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UPSC prelims CSAT maths and data: previous year questionsUPSC प्रीलिम्स CSAT गणित और डेटा: पिछले वर्षों के प्रश्न

114 questions from the prelims papers of 2018 to 2025, newest first. Read the question, try it, then open the answer: it comes from the official UPSC answer key, with a short explanation.2018 से 2025 तक के प्रीलिम्स पेपरों से 114 प्रश्न, नए पहले। प्रश्न पढ़िए, हल कीजिए, फिर उत्तर खोलिए: वह आधिकारिक UPSC उत्तर-कुंजी से है, साथ में छोटी व्याख्या।

Questionsप्रश्न

  1. UPSC 2025 CSAT, question 5प्रश्न 5

    A natural number N is such that it can be expressed as N = p + q + r, where p, q and r are distinct factors of N. How many numbers below 50 have this property?

    1. 6
    2. 7
    3. 8
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 8

    Write the factors as N/a, N/b and N/c. Then N = N/a + N/b + N/c, so 1/a + 1/b + 1/c = 1 with a, b, c distinct. The only such set is (2, 3, 6). So N must be divisible by 6, and N = N/2 + N/3 + N/6. Multiples of 6 below 50 are 6, 12, 18, 24, 30, 36, 42 and 48, giving 8 numbers.

  2. UPSC 2025 CSAT, question 6प्रश्न 6

    Three prime numbers p, q and r, each less than 20, are such that p - q = q - r. How many distinct possible values can we get for (p + q + r)?

    1. 4
    2. 5
    3. 6
    4. More than 6
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) More than 6

    p - q = q - r means p + r = 2q, so p + q + r = 3q. The question does not say the primes are distinct, so p = q = r is allowed, and then every prime below 20 (2, 3, 5, 7, 11, 13, 17, 19) works as q. That gives 8 different values of 3q, which is more than 6. (If the primes had to be distinct, only q = 5, 7, 11, 13 would work, for example 3, 5, 7 and 3, 11, 19, giving 4 values; the official key follows the reading where they need not be distinct.)

  3. UPSC 2025 CSAT, question 7प्रश्न 7

    How 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)?

    1. None
    2. One
    3. Three
    4. More than three
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Three

    Take p <= q <= r. Then 1/p >= 1/3, so p is 2 or 3. If p = 3, then q = r = 3, sum 9. If p = 2, then 1/q + 1/r = 1/2, giving (4, 4) or (3, 6), sums 10 and 11. So the solutions are (3,3,3), (2,4,4) and (2,3,6) with sums 9, 10 and 11: three values.

  4. UPSC 2025 CSAT, question 9प्रश्न 9

    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. 72
    2. 144
    3. 216
    4. Cannot be determined
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) Cannot be determined

    Team Y needed 10% fewer than 20 overs, that is 18 overs, so its run rate was N/18. At a rate 50% higher, 1.5 x N/18 = N/12, the same N runs take exactly 12 overs. This is true for every value of N, so the 12-overs fact gives no new information and N cannot be determined.

  5. UPSC 2025 CSAT, question 10प्रश्न 10

    The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q?

    1. p(10^4 - k^2)^2 = q x 10^8
    2. p(10^4 - k^2)^2 = q x 10^4
    3. p(10^4 - k^2) = q x 10^4
    4. p(10^4 - k^2) = q x 10^8
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) p(10^4 - k^2)^2 = q x 10^8

    q = p(1 + k/100)(1 - k/100)(1 + k/100)(1 - k/100) = p(1 - k^2/10^4)^2 = p(10^4 - k^2)^2 / 10^8. Multiplying both sides by 10^8 gives p(10^4 - k^2)^2 = q x 10^8.

  6. UPSC 2025 CSAT, question 17प्रश्न 17

    The petrol price shot up by 10% as a result of the hike in crude oil prices. The price of petrol before the hike was ₹ 90 per litre. A person travels 2200 km every month and his car gives a mileage of 16 km per litre. By how many km should he reduce his travel if he wants to maintain his expenditure at the previous level?

    1. 180 km
    2. 200 km
    3. 220 km
    4. 240 km
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 200 km

    Old spending: 2200/16 = 137.5 litres x ₹ 90 = ₹ 12,375. New price = ₹ 99 per litre, so the same money buys 12375/99 = 125 litres, enough for 125 x 16 = 2000 km. He must cut his travel by 2200 - 2000 = 200 km. (Shortcut: a 10% price rise means travel must fall to 1/1.1 of 2200 = 2000 km.)

  7. UPSC 2025 CSAT, question 18प्रश्न 18

    A 4-digit number N is such that when divided by 3, 5, 6, 9 leaves a remainder 1, 3, 4, 7 respectively. What is the smallest value of N?

    1. 1068
    2. 1072
    3. 1078
    4. 1082
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 1078

    In each case the remainder is 2 less than the divisor (3 - 1, 5 - 3, 6 - 4, 9 - 7 all equal 2). So N + 2 is divisible by 3, 5, 6 and 9, that is by their LCM 90. N = 90k - 2. The smallest 4-digit value is with k = 12: 1080 - 2 = 1078.

  8. UPSC 2025 CSAT, question 20प्रश्न 20

    What is the unit digit in the multiplication of 1 x 3 x 5 x 7 x 9 x ... x 999?

    1. 1
    2. 3
    3. 5
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 5

    All the factors are odd, and one of them is 5. An odd number multiplied by 5 always ends in 5, so the product of all these odd numbers ends in 5.

  9. UPSC 2025 CSAT, question 25प्रश्न 25

    Consider the first 100 natural numbers. How many of them are not divisible by any one of 2, 3, 5, 7 and 9?

    1. 20
    2. 21
    3. 22
    4. 23
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 22

    Divisibility by 9 is already covered by 3, so we need numbers from 1 to 100 not divisible by 2, 3, 5 or 7. Any such number greater than 1 must have all its prime factors at least 11, and since 11 x 11 = 121 > 100, it must itself be a prime from 11 to 97. There are 25 primes up to 100; removing 2, 3, 5 and 7 leaves 21. Adding the number 1 gives 22.

  10. UPSC 2025 CSAT, question 26प्रश्न 26

    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. 6
    2. 15/2
    3. -15/2
    4. None of the above
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) None of the above

    Maximum of x + y = 8 + 7 = 15. Minimum of x - y uses the smallest x and largest y: 4 - 7 = -3. The ratio is 15 / (-3) = -5, which is not among (a), (b) or (c). So the answer is None of the above.

  11. UPSC 2025 CSAT, question 27प्रश्न 27

    Let both p and k be prime numbers such that (p^2 + k) is also a prime number less than 30. What is the number of possible values of k?

    1. 4
    2. 5
    3. 6
    4. 7
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 5

    If p and k were both odd, p^2 + k would be even and larger than 2, so not prime. So p = 2 or k = 2. With p = 2: 4 + k is prime and below 30 for k = 3, 7, 13, 19 (giving 7, 11, 17, 23). With k = 2: p^2 + 2 = 11 for p = 3 (p = 5 gives 27, not prime). So k can be 2, 3, 7, 13 or 19: 5 values.

  12. UPSC 2025 CSAT, question 28प्रश्न 28

    There are n sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1. What is the value of n?

    1. 6
    2. 7
    3. 8
    4. More than 8
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) More than 8

    1001 = 7 x 11 x 13, so each number is a divisor of 1001. For LCM 1001 and HCF 1, each of the primes 7, 11 and 13 must divide at least one number but not all three. In an ordered triple each prime can go to 6 such combinations of positions, giving 6 x 6 x 6 = 216 ordered triples. Triples where two chosen numbers are equal (say the first two) need each prime to be in both of them or only in the third: 2 x 2 x 2 = 8 for each of 3 pairs, 24 in all (all three equal is impossible). That leaves 192 ordered triples of distinct numbers, which is 192/6 = 32 sets, for example {1, 7, 143} or {7, 11, 13}. So n is more than 8.

  13. UPSC 2025 CSAT, question 29प्रश्न 29

    Let PQR be a 3-digit number, PPT be a 3-digit number and PS be a 2-digit number, where P, Q, R, S, T are distinct non-zero digits. Further, PQR - PS = PPT. If Q = 3 and T < 6, then what is the number of possible values of (R, S)?

    1. 2
    2. 3
    3. 4
    4. More than 4
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 3

    Write the numbers out: (100P + 30 + R) - (10P + S) = 110P + T, which simplifies to R - S - T = 20P - 30. Since R - S - T is at most 7, P must be 1, so S + T = R + 10. Digits must be non-zero and different from P = 1 and Q = 3, and T can be 2, 4 or 5. T = 2 needs S = R + 8, which forces R = 1 (not allowed). T = 4 gives S = R + 6: R = 2, S = 8. T = 5 gives S = R + 5: R = 2, S = 7 or R = 4, S = 9. So (R, S) can be (2, 8), (2, 7) or (4, 9): 3 values. Check: 132 - 18 = 114, 132 - 17 = 115, 134 - 19 = 115.

  14. UPSC 2025 CSAT, question 35प्रश्न 35

    What is the maximum value of n such that 7 x 343 x 385 x 1000 x 2401 x 77777 is divisible by 35^n?

    1. 3
    2. 4
    3. 5
    4. 7
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 4

    35 = 5 x 7, so n is limited by whichever prime appears fewer times. Factors of 5: 385 = 5 x 7 x 11 gives one and 1000 = 2^3 x 5^3 gives three, so 5^4 in all; none of 7, 343, 2401 or 77777 is divisible by 5. Factors of 7: 7 (1) + 343 (3) + 385 (1) + 2401 (4) + 77777 = 7 x 11111 (1) = 10. The product has 5^4 x 7^10, so the largest n is 4.

  15. UPSC 2025 CSAT, question 37प्रश्न 37

    P and Q walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. P walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?

    1. 12
    2. 13
    3. 14
    4. 15
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 13

    Walking in opposite directions, together they cover 5 + 3 = 8 rounds per hour, so they cross every 1/8 hour = 7.5 minutes: at 5:07.5, 5:15, 5:22.5, ... The crossings are at 7.5k minutes after 5:00. Between 5:20 (20 minutes) and 7:00 (120 minutes) the values are k = 3 (22.5 min) up to k = 15 (112.5 min), which is 13 crossings. The crossing at exactly 7:00 (k = 16) is at the end point and is not counted as 'between'.

  16. UPSC 2025 CSAT, question 38प्रश्न 38

    If P = +, Q = -, R = x, S = ÷, then insert the proper notations between the successive numbers in the equation 60_15_3_20_4 = 20:

    1. SPRQ
    2. QRPS
    3. QRSP
    4. SPQR
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) QRPS

    QRPS gives 60 - 15 x 3 + 20 ÷ 4. Doing x and ÷ first: 15 x 3 = 45 and 20 ÷ 4 = 5, so 60 - 45 + 5 = 20. Checking (a): 60 ÷ 15 + 3 x 20 - 4 = 4 + 60 - 4 = 60, which does not work.

  17. UPSC 2025 CSAT, question 39प्रश्न 39

    A tram overtakes 2 persons X and Y walking at an average speed of 3 km/hr and 4 km/hr in the same direction and completely passes them in 8 seconds and 9 seconds respectively. What is the length of the tram?

    1. 15 m
    2. 18 m
    3. 20 m
    4. 24 m
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 20 m

    Let the tram's speed be v km/hr. The length equals relative speed x time, so (v - 3) x 8 = (v - 4) x 9, giving 8v - 24 = 9v - 36 and v = 12 km/hr. Relative speed with X is 9 km/hr = 9 x 5/18 = 2.5 m/s, so the length = 2.5 x 8 = 20 m (check with Y: 8 km/hr = 20/9 m/s x 9 s = 20 m).

  18. UPSC 2025 CSAT, question 40प्रश्न 40

    If N^2 = 12345678987654321, then how many digits does the number N have?

    1. 8
    2. 9
    3. 10
    4. 11
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 9

    Squares of repunits follow the pattern 11^2 = 121, 111^2 = 12321, and so on; 12345678987654321 rises to 9 and falls back, so it equals 111111111^2. N = 111111111 has 9 digits. (Also, N^2 has 17 digits, and a 17-digit square has a 9-digit square root.)

  19. UPSC 2025 CSAT, question 45प्रश्न 45

    A set (X) of 20 pipes can fill 70% of a tank in 14 minutes. Another set (Y) of 10 pipes fills 3/8th of the tank in 6 minutes. A third set (Z) of 16 pipes can empty half of the tank in 20 minutes. If half of the pipes of set X are closed and only half of the pipes of set Y are open, and all pipes of the set (Z) are open, then how long will it take to fill 50% of the tank?

    1. 8 minutes
    2. 10 minutes
    3. 12 minutes
    4. 16 minutes
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 16 minutes

    One X pipe fills 0.7/(14 x 20) = 1/400 of the tank per minute; one Y pipe fills (3/8)/(6 x 10) = 1/160; one Z pipe empties 0.5/(20 x 16) = 1/640. Open pipes: 10 X pipes give 10/400 = 1/40, 5 Y pipes give 5/160 = 1/32, and 16 Z pipes remove 16/640 = 1/40. Net rate = 1/32 of the tank per minute, so 50% takes 0.5 x 32 = 16 minutes.

  20. UPSC 2025 CSAT, question 46प्रश्न 46

    If n is a natural number, then what is the number of distinct remainders of (1^n + 2^n) when divided by 4?

    1. 0
    2. 1
    3. 2
    4. 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 2

    For n = 1: 1 + 2 = 3, remainder 3. For n ≥ 2, 2^n is a multiple of 4, so 1 + 2^n leaves remainder 1. Only two remainders occur (3 and 1).

  21. UPSC 2025 CSAT, question 47प्रश्न 47

    Let P = QQQ be a 3-digit number. What is the HCF of P and 481?

    1. 1
    2. 13
    3. 37
    4. 481
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 37

    QQQ = 111 x Q = 3 x 37 x Q, and 481 = 13 x 37. They share the factor 37. Since Q is a single digit, it can never be a multiple of 13, so the HCF is always 37.

  22. UPSC 2025 CSAT, question 48प्रश्न 48

    What is the 489th digit in the number 123456789101112...?

    1. 0
    2. 3
    3. 6
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 9

    The numbers 1 to 9 use 9 digits and 10 to 99 use 90 x 2 = 180 digits, 189 in total. That leaves 489 - 189 = 300 digits, which is exactly 100 three-digit numbers: 100 to 199. The 489th digit is the last digit of 199, which is 9.

  23. UPSC 2025 CSAT, question 58प्रश्न 58

    The 5-digit number PQRST (all distinct digits) is such that T ≠ 0. P is thrice T. S is greater than Q by 4, while Q is greater than R by 3. How many such 5-digit numbers are possible?

    1. 3
    2. 4
    3. 5
    4. 6
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 4

    P = 3T with T non-zero gives (T, P) = (1, 3), (2, 6) or (3, 9). Q = R + 3 and S = R + 7, so (R, Q, S) = (0, 3, 7), (1, 4, 8) or (2, 5, 9). Checking for distinct digits: T = 1, P = 3 allows only (2, 5, 9), giving 35291; T = 2, P = 6 allows (0, 3, 7) and (1, 4, 8), giving 63072 and 64182; T = 3, P = 9 allows only (1, 4, 8), giving 94183. That is 4 numbers.

  24. UPSC 2025 CSAT, question 59प्रश्न 59

    X can complete one-third of a certain work in 6 days, Y can complete one-third of the same work in 8 days and Z can complete three-fourth of the same work in 12 days. All of them work together for n days and then X and Z quit and Y alone finishes the remaining work in 8 2/3 days. What is n equal to?

    1. 3
    2. 4
    3. 5
    4. 6
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 4

    Full work: X takes 18 days, Y 24 days and Z 16 days. In 8 2/3 = 26/3 days Y does (26/3)/24 = 13/36 of the work, so the three together did 23/36. Their combined daily rate is 1/18 + 1/24 + 1/16 = (8 + 6 + 9)/144 = 23/144. So n = (23/36) / (23/144) = 4.

  25. UPSC 2025 CSAT, question 64प्रश्न 64

    Consider the following statements: I. There exists a natural number which when increased by 50% can have its number of factors unchanged. II. There exists a natural number which when increased by 150% can have its number of factors unchanged. Which of the statements given above is/are correct?

    1. I only
    2. II only
    3. Both I and II
    4. Neither I nor II
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Both I and II

    Increasing by 50% multiplies the number by 3/2; increasing by 150% multiplies it by 5/2. Take N = 2, which has 2 factors. 2 x 3/2 = 3 and 2 x 5/2 = 5 are both primes with 2 factors each. So such numbers exist in both cases and both statements are correct.

  26. UPSC 2025 CSAT, question 65प्रश्न 65

    There are 7 places A, B, C, D, E, F and G in a city connected by various roads AB, AC, CD, DE, BF, EG and FG. A is 6 km south of B. A is 10 km west of C. D is 5 km east of E. C is 6 km north of D. F is 9 km west of B. F is 12 km north of G. A person travels from D to F through these roads. What is the distance covered by the person?

    1. 20 km
    2. 25 km
    3. 31 km
    4. 37 km
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 31 km

    Put A at (0, 0). Then B = (0, 6), C = (10, 0), D = (10, -6), E = (5, -6), F = (-9, 6) and G = (-9, -6). There are two routes from D to F. D-C-A-B-F: 6 + 10 + 6 + 9 = 31 km. D-E-G-F: 5 + 14 + 12 = 31 km (E to G is from x = 5 to x = -9). Either way the distance is 31 km.

  27. UPSC 2025 CSAT, question 67प्रश्न 67

    What is the remainder when 9^3 + 9^4 + 9^5 + 9^6 + ... + 9^100 is divided by 6?

    1. 0
    2. 1
    3. 2
    4. 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 0

    Every power of 9 is odd and divisible by 3, so each leaves remainder 3 when divided by 6. There are 100 - 3 + 1 = 98 terms, so the sum leaves the same remainder as 98 x 3 = 294, and 294 = 6 x 49. The remainder is 0.

  28. UPSC 2025 CSAT, question 69प्रश्न 69

    A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: Let P, Q, R, S be distinct non-zero digits. If PP x PQ = RRSS, where P ≤ 3 and Q ≤ 4, then what is Q equal to? Statement I: R = 1. Statement II: S = 2. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
    2. The Question can be answered by using either Statement alone.
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
    4. The Question can be answered even without using any of the Statements.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) The Question can be answered even without using any of the Statements.

    For P = 1 or 2 the product is at most 11 x 14 = 154 or 22 x 24 = 528, not a 4-digit number. So P = 3, and Q can be 1, 2 or 4 (distinct from P). 33 x 31 = 1023 and 33 x 32 = 1056 are not of the form RRSS, but 33 x 34 = 1122 is. So Q = 4 from the question alone; the statements are not needed.

  29. UPSC 2025 CSAT, question 74प्रश्न 74

    A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option. Question: Is (p + q)^2 - 4pq, where p, q are natural numbers, positive? Statement I: p < q. Statement II: p > q. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other statement alone.
    2. The Question can be answered by using either Statement alone.
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
    4. The Question can be answered even without using any of the Statements.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) The Question can be answered by using either Statement alone.

    (p + q)^2 - 4pq = (p - q)^2, which is positive whenever p ≠ q and zero when p = q. Without the statements we cannot tell. Statement I (p < q) alone shows p ≠ q, so the answer is yes; Statement II (p > q) alone also shows p ≠ q. Either statement alone is enough.

  30. UPSC 2025 CSAT, question 75प्रश्न 75

    In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by X to the number of runs scored by Y is equal to ratio of number of runs scored by Y to number of runs scored by Z. Value-I = Runs scored by X Value-II = Runs scored by Y Value-III = Runs scored by Z Which one of the following is correct?

    1. Value-I < Value-II < Value-III
    2. Value-III < Value-II < Value-I
    3. Value-I < Value-III < Value-II
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) Cannot be determined due to insufficient data

    X : Y = Y : Z means the runs are in geometric progression: Y^2 = XZ. With whole-number runs adding to 37, the only possibility is the set 9, 12, 16 (9 x 16 = 144 = 12^2, and 9 + 12 + 16 = 37). But it can be X = 9, Y = 12, Z = 16 (increasing) or X = 16, Y = 12, Z = 9 (decreasing). Both fit, so the order cannot be determined.

  31. UPSC 2025 CSAT, question 76प्रश्न 76

    Let 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?

    1. Value-I < Value-II
    2. Value-II < Value-I
    3. Value-I = Value-II
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Value-I = Value-II

    For a fixed sum, the product is largest when the two numbers are equal or as close as possible. With p + q = 10, p = q = 5 gives 25, so Value-I = 25. In the second case p can range from -6 to 14, which still includes p = q = 5, and any other choice gives a smaller product (for example 4 x 6 = 24, -6 x 16 = -96). So Value-II = 25 and the two values are equal.

  32. UPSC 2025 CSAT, question 77प्रश्न 77

    Consider a set of 11 numbers: Value-I = Minimum value of the average of the numbers of the set when they are consecutive integers ≥ -5. Value-II = Minimum value of the product of the numbers of the set when they are consecutive non-negative integers. Which one of the following is correct?

    1. Value-I < Value-II
    2. Value-II < Value-I
    3. Value-I = Value-II
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Value-I = Value-II

    Value-I: the smallest set of 11 consecutive integers starting at -5 is -5, -4, ..., 5, whose average is 0. Value-II: 11 consecutive non-negative integers have the smallest product when 0 is included (0, 1, ..., 10), giving a product of 0. Both values are 0, so they are equal.

  33. UPSC 2025 CSAT, question 78प्रश्न 78

    The average of three numbers p, q and r is k. p is as much more than the average as q is less than the average. What is the value of r?

    1. k
    2. k - 1
    3. k + 1
    4. k/2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) k

    p - k = k - q, so p + q = 2k. Since p + q + r = 3k, r = 3k - 2k = k.

  34. UPSC 2025 CSAT, question 79प्रश्न 79

    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 > √x. Select the correct answer using the code given below:

    1. I only
    2. II only
    3. Both I and II
    4. Neither I nor II
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) I only

    For 0 < x < 1, higher powers get smaller: x^3 = x^2 x x < x^2, so I is correct. Also √x > x for such x (for example x = 0.25 gives √x = 0.5), so II is wrong.

  35. UPSC 2025 CSAT, question 80प्रश्न 80

    The difference between any two natural numbers is 10. What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?

    1. There is only one such number.
    2. There are only two such numbers.
    3. There can be more than one such number.
    4. No such number exists.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) There can be more than one such number.

    Take examples. Between 1 and 11 lie 5 and 10, two multiples of 5. Between 5 and 15 (both multiples of 5) only 10 lies strictly between them. So the count can be one or two; neither 'only one' nor 'only two' is always true, and there is always at least one. The best description is that there can be more than one such number.

  36. UPSC 2024 CSAT, question 6प्रश्न 6

    In the expression 5 * 4 * 3 * 2 * 1, * is chosen from +, -, x each at most two times. What is the smallest non-negative value of the expression?

    1. 3
    2. 2
    3. 1
    4. 0
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 0

    Try to reach 0. Choose the four signs as -, -, + and x: 5 - 4 - 3 + 2 x 1 = 5 - 4 - 3 + 2 = 0. Here minus is used twice, plus once and times once, which is allowed. Since 0 is the smallest non-negative value possible, the answer is 0.

  37. UPSC 2024 CSAT, question 7प्रश्न 7

    A certain number of men can complete a piece of work in 6k days, where k is a natural number. By what percent should the number of men be increased so that the work can be completed in 5k days?

    1. 10%
    2. (50/3)%
    3. 20%
    4. 25%
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 20%

    For a fixed job, the number of men is inversely proportional to the number of days. So new men : old men = 6k : 5k = 6 : 5. The increase is 1 part on 5, that is 1/5 = 20%.

  38. UPSC 2024 CSAT, question 8प्रश्न 8

    X, Y and Z can complete a piece of work individually in 6 hours, 8 hours and 8 hours respectively. However, only one person at a time can work in each hour and nobody can work for two consecutive hours. All are engaged to finish the work. What is the minimum amount of time that they will take to finish the work?

    1. 6 hours 15 minutes
    2. 6 hours 30 minutes
    3. 6 hours 45 minutes
    4. 7 hours
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 6 hours 45 minutes

    Take the work as 24 units (LCM of 6 and 8). X does 4 units per hour, Y and Z do 3 units per hour each. To finish fastest, use X, the quickest, in every other hour: X, Y, X, Z, X, Y. In 6 hours they do 4 + 3 + 4 + 3 + 4 + 3 = 21 units. The 7th hour is X's turn; the remaining 3 units take X 3/4 hour = 45 minutes. Total = 6 hours 45 minutes.

  39. UPSC 2024 CSAT, question 9प्रश्न 9

    How many consecutive zeros are there at the end of the integer obtained in the product 1^2 x 2^4 x 3^6 x 4^8 x ... x 25^50?

    1. 50
    2. 55
    3. 100
    4. 200
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 200

    The general term is n^(2n) for n = 1 to 25. Trailing zeros come from pairs of 2 and 5, and 5s are far fewer than 2s, so count the 5s. Only multiples of 5 contribute: 5^10 gives 10, 10^20 gives 20, 15^30 gives 30, 20^40 gives 40, and 25^50 = 5^100 gives 100. Total = 10 + 20 + 30 + 40 + 100 = 200 zeros.

  40. UPSC 2024 CSAT, question 10प्रश्न 10

    On January 1st, 2023, a person saved ₹ 1. On January 2nd, 2023, he saved ₹ 2 more than that on the previous day. On January 3rd, 2023, he saved ₹ 2 more than that on the previous day and so on. At the end of which date was his total savings a perfect square as well a perfect cube?

    1. 7th January, 2023
    2. 8th January, 2023
    3. 9th January, 2023
    4. Not possible
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 8th January, 2023

    He saves 1, 3, 5, 7, ... rupees, the odd numbers. The sum of the first n odd numbers is n^2, so after n days the total is n^2, always a perfect square. For n^2 to also be a perfect cube it must be a sixth power, so n itself must be a perfect cube. Apart from the trivial n = 1 (total 1), the first such day is n = 8: total 64 = 8^2 = 4^3. So the date is 8th January, 2023.

  41. UPSC 2024 CSAT, question 15प्रश्न 15

    222^333 + 333^222 is divisible by which of the following numbers?

    1. 2 and 3 but not 37
    2. 3 and 37 but not 2
    3. 2 and 37 but not 3
    4. 2, 3 and 37
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 3 and 37 but not 2

    222 = 2 x 3 x 37 and 333 = 3 x 3 x 37. Both terms are multiples of 3 and of 37, so their sum is divisible by 3 and 37. For 2: 222^333 is even but 333^222 is odd, so the sum is odd and not divisible by 2.

  42. UPSC 2024 CSAT, question 16प्रश्न 16

    What percent of water must be mixed with honey so as to gain 20% by selling the mixture at the cost price of honey?

    1. 20%
    2. 10%
    3. 5%
    4. 4%
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 20%

    Let honey cost ₹ 1 per litre and water be free. To gain 20%, the selling price must be 1.2 times the cost. Selling at ₹ 1 per litre, ₹ 1.2 is received for 1.2 litres of mixture that contains only 1 litre of honey (cost ₹ 1). So 0.2 litre of water is added per litre of honey, which is 20% of the honey.

  43. UPSC 2024 CSAT, question 17प्रश्न 17

    What is the rightmost digit preceding the zeros in the value of 30^30?

    1. 1
    2. 3
    3. 7
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 9

    30^30 = 3^30 x 10^30. The 10^30 gives the zeros, so the digit before them is the units digit of 3^30. Units digits of powers of 3 repeat in the cycle 3, 9, 7, 1. Since 30 leaves remainder 2 when divided by 4, the units digit is 9.

  44. UPSC 2024 CSAT, question 18प्रश्न 18

    421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1?

    1. 1
    2. 2
    3. 3
    4. 4
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 3

    The divisor must divide 421 - 1 = 420 and 427 - 1 = 426, so it divides HCF(420, 426) = 6. The divisors of 6 are 1, 2, 3, 6, but a remainder of 1 needs a divisor greater than 1. So 2, 3 and 6 work: 3 numbers.

  45. UPSC 2024 CSAT, question 19प्रश्न 19

    A can X contains 399 litres of petrol and a can Y contains 532 litres of diesel. They are to be bottled in bottles of equal size so that whole of petrol and diesel would be separately bottled. The bottle capacity in terms of litres is an integer. How many different bottle sizes are possible?

    1. 3
    2. 4
    3. 5
    4. 6
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 4

    The bottle size must divide both 399 and 532. 399 = 3 x 7 x 19 and 532 = 4 x 7 x 19, so HCF = 133 = 7 x 19. Its divisors are 1, 7, 19 and 133, giving 4 possible bottle sizes.

  46. UPSC 2024 CSAT, question 20प्रश्न 20

    Consider the following statements in respect of the sum S = x + y + z, where x, y and z are distinct prime numbers each less than 10: 1. The unit digit of S can be 0. 2. The unit digit of S can be 9. 3. The unit digit of S can be 5. Which of the statements given above are correct?

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 1 and 3 only

    The primes below 10 are 2, 3, 5 and 7. The possible sums of three distinct ones are 2 + 3 + 5 = 10, 2 + 3 + 7 = 12, 2 + 5 + 7 = 14 and 3 + 5 + 7 = 15. The unit digit can be 0 (from 10) and 5 (from 15) but never 9. So statements 1 and 3 are correct.

  47. UPSC 2024 CSAT, question 25प्रश्न 25

    Two persons P and Q enter into a business. P puts ₹ 14,000 more than Q, but P has invested for 8 months and Q has invested for 10 months. If P's share is ₹ 400 more than Q's share out of the total profit of ₹ 2,000, what is the capital contributed by P?

    1. ₹ 30,000
    2. ₹ 26,000
    3. ₹ 24,000
    4. ₹ 20,000
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) ₹ 30,000

    Profit shares: P + Q = 2000 and P - Q = 400, so P gets 1200 and Q gets 800, a ratio of 3 : 2. Profit is in the ratio of capital x time: 8P : 10Q = 3 : 2, so 16P = 30Q and P = 15Q/8. Then P - Q = 7Q/8 = 14,000, so Q = 16,000 and P = 30,000.

  48. UPSC 2024 CSAT, question 26प्रश्न 26

    P's salary is 20% lower than Q's salary which is 20% lower than R's salary. By how much percent is R's salary more than P's salary?

    1. 48.75%
    2. 56.25%
    3. 60.50%
    4. 62.25%
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 56.25%

    Let R = 100. Then Q = 80 and P = 80 x 0.8 = 64. R is more than P by 36, and 36/64 x 100 = 56.25%.

  49. UPSC 2024 CSAT, question 27प्रश्न 27

    A number is mistakenly divided by 4 instead of multiplying by 4. What is the percentage change in the result due to this mistake?

    1. 25%
    2. 50%
    3. 72.75%
    4. 93.75%
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 93.75%

    Correct result = 4x; wrong result = x/4. Change = 4x - x/4 = 15x/4. As a percentage of the correct result: (15x/4) / 4x = 15/16 = 93.75% (a decrease).

  50. UPSC 2024 CSAT, question 28प्रश्न 28

    In an examination, 80% of students passed in English, 70% of students passed in Hindi and 15% failed in both the subjects. What is the percentage of students who failed in only one subject?

    1. 15%
    2. 20%
    3. 25%
    4. 35%
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 20%

    Passed in at least one subject = 100 - 15 = 85%. Passed in both = 80 + 70 - 85 = 65%. Students who failed in exactly one subject are those who passed exactly one: 85 - 65 = 20%.

  51. UPSC 2024 CSAT, question 29प्रश्न 29

    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. 30 years
    2. 32 years
    3. 34 years
    4. 36 years
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 30 years

    Let present ages be F and S. From F - n = S, the age gap is n = F - S. From F = 4(S - n) = 4(S - F + S) = 4(2S - F), we get 5F = 8S. With F + S = 130: S = 50 and F = 80. Difference = 30 years.

  52. UPSC 2024 CSAT, question 30प्रश्न 30

    Consider the following: 1. 1000 litres = 1 m^3 2. 1 metric ton = 1000 kg 3. 1 hectare = 10000 m^2 Which of the above are correct?

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 1, 2 and 3

    1 m^3 = 100 cm x 100 cm x 100 cm = 10^6 cm^3 = 1000 litres. A metric ton (tonne) is 1000 kg. A hectare is 100 m x 100 m = 10,000 m^2. All three are correct.

  53. UPSC 2024 CSAT, question 35प्रश्न 35

    Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X?

    1. 2
    2. 4
    3. 6
    4. 8
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 8

    If X = 10a + b, then Y = 10b + a and X + Y = 11(a + b). The greatest two-digit number is 99, so a + b = 9. Since Y must also be a two-digit number, b cannot be 0, so neither digit is 0. The possibilities are 18, 27, 36, 45, 54, 63, 72, 81: 8 values.

  54. UPSC 2024 CSAT, question 36प्रश्न 36

    Consider the following: Weight of 6 boys = Weight of 7 girls = Weight of 3 men = Weight of 4 women If the average weight of the women is 63 kg, then what is the average weight of the boys?

    1. 40 kg
    2. 42 kg
    3. 45 kg
    4. 63 kg
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 42 kg

    Weight of 4 women = 4 x 63 = 252 kg. This equals the weight of 6 boys, so the average weight of a boy = 252 / 6 = 42 kg.

  55. UPSC 2024 CSAT, question 39प्रश्न 39

    Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements: 1. (p - r)^2 (qs) is even. 2. (q - s)q^2 s is even. 3. (q + r)^2 (p + s) is odd. Which of the statements given above are correct?

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 1, 2 and 3

    1: qs contains the even factor s, so the product is even. 2: the factor s is even, so the product is even. 3: q + r is odd + even = odd, so its square is odd; p + s is odd + even = odd; odd x odd = odd. All three statements are correct.

  56. UPSC 2024 CSAT, question 45प्रश्न 45

    What is the number of fives used in numbering a 260-page book?

    1. 55
    2. 56
    3. 57
    4. 60
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 56

    Count the digit 5 in page numbers 1 to 260. In the units place: 5, 15, 25, ..., 255, which is 26 numbers. In the tens place: 50 to 59, 150 to 159 and 250 to 259, which is 30 numbers. No page number from 1 to 260 has 5 in the hundreds place. Total = 26 + 30 = 56.

  57. UPSC 2024 CSAT, question 46प्रश्न 46

    What is the sum of the first 28 terms in the following sequence? 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, ...

    1. 83
    2. 84
    3. 85
    4. 86
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 84

    Group the terms as 1 | 1 | 2, 1 | 3, 2, 1 | 4, 3, 2, 1 | 5, 4, 3, 2, 1 | 6, 5, 4, 3, 2, 1 | 7, 6, ... The first seven groups have 1 + 1 + 2 + 3 + 4 + 5 + 6 = 22 terms with sum 1 + 1 + 3 + 6 + 10 + 15 + 21 = 57. Terms 23 to 28 are 7, 6, 5, 4, 3, 2 with sum 27. Total = 57 + 27 = 84.

  58. UPSC 2024 CSAT, question 47प्रश्न 47

    A person buys three articles P, Q and R for ₹ 3,330. If P costs 25% more than R and R costs 20% more than Q, then what is the cost of P?

    1. ₹ 1,000
    2. ₹ 1,200
    3. ₹ 1,250
    4. ₹ 1,350
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) ₹ 1,350

    Let Q = x. Then R = 1.2x and P = 1.25 x 1.2x = 1.5x. Total = x + 1.2x + 1.5x = 3.7x = 3330, so x = 900. P = 1.5 x 900 = ₹ 1,350.

  59. UPSC 2024 CSAT, question 48प्रश्न 48

    If the sum of the two-digit numbers AB and CD is the three-digit number 1CE, where the letters A, B, C, D, E denote distinct digits, then what is the value of A?

    1. 9
    2. 8
    3. 7
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 9

    AB + CD = 1CE. In the tens column, A + C + (carry from units) must give C in the tens place and 1 carried to the hundreds, so A + carry = 10. A is a single digit, so the carry must be 1 and A = 9. For example, 96 + 47 = 143 fits, with A = 9, B = 6, C = 4, D = 7, E = 3.

  60. UPSC 2024 CSAT, question 49प्रश्न 49

    Three numbers x, y, z are selected from the set of the first seven natural numbers such that x > 2y > 3z. How many such distinct triplets (x, y, z) are possible?

    1. One triplet
    2. Two triplets
    3. Three triplets
    4. Four triplets
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) Four triplets

    x is at most 7, so 2y is at most 6 and y is at most 3. If z = 1, then 2y > 3 gives y = 2 or 3. For y = 2, x > 4 gives x = 5, 6, 7 (3 triplets). For y = 3, x > 6 gives x = 7 (1 triplet). If z = 2, 2y > 6 needs y at least 4, which is impossible. Total = 4 triplets.

  61. UPSC 2024 CSAT, question 50प्रश्न 50

    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. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Both 1 and 2

    Let the prices be o, m, a. 4o + 6m + 8a = 2o + 4m + 10a gives 2o + 2m = 2a, so o + m = a (statement 2 is correct). Then (4o + 6m + 8a) - (3o + 5m + 9a) = o + m - a = 0, so the two totals are equal (statement 1 is correct). Both are correct.

  62. UPSC 2024 CSAT, question 54प्रश्न 54

    32^5 + 2^27 is divisible by

    1. 3
    2. 7
    3. 10
    4. 11
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 10

    32 = 2^5, so 32^5 = 2^25. Then 2^25 + 2^27 = 2^25 (1 + 2^2) = 2^25 x 5 = 2^24 x 10. So the number is divisible by 10.

  63. UPSC 2024 CSAT, question 55प्रश्न 55

    Let p and q be positive integers satisfying p < q and p + q = k. What is the smallest value of k that does not determine p and q uniquely?

    1. 3
    2. 4
    3. 5
    4. 6
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 5

    k = 3: only (1, 2). k = 4: only (1, 3), since (2, 2) fails p < q. k = 5: (1, 4) and (2, 3), two possibilities. So 5 is the smallest k for which p and q are not unique.

  64. UPSC 2024 CSAT, question 59प्रश्न 59

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What are the values of m and n, where m and n are natural numbers? Statement-I: m + n > mn and m > n. Statement-II: The product of m and n is 24. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

    Statement I: m + n > mn means mn - m - n + 1 < 1, that is (m - 1)(n - 1) < 1, so (m - 1)(n - 1) = 0 and one of them is 1. With m > n, n = 1 and m can be any number from 2 upward, so I alone is not enough. Statement II alone allows (1, 24), (2, 12), (3, 8), (4, 6) and so on, so it is not enough. Together: n = 1 and mn = 24 give m = 24, n = 1 (check: 25 > 24). Both statements together are needed.

  65. UPSC 2024 CSAT, question 60प्रश्न 60

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What is the time required to download the software? Statement-I: The size of the software is 12 megabytes. Statement-II: The transfer rate is 2.4 kilobytes per second. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

    Time = size / transfer rate. Statement I gives only the size and Statement II only the rate, so neither alone is enough. Together: 12 megabytes = 12 x 1024 = 12,288 kilobytes, and 12,288 / 2.4 = 5,120 seconds (about 85 minutes). So both statements together answer the question.

  66. UPSC 2024 CSAT, question 64प्रश्न 64

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What are the unique values of x and y, where x, y are distinct natural numbers? Statement-I: x / y is odd. Statement-II: xy = 12 Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

    Statement I alone allows endless pairs such as (3, 1), (5, 1), (6, 2). Statement II alone allows (1, 12), (12, 1), (2, 6), (6, 2), (3, 4), (4, 3). Together, x/y must be an odd whole number: 12/1 = 12 is even, 6/2 = 3 is odd, and the other ratios are fractions. So x = 6, y = 2 uniquely. Both statements together are needed.

  67. UPSC 2024 CSAT, question 65प्रश्न 65

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. A certain amount was distributed among X, Y and Z. Question: Who received the least amount? Statement-I: X received 4/5 of what Y and Z together received. Statement-II: Y received 2/7 of what X and Z together received. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

    Let the total be T. Statement I gives X = 4/9 of T but says nothing about how Y and Z share the rest. Statement II gives Y = 2/9 of T but nothing about X and Z separately. Together: X = 4T/9, Y = 2T/9, so Z = 3T/9. Y received the least. Both statements are needed.

  68. UPSC 2024 CSAT, question 66प्रश्न 66

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: If the average marks in a class are 60, then what is the number of students in the class? Statement-I: The highest marks in the class are 70 and the lowest marks are 50. Statement-II: Exclusion of highest and lowest marks from the class does not change the average. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) The Question cannot be answered even by using both the Statements together

    Neither statement gives a total of marks or any count. Even together, the highest and lowest marks (70 and 50) themselves average 60, so removing them leaves the average at 60 whatever the class size is. A class of 3, 10 or 50 students fits equally well, so the number of students cannot be found.

  69. UPSC 2024 CSAT, question 67प्रश्न 67

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. There are three distinct prime numbers whose sum is a prime number. Question: What are those three numbers? Statement-I: Their sum is less than 23. Statement-II: One of the numbers is 5. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

    If 2 were one of the primes, the sum would be even and greater than 2, so not prime; hence all three are odd primes. Statement I: odd-prime triples with sum below 23 are 3 + 5 + 7 = 15, 3 + 5 + 11 = 19, 3 + 5 + 13 = 21 and 3 + 7 + 11 = 21; only 19 is prime, so the numbers are 3, 5, 11 uniquely. Statement II alone is not enough: 3 + 5 + 11 = 19 and 5 + 7 + 11 = 23 both work. So Statement I alone answers the question but Statement II alone does not.

  70. UPSC 2024 CSAT, question 68प्रश्न 68

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: Is (x + y) an integer? Statement-I: (2x + y) is an integer. Statement-II: (x + 2y) is an integer. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) The Question cannot be answered even by using both the Statements together

    Adding the two expressions gives 3(x + y), so together they only tell us that 3(x + y) is an integer. Take x = y = 1/3: 2x + y = 1 and x + 2y = 1 are integers but x + y = 2/3 is not. Take x = y = 1: both are integers and x + y = 2 is an integer. Since both answers are possible, the question cannot be answered even with both statements.

  71. UPSC 2024 CSAT, question 69प्रश्न 69

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. A person buys three articles p, q and r for ₹ 50. The price of the article q is ₹ 16 which is the least. Question: What is the price of the article p? Statement-I: The cost of p is not more than that of r. Statement-II: The cost of r is not more than that of p. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone

    p + r = 50 - 16 = 34, and both p and r are more than 16. Statement I (p not more than r) only gives p at most 17, so p can be any price above 16 up to 17, such as 16.50 or 17. Statement II (r not more than p) only gives p at least 17. Together p is both at most and at least r, so p = r = 34/2 = ₹ 17. Both statements are needed. (This reading treats prices as not necessarily whole rupees, as the official key does.)

  72. UPSC 2024 CSAT, question 70प्रश्न 70

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. P, Q, R and S appeared in a test. Question: Has P scored more marks than Q? Statement-I: The sum of the marks scored by P and Q is equal to the sum of the marks scored by R and S. Statement-II: The sum of the marks scored by P and S is more than the sum of the marks scored by Q and R. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) The Question cannot be answered even by using both the Statements together

    Combining the two statements only tells us P > R and S > Q; it says nothing about P against Q. Example 1: P = 10, Q = 5, R = 4, S = 11 gives 15 = 15 and 21 > 9, with P > Q. Example 2: P = 5, Q = 10, R = 4, S = 11 gives 15 = 15 and 16 > 14, with P < Q. Both fit the statements, so the question cannot be answered even with both.

  73. UPSC 2024 CSAT, question 73प्रश्न 73

    A Question is given followed by two Statements I and II. Consider the Question and the Statements. Age of each of P and Q is less than 100 years but more than 10 years. If you interchange the digits of the age of P, the number represents the age of Q. Question: What is the difference of their ages? Statement-I: The age of P is greater than the age of Q. Statement-II: The sum of their ages is 11/6 times their difference. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
    2. The Question can be answered by using either Statement alone
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
    4. The Question cannot be answered even by using both the Statements together
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone

    Let the ages be 10a + b and 10b + a (both digits non-zero, since both ages are two-digit). Sum = 11(a + b) and difference = 9 x (difference of digits). Statement II: 11(a + b) = (11/6) x 9 x (difference of digits), so 2(a + b) = 3 x (difference of digits). Taking the larger digit as 5 times the smaller, the only digits that work are 5 and 1, so the ages are 51 and 15 and the difference is 36, whichever is older. So II alone answers the question. Statement I alone only says P is older, which allows many pairs (21 and 12, 31 and 13, ...), so it is not enough.

  74. UPSC 2024 CSAT, question 75प्रश्न 75

    If a + b means a - b; a - b means a x b; a x b means a ÷ b; a ÷ b means a + b, then what is the value of 10 + 30 - 100 x 50 ÷ 25? (Operations are to be replaced simultaneously)

    1. 15
    2. 0
    3. -15
    4. -25
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) -25

    Replace each sign at once: + becomes -, - becomes x, x becomes ÷, ÷ becomes +. The expression becomes 10 - 30 x 100 ÷ 50 + 25. Using BODMAS: 30 x 100 = 3000, 3000 ÷ 50 = 60, so 10 - 60 + 25 = -25.

  75. UPSC 2024 CSAT, question 79प्रश्न 79

    What will come in place of * in the sequence 3, 14, 39, 84, *, 258?

    1. 150
    2. 155
    3. 160
    4. 176
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 155

    The n-th term is n(n^2 + n + 1): 1 x 3 = 3, 2 x 7 = 14, 3 x 13 = 39, 4 x 21 = 84, 5 x 31 = 155, 6 x 43 = 258. (Equivalently, the differences 11, 25, 45, 71, 103 grow by 14, 20, 26, 32.) So * = 155.

  76. UPSC 2024 CSAT, question 80प्रश्न 80

    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. 1
    2. 2
    3. 3
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 2

    Q - S = 2S means Q = 3S, so (S, Q) is (4, 12) or (5, 15). If S = 4, Q = 12, the rest are 5, 10, 15 and T = R + 7 has no solution among them. If S = 5, Q = 15, the rest are 4, 10, 12 and T = R + 8 gives R = 4, T = 12, leaving P = 10. So P + R - T = 10 + 4 - 12 = 2.

  77. UPSC 2023 CSAT, question 4प्रश्न 4

    Raj has ten pairs of red, nine pairs of white and eight pairs of black shoes in a box. If he randomly picks shoes one by one (without replacement) from the box to get a red pair of shoes to wear, what is the maximum number of attempts he has to make?

    1. 27
    2. 36
    3. 44
    4. 45
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 45

    Take the worst case. He could first draw every white and black shoe: 18 + 16 = 34 shoes. He could then draw all 10 red shoes of one foot (say all left shoes) without completing a wearable pair. The next shoe must be a red shoe of the other foot, completing a red pair. Maximum attempts = 34 + 10 + 1 = 45.

  78. UPSC 2023 CSAT, question 5प्रश्न 5

    In how many ways can a batsman score exactly 25 runs by scoring single runs, fours and sixes only, irrespective of the sequence of scoring shots?

    1. 18
    2. 19
    3. 20
    4. 21
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 19

    Let the number of sixes be z and fours be y; the singles then fill the rest, so we need 6z + 4y <= 25. z = 0: y = 0 to 6 (7 ways). z = 1: 4y <= 19, y = 0 to 4 (5 ways). z = 2: 4y <= 13, y = 0 to 3 (4 ways). z = 3: 4y <= 7, y = 0 or 1 (2 ways). z = 4: 4y <= 1, y = 0 (1 way). Total = 7 + 5 + 4 + 2 + 1 = 19.

  79. UPSC 2023 CSAT, question 7प्रश्न 7

    What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100?

    1. 0
    2. 1
    3. 2
    4. 4
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 0

    100 = 4 x 25. The factors 85 = 5 x 17 and 95 = 5 x 19 together give 5 x 5 = 25, and 96 = 32 x 3 gives at least 4. So the product contains 25 x 4 = 100 as a factor and is exactly divisible by 100. The remainder is 0.

  80. UPSC 2023 CSAT, question 8प्रश्न 8

    What is the unit digit in the expansion of (57242)^(9×7×5×3×1)?

    1. 2
    2. 4
    3. 6
    4. 8
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 2

    The unit digit depends only on the last digit 2. Powers of 2 end in 2, 4, 8, 6 and repeat every 4. The exponent is 9 x 7 x 5 x 3 x 1 = 945, and 945 leaves remainder 1 when divided by 4. So the unit digit is the same as that of 2^1, which is 2.

  81. UPSC 2023 CSAT, question 9प्रश्न 9

    If ABC and DEF are both 3-digit numbers such that A, B, C, D, E and F are distinct non-zero digits such that ABC + DEF = 1111, then what is the value of A + B + C + D + E + F?

    1. 28
    2. 29
    3. 30
    4. 31
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 31

    Units column: C + F must end in 1. Since both are non-zero and distinct, C + F cannot be 1, so C + F = 11 with carry 1. Tens column: B + E + 1 must end in 1, so B + E = 10 with carry 1. Hundreds column: A + D + 1 = 11, so A + D = 10. Total = 11 + 10 + 10 = 31. (Example: 124 + 987 = 1111.)

  82. UPSC 2023 CSAT, question 10प्रश्न 10

    D is a 3-digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the digit at the hundred's place and the digit at the unit's place of D?

    1. 0
    2. 7
    3. 8
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 8

    To make number / digit sum small, keep the hundreds digit as small as possible (1) and the other digits as large as possible. 199 / 19 = 10.47, while 189 / 18 = 10.5, 198 / 18 = 11 and 299 / 20 = 14.95, so 199 gives the least ratio. Its hundreds digit is 1 and units digit is 9, a difference of 9 - 1 = 8.

  83. UPSC 2023 CSAT, question 14प्रश्न 14

    Three of the five positive integers p, q, r, s, t are even and two of them are odd (not necessarily in order). Consider the following: 1. p + q + r - s - t is definitely even. 2. 2p + q + 2r - 2s + t is definitely odd. Which of the above statements is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 1 only

    Subtracting a number has the same effect on parity as adding it, so p + q + r - s - t has the same parity as p + q + r + s + t. With exactly two odd numbers, that sum is even, so statement 1 is correct. In 2p + q + 2r - 2s + t the terms 2p, 2r and 2s are even, so the parity depends only on q + t. If q and t are both odd (or both even) this is even, so the expression is not definitely odd. Only statement 1 is correct.

  84. UPSC 2023 CSAT, question 15प्रश्न 15

    Consider the following in respect of prime number p and composite number c. 1. (p + c)/(p - c) can be even. 2. 2p + c can be odd. 3. pc can be odd. Which of the statements given above are correct?

    1. 1 and 2 only
    2. 2 and 3 only
    3. 1 and 3 only
    4. 1, 2 and 3
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 1, 2 and 3

    1: take p = 2, c = 6: (2 + 6)/(2 - 6) = 8/(-4) = -2, an even number. 2: 2p is always even, so 2p + c is odd whenever c is an odd composite, for example 2 x 3 + 9 = 15. 3: an odd prime times an odd composite is odd, for example 3 x 9 = 27. All three can happen.

  85. UPSC 2023 CSAT, question 16प्रश्न 16

    A 3-digit number ABC, on multiplication with D gives 37DD where A, B, C and D are different non-zero digits. What is the value of A + B + C?

    1. 18
    2. 16
    3. 15
    4. Cannot be determined due to insufficient data
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 18

    37DD = 3700 + 11D. For this to be divisible by D, D must divide 3700, so D is 1, 2, 4 or 5. D = 1 gives ABC = 3711 and D = 2 gives 1861, both 4-digit. D = 5 gives 3755 / 5 = 751, which repeats the digit 5. D = 4 gives 3744 / 4 = 936, with digits 9, 3, 6 all different from 4. So A + B + C = 9 + 3 + 6 = 18.

  86. UPSC 2023 CSAT, question 17प्रश्न 17

    For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by:

    1. 7 and 11 only
    2. 11 and 13 only
    3. 7 and 13 only
    4. 7, 11 and 13
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 7, 11 and 13

    XYZXYZ = XYZ x 1000 + XYZ = XYZ x 1001, and 1001 = 7 x 11 x 13. So every such number is divisible by 7, 11 and 13.

  87. UPSC 2023 CSAT, question 18प्रश्न 18

    125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side?

    1. 27
    2. 25
    3. 21
    4. 18
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 27

    125 cubes form a 5 x 5 x 5 block. A cube is surrounded on every side only if it is not on any outer face. Removing the outer layer leaves an inner block of (5 - 2) x (5 - 2) x (5 - 2) = 3 x 3 x 3 = 27 cubes.

  88. UPSC 2023 CSAT, question 19प्रश्न 19

    How many distinct 8-digit numbers can be formed by rearranging the digits of the number 11223344 such that odd digits occupy odd positions and even digits occupy even positions?

    1. 12
    2. 18
    3. 36
    4. 72
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 36

    The odd digits 1, 1, 3, 3 go into the 4 odd positions: 4!/(2! x 2!) = 6 ways. The even digits 2, 2, 4, 4 go into the 4 even positions: also 6 ways. Total = 6 x 6 = 36.

  89. UPSC 2023 CSAT, question 20प्रश्न 20

    A, B, C working independently can do a piece of work in 8, 16 and 12 days respectively. A alone works on Monday, B alone works on Tuesday, C alone works on Wednesday; A alone, again works on Thursday and so on. Consider the following statements: 1. The work will be finished on Thursday. 2. The work will be finished in 10 days. Which of the above statements is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 1 only

    Take total work = 48 units. A does 6, B does 3 and C does 4 units a day, so each 3-day round gives 13 units. After 3 rounds (9 days) 39 units are done. Day 10 (A) brings it to 45 and day 11 (B) to 48, so the work ends on day 11. Day 1 is Monday, so day 11 is Monday + 10 days = Thursday. Statement 1 is correct; statement 2 is wrong because it takes 11 days.

  90. UPSC 2023 CSAT, question 25प्रश्न 25

    Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?

    1. 10
    2. 11
    3. 12
    4. Infinitely many
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 12

    (7x + 96)/x = 7 + 96/x, so x must be a divisor of 96. Since 96 = 2^5 x 3, it has (5 + 1)(1 + 1) = 12 positive divisors. So 12 values of x are possible.

  91. UPSC 2023 CSAT, question 26प्रश्न 26

    If p, q, r and s are distinct single digit positive numbers, then what is the greatest value of (p + q)(r + s)?

    1. 230
    2. 225
    3. 224
    4. 221
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 225

    Use the four largest digits 9, 8, 7 and 6, whose total is 30. For a fixed sum, a product of two parts is largest when the parts are equal, so split them into 15 and 15: (9 + 6)(8 + 7) = 15 x 15 = 225.

  92. UPSC 2023 CSAT, question 27प्रश्न 27

    A number N is formed by writing 9 for 99 times. What is the remainder if N is divided by 13?

    1. 11
    2. 9
    3. 7
    4. 1
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 11

    N = 10^99 - 1. Since 999999 = 13 x 76923, 10^6 leaves remainder 1 on division by 13. 99 = 6 x 16 + 3, so 10^99 leaves the same remainder as 10^3 = 1000 = 13 x 76 + 12, that is 12. Then N = 10^99 - 1 leaves 12 - 1 = 11.

  93. UPSC 2023 CSAT, question 28प्रश्न 28

    Each digit of a 9-digit number is 1. It is multiplied by itself. What is the sum of the digits of the resulting number?

    1. 64
    2. 80
    3. 81
    4. 100
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 81

    111111111 x 111111111 = 12345678987654321. The sum of its digits is (1 + 2 + ... + 8) x 2 + 9 = 36 x 2 + 9 = 81.

  94. UPSC 2023 CSAT, question 29प्रश्न 29

    What is the sum of all digits which appear in all the integers from 10 to 100?

    1. 855
    2. 856
    3. 910
    4. 911
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 856

    From 10 to 99, each tens digit 1 to 9 appears 10 times: 45 x 10 = 450. Each units digit 0 to 9 appears 9 times: 45 x 9 = 405. That gives 855, and 100 adds 1 more. Total = 856.

  95. UPSC 2023 CSAT, question 30प्रश्न 30

    ABCD is a square. One point on each of AB and CD; and two distinct points on each of BC and DA are chosen. How many distinct triangles can be drawn using any three points as vertices out of these six points?

    1. 16
    2. 18
    3. 20
    4. 24
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 20

    There are 6 points and C(6, 3) = 20 ways to pick three of them. Three points can fail to form a triangle only if they lie on one straight line, which needs three points on the same side, but no side has more than two chosen points. So all 20 selections give triangles.

  96. UPSC 2023 CSAT, question 36प्रश्न 36

    A box contains 14 black balls, 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls and 54 white balls. Consider the following statements: 1. The smallest number n such that any n balls drawn from the box randomly must contain one full group of at least one colour is 175. 2. The smallest number m such that any m balls drawn from the box randomly must contain at least one ball of each colour is 167. Which of the above statements is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Both 1 and 2

    Statement 1: in the worst case we draw one less than every full group: 13 + 19 + 25 + 27 + 37 + 53 = 174 balls with no colour complete. The 175th ball must complete some colour, so n = 175. Statement 2: in the worst case we first draw every ball except the smallest colour (black): 20 + 26 + 28 + 38 + 54 = 166. The 167th ball must be black, so m = 167. Both are correct.

  97. UPSC 2023 CSAT, question 44प्रश्न 44

    A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct?

    1. R = S
    2. R > S
    3. R < S
    4. R ≤ S
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) R < S

    Half-yearly: Q = P(1 + R/200)^2 = P(1 + R/100 + R^2/40000). Annually: Q = P(1 + S/100). Equating, S = R + R^2/400, which is greater than R for any positive R. Half-yearly compounding earns interest on interest, so a lower nominal rate R gives the same amount as the annual rate S. Hence R < S.

  98. UPSC 2023 CSAT, question 45प्रश्न 45

    How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?

    1. 6
    2. 7
    3. 8
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 9

    The divisor must divide 1186 - 31 = 1155 and must be greater than 31 (a remainder is always smaller than the divisor). 1155 = 3 x 5 x 7 x 11, whose divisors are 1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385 and 1155. Those greater than 31 are 33, 35, 55, 77, 105, 165, 231, 385 and 1155, which is 9 numbers.

  99. UPSC 2023 CSAT, question 46प्रश्न 46

    Let pp, qq and rr be 2-digit numbers where p < q < r. If pp + qq + rr = tt0, where tt0 is a 3-digit number ending with zero, consider the following statements: 1. The number of possible values of p is 5. 2. The number of possible values of q is 6. Which of the above statements is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Both 1 and 2

    pp + qq + rr = 11(p + q + r) and tt0 = 110t, so p + q + r = 10t. With distinct digits 1 to 9 the sum is at most 7 + 8 + 9 = 24, so the sum is 10 or 20. Sum 10: (1,2,7), (1,3,6), (1,4,5), (2,3,5). Sum 20: (3,8,9), (4,7,9), (5,6,9), (5,7,8). Possible p: 1, 2, 3, 4, 5, five values. Possible q: 2, 3, 4, 6, 7, 8, six values. Both statements are correct.

  100. UPSC 2023 CSAT, question 47प्रश्न 47

    What is the sum of all 4-digit numbers less than 2000 formed by the digits 1, 2, 3 and 4, where none of the digits is repeated?

    1. 7998
    2. 8028
    3. 8878
    4. 9238
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 7998

    A number below 2000 must start with 1, and the other three places take 2, 3 and 4 in some order: 3! = 6 numbers. Thousands place: 6 x 1000 = 6000. In each of the other three places, each of 2, 3, 4 appears twice, giving 2 x (2 + 3 + 4) = 18 per place, so 18 x 111 = 1998. Total = 6000 + 1998 = 7998.

  101. UPSC 2023 CSAT, question 48प्रश्न 48

    What is the number of selections of 10 consecutive things out of 12 things in a circle taken in the clockwise direction?

    1. 3
    2. 11
    3. 12
    4. 66
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 12

    In a circle, a block of 10 consecutive things taken clockwise is fixed by its starting point. Any of the 12 things can be the start, and each start gives a different block, so there are 12 selections.

  102. UPSC 2023 CSAT, question 50प्रश्न 50

    There are three traffic signals. Each signal changes colour from green to red and then from red to green. The first signal takes 25 seconds, the second signal takes 39 seconds and the third signal takes 60 seconds to change the colour from green to red. The durations for green and red colours are same. At 2:00 p.m, they together turn green. At what time will they change to green next, simultaneously?

    1. 4:00 p.m.
    2. 4:10 p.m.
    3. 4:20 p.m.
    4. 4:30 p.m.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 4:10 p.m.

    Each signal stays green and red for equal times, so a full green-to-green cycle is twice the given time: 50, 78 and 120 seconds. LCM(50, 78, 120): 50 = 2 x 5^2, 78 = 2 x 3 x 13, 120 = 2^3 x 3 x 5, so LCM = 2^3 x 3 x 5^2 x 13 = 7800 seconds = 130 minutes = 2 hours 10 minutes. They next turn green together at 4:10 p.m.

  103. UPSC 2023 CSAT, question 54प्रश्न 54

    40 children are standing in a circle and one of them (say child-1) has a ring. The ring is passed clockwise. Child-1 passes on to child-2, child-2 passes on to child-4, child-4 passes on to child-7 and so on. After how many such changes (including child-1) will the ring be in the hands of child-1 again?

    1. 14
    2. 15
    3. 16
    4. 17
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) 15

    The jumps are 1, 2, 3, 4, ... places, so after k passes the ring has moved 1 + 2 + ... + k = k(k + 1)/2 places. It is back with child-1 when this is a multiple of 40. The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, and 120 = 3 x 40 is the first multiple of 40, reached at k = 15. So after 15 changes.

  104. UPSC 2023 CSAT, question 58प्रश्न 58

    In a party, 75 persons took tea, 60 persons took coffee and 15 persons took both tea and coffee. No one taking milk takes tea. Each person takes at least one drink. Question: How many persons attended the party? Statement-1: 50 persons took milk. Statement-2: Number of persons who attended the party is five times the number of persons who took milk only. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
    2. The Question can be answered by using either Statement alone.
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
    4. The Question cannot be answered even by using both the Statements together.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

    Persons taking tea or coffee = 75 + 60 - 15 = 120. Everyone else took only milk, so total = 120 + (milk only). Statement 1 alone fails: some of the 50 milk takers may also take coffee, so milk-only is unknown. Statement 2 alone works: if milk-only = m, then 120 + m = 5m, so m = 30 and total = 150. So one statement (statement 2) is enough and the other is not.

  105. UPSC 2023 CSAT, question 59प्रश्न 59

    Consider a 3-digit number. Question: What is the number? Statement-1: The sum of the digits of the number is equal to the product of the digits. Statement-2: The number is divisible by the sum of the digits of the number. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
    2. The Question can be answered by using either Statement alone.
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
    4. The Question cannot be answered even by using both the Statements together.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) The Question cannot be answered even by using both the Statements together.

    Statement 1: a zero digit would make the product 0 but the sum positive, so all digits are non-zero, and the only digits with sum = product are 1, 2, 3 (1 + 2 + 3 = 1 x 2 x 3 = 6). That allows 123, 132, 213, 231, 312, 321. Statement 2 then needs divisibility by 6, which means the number must be even: 132 and 312 both work (132 = 6 x 22, 312 = 6 x 52). Two numbers remain, so the question cannot be answered even with both statements.

  106. UPSC 2023 CSAT, question 60प्रश्न 60

    For five children with ages a < b < c < d < e; any two successive ages differ by 2 years. Question: What is the age of the youngest child? Statement-1: The age of the eldest is 3 times the youngest. Statement-2: The average age of the children is 8 years. Which one of the following is correct in respect of the above Question and the Statements?

    1. The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
    2. The Question can be answered by using either Statement alone.
    3. The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.
    4. The Question cannot be answered even by using both the Statements together.
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (b) The Question can be answered by using either Statement alone.

    The ages are a, a + 2, a + 4, a + 6, a + 8. Statement 1: a + 8 = 3a, so a = 4. Statement 2: the average of these five ages is the middle one, a + 4 = 8, so a = 4. Each statement alone gives the answer.

  107. UPSC 2023 CSAT, question 65प्रश्न 65

    In an examination, the maximum marks for each of the four papers namely P, Q, R and S are 100. Marks scored by the students are in integers. A student can score 99% in n different ways. What is the value of n?

    1. 16
    2. 17
    3. 23
    4. 35
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 35

    99% of 400 is 396, so the four papers together fall short of full marks by exactly 4. Let the shortfalls be a, b, c, d >= 0 with a + b + c + d = 4 (none can exceed 100, so no other limit applies). The number of non-negative solutions is C(4 + 3, 3) = C(7, 3) = 35.

  108. UPSC 2023 CSAT, question 66प्रश्न 66

    A flag has to be designed with 4 horizontal stripes using some or all of the colours red, green and yellow. What is the number of different ways in which this can be done so that no two adjacent stripes have the same colour?

    1. 12
    2. 18
    3. 24
    4. 36
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) 24

    The top stripe can be any of 3 colours. Each following stripe can be any colour except the one just above it, so 2 choices each. Total = 3 x 2 x 2 x 2 = 24.

  109. UPSC 2023 CSAT, question 67प्रश्न 67

    A rectangular floor measures 4 m in length and 2.2 m in breadth. Tiles of size 140 cm by 60 cm have to be laid such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. What is the maximum number of tiles that can be accommodated on the floor?

    1. 6
    2. 7
    3. 8
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 9

    By area, at most 88000 / 8400 = 10.47, so no more than 10 tiles, and 10 cannot actually be fitted. A simple layout of equal rows gives only 8, but mixing orientations gives 9. Measure x along the 400 cm length and y along the 220 cm breadth. Lay 3 tiles flat (140 along x) in the strip x = 0 to 140, at y = 0, 60 and 120. Lay 1 flat tile at x = 140 to 280, y = 0 to 60. Stand 2 tiles upright (60 along x, 140 along y) at x = 140 to 200 and 200 to 260, both covering y = 60 to 200. Stand 2 more upright at x = 280 to 340 and 340 to 400, covering y = 0 to 140. Lay 1 flat tile at x = 260 to 400, y = 140 to 200. That is 3 + 1 + 2 + 2 + 1 = 9 tiles with no overlap.

  110. UPSC 2023 CSAT, question 68प्रश्न 68

    There are five persons P, Q, R, S and T each one of whom has to be assigned one task. Neither P nor Q can be assigned Task-1. Task-2 must be assigned to either R or S. In how many ways can the assignment be done?

    1. 6
    2. 12
    3. 18
    4. 24
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 24

    Treat it as 5 tasks for 5 persons. Task-2 goes to R or S: 2 ways. Task-1 cannot go to P or Q, nor to the person already given Task-2, so it goes to one of the remaining two among R, S, T: 2 ways. The other 3 tasks go to the remaining 3 persons in 3! = 6 ways. Total = 2 x 2 x 6 = 24.

  111. UPSC 2023 CSAT, question 69प्रश्न 69

    There are large number of silver coins weighing 2 gm, 5 gm, 10 gm, 25 gm, 50 gm each. Consider the following statements: 1. To buy 78 gm of coins one must buy at least 7 coins. 2. To weigh 78 gm using these coins one can use less than 7 coins. Which of the statements given above is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (c) Both 1 and 2

    Statement 1: the coins bought must add up to exactly 78. 50 + 25 leaves 3, which cannot be made, so try 50 alone: the remaining 28 needs at least 6 coins (10 + 10 + 2 + 2 + 2 + 2), giving 7 coins. Other combinations, such as 25 + 25 with 28 more, or using 5 gm coins (which must come in pairs to keep the total even), need more coins. So at least 7 coins are needed. Statement 2: when weighing on a balance, coins can go on both pans. Putting 50 + 25 + 5 = 80 gm on one pan and a 2 gm coin with the silver on the other weighs out 78 gm using only 4 coins. Both statements are correct.

  112. UPSC 2023 CSAT, question 74प्रश्न 74

    What is the remainder if 2^192 is divided by 6?

    1. 0
    2. 1
    3. 2
    4. 4
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (d) 4

    Powers of 2 divided by 6 leave remainders 2, 4, 2, 4, ... (2, 4, 8 = 6 + 2, 16 = 12 + 4), so odd powers leave 2 and even powers leave 4. Since 192 is even, 2^192 leaves remainder 4.

  113. UPSC 2023 CSAT, question 76प्रश्न 76

    AB and CD are 2-digit numbers. Multiplying AB with CD results in a 3-digit number DEF. Adding DEF to another 3-digit number GHI results in 975. Further A, B, C, D, E, F, G, H, I are distinct digits. If E = 0, F = 8, then what is A + B + C equal to?

    1. 6
    2. 7
    3. 8
    4. 9
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 6

    DEF = D08. In D08 + GHI = 975: units 8 + I = 15 gives I = 7 with carry 1; tens 0 + H + 1 = 7 gives H = 6; hundreds D + G = 9. With 0, 6, 7, 8 used, D + G = 9 is only possible as 4 + 5. If D = 5, AB x C5 = 508 = 4 x 127, which has no 2-digit factor ending in 5. So D = 4 and AB x C4 = 408. The 2-digit factors of 408 ending in 4 are 24 (408 / 24 = 17, but 7 is already I) and 34 (408 / 34 = 12). So AB = 12, CD = 34, G = 5, and all nine digits are distinct. A + B + C = 1 + 2 + 3 = 6.

  114. UPSC 2023 CSAT, question 79प्रश्न 79

    A cuboid of dimensions 7 cm × 5 cm × 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct? 1. There are exactly 15 small cubes with no paint on any face. 2. There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green. Select the correct answer using the code given below:

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer and explanationउत्तर और व्याख्या

    Answer:उत्तर: (a) 1 only

    Statement 1: unpainted cubes form the inner block (7 - 2) x (5 - 2) x (3 - 2) = 5 x 3 x 1 = 15, so it is correct. Statement 2: a green face (5 x 3) meets a blue face (7 x 3) along an edge of length 3 cm, and there are 4 such edges. Leaving out the two corner cubes on each edge, each edge has 3 - 2 = 1 cube with exactly a green and a blue face, giving 4 such cubes, not 6. Only statement 1 is correct.

Questions and answer keys are published by the Union Public Service Commission (upsc.gov.in). Explanations are written by IndiaDigital; if one looks wrong, use the report link under it.प्रश्न और उत्तर-कुंजियाँ संघ लोक सेवा आयोग (upsc.gov.in) द्वारा प्रकाशित हैं। व्याख्याएँ IndiaDigital ने लिखी हैं; कोई गलत लगे तो उसके नीचे दिए लिंक से बताइए।

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