found.
Answer: option (d).Source: UPSC CSAT 2025 question; answer key checked by independent computation
- #8UPSC CSAT 2025If 4 <= x <= 8 and 2 <= y <= 7, then what is the ratio of maximum value of (x + y) to minimum value of (x - y)?
- A6
- B15/2
- C-15/2
- DNone of the above
Answer: (D) None of the aboveExplanation
Maximum of x + y = 8 + 7 = 15. Minimum of x - y = 4 - 7 = -3.
The ratio is 15/(-3) = -5, which matches none of 6, 15/2 or -15/2.
Answer: option (d).Source: UPSC CSAT 2025 question; answer key checked by independent computation
- #9UPSC CSAT 2025Let p + q = 10, where p, q are integers.Value-I = Maximum value of p x q when p, q are positive integers.Value-II = Maximum value of p x q when p >= -6, q >= -4.Which one of the following is correct?
- AValue-I < Value-II
- BValue-II < Value-I
- CValue-I = Value-II
- DCannot be determined due to insufficient data
Answer: (C) Value-I = Value-IIExplanation
pq = p(10 - p) is largest at p = q = 5, giving 25.
This point satisfies both p >= -6 and q >= -4, so both values equal 25.
Answer: option (c).Source: UPSC CSAT 2025 question; answer key checked by independent computation
- #10UPSC CSAT 2025Let x be a real number between 0 and 1. Which of the following statements is/are correct?I. x^2 > x^3II. x > sqrt(x)Select the correct answer using the code given below:
- AI only
- BII only
- CBoth I and II
- DNeither I nor II
Answer: (A) I onlyExplanation
For 0 < x < 1, x^2 - x^3 = x^2(1 - x) > 0, so I is true.
For example x = 0.25 gives sqrt(x) = 0.5 > x, so II is false.
Only I is correct.
Answer: option (a).Source: UPSC CSAT 2025 question; answer key checked by independent computation
- #11UPSC CSAT 2024The 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?
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (C) Both 1 and 2Explanation
4O + 6M + 8A = 2O + 4M + 10A simplifies to O + M = A, so statement 2 is true.
Statement 1: (4O + 6M + 8A) - (3O + 5M + 9A) = O + M - A = 0, so the two totals are equal.
Both statements hold. Answer: option (c).Source: UPSC CSAT 2024 question; answer key checked by independent computation
- #12UPSC CSAT 2022A bill for Rs 1,840 is paid in the denominations of Rs 50, Rs 20 and Rs 10 notes. 50 notes in all are used. Consider the following statements:1. 25 notes of Rs 50 are used and the remaining are in the denominations of Rs 20 and Rs 10.2. 35 notes of Rs 20 are used and the remaining are in the denominations of Rs 50 and Rs 10.3. 20 notes of Rs 10 are used and the remaining are in the denominations of Rs 50 and Rs 20.Which of the above statements are not correct?
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- D1, 2 and 3
Answer: (D) 1, 2 and 3Explanation
Statement 1: 25 x 50 = 1250, leaving 590 from 25 notes of Rs 20/10, whose maximum is 500. Impossible.
Statement 2: 35 x 20 = 700, leaving 1140 from 15 notes of Rs 50/10, maximum 750. Impossible.
Statement 3: 20 x 10 = 200, leaving 1640 from 30 notes of Rs 50/20, maximum 1500. Impossible.
So all three are incorrect.
Answer: option (d).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #13UPSC CSAT 2022A 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?
- A76
- B68
- C60
- D52
Answer: (D) 52Explanation
Let A have x. B receives x/2 + 2. C receives (x/2+2)/2 + 2 = x/4 + 3.
D receives (x/4+3)/2 + 2 = x/8 + 7/2.
Set x/8 + 7/2 = 10 (smallest two-digit number): x = 52.
Answer: option (d).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #14UPSC CSAT 2022Five 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 Rs.40, then what is the total cost of all the notebooks?
- A2,600
- B2,400
- C2,360
- D2,320
Answer: (A) 2,600Explanation
P + Q = 21 and P - Q = 5 give P = 13, Q = 8.
X = 13 - 5 = 8, Z = 8 + 8 = 16, Y = 28 - 8 = 20.
Total = 13 + 8 + 8 + 16 + 20 = 65 notebooks; 65 x 40 = Rs 2,600.
Answer: option (a).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #15UPSC CSAT 2022The sum of three consecutive integers is equal to their product. How many such possibilities are there?
- AOnly one
- BOnly two
- COnly three
- DNo such possibility is there
Answer: (C) Only threeExplanation
Let the integers be a-1, a, a+1. Sum = 3a, product = a^3 - a.
So a^3 - 4a = 0, giving a = 0, 2 or -2.
The triples are (-1,0,1), (1,2,3) and (-3,-2,-1): three possibilities.
Answer: option (c).Source: UPSC CSAT 2022 question; answer key checked by independent computation
- #16UPSC CSAT 2021From January 1, 2021, the price of petrol (in Rupees per litre) on m-th 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 n-th 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?
- A21st May
- B20th May
- C19th May
- D18th May
Answer: (B) 20th MayExplanation
From day 100 petrol stays at 80 + 10 = Rs 90.
Diesel = 90 when 0.15n = 21, so n = 140.
Day 140 of 2021 (31+28+31+30 = 120 days to April end) is 20th May.
Answer: option (b).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #17UPSC CSAT 2021A 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?
- AP = 12 + 50n
- BP = 50 + 12n
- CP = 50 (2)^(12n)
- DP = 50 (2)^(n/12)
Answer: (D) P = 50 (2)^(n/12)Explanation
Doubling every 12 years means P = 50 x 2^(n/12).
Check n = 12 gives 100 (doubled) and n = 0 gives 50.
The linear options give wrong values and 2^(12n) grows far too fast.
Answer: option (d).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #18UPSC CSAT 2021The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.Consider the following statements:1. The sum of the two digits of the number can be determined only if the product of the two digits is known.2. The difference between the two digits of the number can be determined.Which of the above statements is/are correct?
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (B) 2 onlyExplanation
Let the number be 10a + b. Then 9(a - b) = 54, so a - b = 6 always; statement 2 is true.
Possible numbers are 71, 82, 93 (and 60), with digit sums 8, 10, 12, so the sum is not fixed by the given data alone.
Statement 1 is read as a false claim about the sum being determinable only through the product, so the book key keeps only statement 2.
Answer: option (b).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #19UPSC CSAT 2021X 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?
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Answer: (B) 2 onlyExplanation
When Y was born X was 42 - y, which equals 2y, so y = 14.
Eight years ago: X = 34, Y = 6; 34 is not 5 x 6, so statement 1 is false.
After 14 years: X = 56, Y = 28; 56 = 2 x 28, so statement 2 is true.
Answer: option (b).Source: UPSC CSAT 2021 question; answer key checked by independent computation
- #20UPSC CSAT 2021Jay 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?
- AA pencil costs more than a pen.
- BThe price of a pencil is equal to that of a pen
- CThe price of a pen is two times the price of a pencil
- DThe price of a pen is three times the price of a pencil
Answer: (C) The price of a pen is two times the price of a pencilExplanation
Equal spending: 3P + 5C = 2P + 7C.
So P = 2C: a pen costs twice a pencil.
Answer: option (c).Source: UPSC CSAT 2021 question; answer key checked by independent computation
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