Coding-decoding, series, patterns, cubes, dice and visual reasoning: previous-year questions
95 real UPSC Prelims questions (2011–2026) with the official answer and a clear explanation. 34 of them came in the last five papers.
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A pattern formed by two characters a and b is repeated more than once in the following string :
× b × a × a × × a × a × bab
What is × × in the 7th and 8th positions from the left in the above string?
Answer (d) & explanation
Positions 7 and 8 are bb (d). Step 1: the string has 15 places, so try a block of 5 letters repeated three times: (× b × a ×)(a × × a ×)(a × b a b). Step 2: read each column across the three blocks. Position 1: ×, a, a → a. Position 2: b, ×, × → b. Position 3: ×, ×, b → b. Position 4: a, a, a → a. Position 5: ×, ×, b → b. Every known letter fits, so the block is abbab. Step 3: places 7 and 8 are the 2nd and 3rd letters of the second block: b, b. Options (a), (b) and (c) fail.
Tip: for repeating patterns, try block sizes that divide the total length. Stack the blocks and fill each column from the known letters.
a block of 3 fails: the 2nd column would need both b and a.
Answer as per the official UPSC answer key.
A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by xₙ. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for x₄?
Answer (a) & explanation
16 is not possible (a). Step 1: three plane cuts that leave identical pieces can give 4 slabs (all parallel), 6 equal pieces (two parallel plus one across), or 8 small cubes (one in each direction). Step 2: a fourth cut can add at most one piece for each piece it passes through. From 4 slabs: 5 to 8 pieces. From 6 pieces: up to 12. From 8 cubes: a single plane can pass through at most 7 of the 8 small cubes, so at most 15. Step 3: so 5, 8 and 12 are possible, but 16 would need the cut to split all 8 cubes, which no plane can do. Options (b), (c) and (d) are possible values.
Tip: a plane can never cut all 8 octants of a cube at once; it misses at least one.
Answer as per the official UPSC answer key.
In a sequence of numbers, each number other than the first two is the sum of the two immediately preceding numbers from it. If the first two numbers in the sequence are 4 and 7, then the sixth number is
Answer (d) & explanation
The sixth number is 47 (d). Step 1: the rule: each number after the first two is the sum of the two before it. Step 2: start with 4 and 7. Third: 4 + 7 = 11. Fourth: 7 + 11 = 18. Fifth: 11 + 18 = 29. Step 3: sixth: 18 + 29 = 47. Option (a), 29, is the fifth number, a common slip from miscounting positions. Options (b) and (c) are wrong sums.
Tip: this is a Fibonacci-style sequence. Write out every term and number its position so you do not stop one step early. The classic Fibonacci series starts 1, 1, 2, 3, 5, 8.
Answer as per the official UPSC answer key.
If ZERO is encoded as ADSN, then how do you encode STOP?
Answer (b) & explanation
STOP is encoded as TSPO (b). Step 1: compare ZERO and ADSN letter by letter: Z → A (+1, wrapping round), E → D (−1), R → S (+1), O → N (−1). The moves alternate +1, −1. Step 2: apply to STOP: S + 1 = T, T − 1 = S, O + 1 = P, P − 1 = O. Step 3: so STOP becomes TSPO. Options (a) SPOT, (c) TSOP and (d) POST are rearrangements that do not follow the rule.
Tip: in letter codes, write the shift for each position. Remember that after Z comes A again. Options that are real words are often traps.
Answer as per the official UPSC answer key.
What is X in the sequence 24, X, 12, 18, 36, 90?
Answer (b) & explanation
X is 12 (b). Step 1: look at the later terms: 12 → 18 is × 1.5, 18 → 36 is × 2, 36 → 90 is × 2.5. The multiplier rises by 0.5 each time. Step 2: go backwards: before 1.5 comes 1, and before that 0.5. Step 3: so 24 × 0.5 = 12 = X, and 12 × 1 = 12 ✓. Options (a) 18, (c) 9 and (d) 6 break the pattern.
Tip: if numbers rise unevenly, try ratios instead of differences. When the start of a pattern is missing, work backwards from the known part.
Answer as per the official UPSC answer key.
If P = +, Q = –, R = ×, S = ÷, then insert the proper notations between the successive numbers in the equation 60 _ 15 _ 3 _ 20 _ 4 = 20 :
Answer (b) & explanation
The correct order is QRPS (b). Step 1: decode: P = +, Q = −, R = ×, S = ÷. Step 2: test (b) QRPS: 60 − 15 × 3 + 20 ÷ 4. Do × and ÷ first: 15 × 3 = 45 and 20 ÷ 4 = 5. Then 60 − 45 + 5 = 20 ✓. Step 3: test (a) SPRQ: 60 ÷ 15 + 3 × 20 − 4 = 4 + 60 − 4 = 60 ✗. Option (c) QRSP gives 60 − 15 × 3 ÷ 20 + 4 = 61.75, and (d) SPQR gives 4 + 3 − 80 = −73.
Tip: test the options rather than building the answer yourself. Always apply BODMAS: × and ÷ before + and −.
Answer as per the official UPSC answer key.
If NO is coded as 210, NOT is coded as 4200 and NOTE is coded as 21000, then how is NOTES coded?
Answer (a) & explanation
NOTES is coded as 399000 (a). Step 1: use alphabet positions: N = 14, O = 15. 14 × 15 = 210 = NO ✓. Step 2: T = 20: 210 × 20 = 4200 = NOT ✓. E = 5: 4200 × 5 = 21000 = NOTE ✓. The rule is the product of positions. Step 3: S = 19: 21000 × 19 = 399000. Options (b), (c) and (d) multiply by 20, 21 or 30 instead of 19.
Tip: when each longer word's code is a multiple of the shorter one's, the rule is likely multiplication by the new letter's position. Know positions by heart: N = 14, S = 19.
Answer as per the official UPSC answer key.
If FRANCE is coded as 654321 and GERMANY is coded as 9158437, then how is YEMEN coded ?
Answer (d) & explanation
YEMEN is coded as 71813 (d). Step 1: match letters to digits from FRANCE = 654321: F = 6, R = 5, A = 4, N = 3, C = 2, E = 1. Step 2: from GERMANY = 9158437: G = 9, E = 1 ✓, R = 5 ✓, M = 8, A = 4 ✓, N = 3 ✓, Y = 7. Step 3: YEMEN = Y E M E N = 7, 1, 8, 1, 3 = 71813. Option (c), 71913, takes M as 9 (that is G). Options (a) and (b) are wrong.
Tip: in letter-to-digit codes, line up each word with its code and check shared letters (here E, R, A, N) to confirm the mapping before answering.
Answer as per the official UPSC answer key.
What is X in the sequence 1, 3, 6, 11, 18, X, 42?
Answer (c) & explanation
X is 29 (c). Step 1: find the gaps: 3 − 1 = 2, 6 − 3 = 3, 11 − 6 = 5, 18 − 11 = 7. Step 2: the gaps are the prime numbers in order: 2, 3, 5, 7, so next come 11 and 13. Step 3: X = 18 + 11 = 29, and 29 + 13 = 42 ✓. Options (a) 26, (b) 27 and (d) 30 do not give 42 with a prime gap.
Tip: when gaps grow unevenly, check whether they are primes, squares or Fibonacci numbers. Always confirm with the last given term.
Answer as per the official UPSC answer key.
In a certain code if 64 is written as 343 and 216 is written as 729, then how is 512 written in that code?
Answer (b) & explanation
512 is written as 1331 (b). Step 1: notice the cubes: 64 = 4³ and 343 = 7³. 216 = 6³ and 729 = 9³. Step 2: the rule: add 3 to the cube root, then cube again: 4 → 7, 6 → 9. Step 3: 512 = 8³, so 8 + 3 = 11, and 11³ = 1331. Option (a), 1000 = 10³, adds only 2. Options (c) 12³ and (d) 13³ add too much.
Tip: learn cubes up to 15 (1331 = 11³, 1728 = 12³, 2197 = 13³). Spotting squares and cubes solves many number codes quickly.
Answer as per the official UPSC answer key.
What comes at X and Y respectively in the following sequence?
January, January, December, October, X, March, October, Y, September
Answer (b) & explanation
X is July and Y is April (b). Step 1: write the months as numbers: 1, 1, 12, 10, X, 3, 10, Y, 9. Step 2: each step goes back by one more month than before: back 0 (Jan to Jan), back 1 (Jan to Dec), back 2 (Dec to Oct), back 3 (Oct to July, X = 7), back 4 (July to March) ✓. Step 3: continue: back 5 (March to October) ✓, back 6 (October to April, Y = 4), back 7 (April to September) ✓. Options (a), (c) and (d) break the pattern.
Tip: turn months or letters into numbers, and count around the circle (after 1 comes 12). Then look for steps that grow by one each time.
Answer as per the official UPSC answer key.
A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?
- I. The minimum number of cuts is 9.
- II. The number of smaller pieces which are not painted on any face is 6.
Select the correct answer using the code given below:
Answer (c) & explanation
Both statements are correct (c). Step 1: for the fewest cuts, split the 60 pieces as evenly as possible across three directions: 60 = 3 × 4 × 5. Cuts needed = (3 − 1) + (4 − 1) + (5 − 1) = 2 + 3 + 4 = 9. So I is right. Step 2: pieces with no paint come from the inside: remove one layer from each side in every direction: (3 − 2) × (4 − 2) × (5 − 2) = 1 × 2 × 3 = 6. So II is right. Options (a), (b) and (d) fail.
Tip: for the minimum cuts, keep the three factors as close together as possible. Unpainted inner pieces = (a − 2)(b − 2)(c − 2).
Answer as per the official UPSC answer key.
If 7 * 24 = 25 and 12 * 16 = 20, then what is 16 * 63 equal to?
Answer (c) & explanation
1. Look at the first two: 7 × 7 + 24 × 24 = 49 + 576 = 625 = 25 × 25. And 12 × 12 + 16 × 16 = 144 + 256 = 400 = 20 × 20.
2. So the rule is: square both numbers, add, and find the number whose square is the total.
3. 16 × 16 + 63 × 63 = 256 + 3969 = 4225 = 65 × 65. Answer 65.
Answer as per the official UPSC answer key.
Consider the sequence AB_CC_A_BCCC_BBC_C that follows a certain pattern. Which one of the following completes the sequence?
Answer (c) & explanation
The blanks are B, C, B, A, C (c). Step 1: put option (c) into the five gaps: ABBCCCABBCCCABBCCC. Step 2: it reads ABBCCC ABBCCC ABBCCC: one A, two Bs and three Cs, repeated three times. A clear pattern. Step 3: the other options do not give a repeating block. For example, (a) puts A in the last gap, breaking the CCC at the end. So (a), (b) and (d) fail.
Tip: in letter-gap series, first count the total letters (here 18) and try block sizes that divide it. Then test options quickly rather than solving from scratch.
Answer as per the official UPSC answer key.
Consider the sequence
A _ BCD _ BBCDABC _ DABC _ D
that follows a certain pattern. Which one of the following completes the sequence?
Answer (c) & explanation
The blanks are A, A, C, D (c). Step 1: break the sequence into blocks of five letters: A_BCD, _BBCD, ABC_D, ABC_D. Step 2: look for the rule: each block is ABCD with one letter doubled, and the doubled letter moves along: AABCD, ABBCD, ABCCD, ABCDD. Step 3: so the blanks are A, A, C, D. Option (d), A, A, D, C, swaps the last two. Options (a) and (b) break the first blocks.
Tip: in letter-series blanks, try splitting the line into equal blocks of 4, 5 or 6. A pattern usually shows up once the blocks line up.
Answer as per the official UPSC answer key.
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, …
Answer (b) & explanation
The sum of the first 28 terms is 84 (b). Step 1: group the terms: (1), (1, 2), (1, 3, 2), (1, 4, 3, 2), (1, 5, 4, 3, 2) … Group n has n terms and its sum is 1 + 2 + … + n. Step 2: 1 + 2 + … + 7 = 28 terms, so the first 28 terms are exactly groups 1 to 7. Step 3: sum = 1 + 3 + 6 + 10 + 15 + 21 + 28 = 84. Options (a), (c) and (d) are off by one.
Tip: odd-looking sequences often break into groups. Find where each group starts (here, every new 1) and the pattern becomes clear.
Answer as per the official UPSC answer key.
If a + b means a − b; a − b means a × b; a × b means a ÷ b; a ÷ b means a + b, then what is the value of 10 + 30 − 100 × 50 ÷ 25? (Operations are to be replaced simultaneously)
Answer (d) & explanation
The value is −25 (d). Step 1: replace every sign at the same time: + becomes −, − becomes ×, × becomes ÷, ÷ becomes +. So 10 + 30 − 100 × 50 ÷ 25 becomes 10 − 30 × 100 ÷ 50 + 25. Step 2: do × and ÷ first (BODMAS): 30 × 100 = 3000, ÷ 50 = 60. Step 3: 10 − 60 + 25 = −25. Options (a), (b) and (c) come from changing signs one at a time or ignoring the order of operations.
Tip: rewrite the whole expression with new signs before calculating. Then apply BODMAS: division and multiplication before addition and subtraction.
replacing one sign at a time changes signs you have already replaced. Swap all of them together.
One trap spotted is one mark saved. The PYQ Book lines up every trap UPSC has set, so you walk in prepared. Try the free sample
Answer as per the official UPSC answer key.
If P means 'greater than (>)'; Q means 'less than (<)'; R means 'not greater than (≯)'; S means 'not less than (≮)' and T means 'equal to (=)', then consider the following statements :
- 1. If 2x(S)3y and 3x(T)4z, then 9y(P)8z.
- 2. If x(Q)2y and y(R)z, then x(R)z.
Which of the statements given above is/are correct?
Answer (d) & explanation
Neither statement is correct (d). Step 1: decode: P = >, Q = <, R = ≤ (not greater), S = ≥ (not less), T = =. Step 2: statement 1: 2x ≥ 3y and 3x = 4z, so x = 4z/3. Then 8z/3 ≥ 3y, that is 8z ≥ 9y. The claim 9y > 8z is the opposite, so 1 is wrong. Step 3: statement 2: x < 2y and y ≤ z give only x < 2z. Try y = z = 5, x = 8: x < 10 holds, but x > z. So x ≤ z need not hold; 2 is wrong. Options (a), (b) and (c) fail.
Tip: decode symbols first, then test a counter-example. One case that breaks a statement is enough to reject it.
Answer as per the official UPSC answer key.
If in a certain code, 'ABCD' is written as 24 and 'EFGH' is written as 1680, then how is 'IJKL' written in that code?
Answer (a) & explanation
IJKL is written as 11880 (a). Step 1: use alphabet positions: A = 1, B = 2, C = 3, D = 4. ABCD = 1 × 2 × 3 × 4 = 24. Step 2: check: E, F, G, H = 5, 6, 7, 8. 5 × 6 × 7 × 8 = 1680. The rule is the product of positions. Step 3: I, J, K, L = 9, 10, 11, 12. 9 × 10 × 11 × 12 = 90 × 132 = 11880. Options (b), (c) and (d) are wrong products.
Tip: in number codes for letters, first try the sum of positions, then the product. Learn the positions: E = 5, J = 10, O = 15, T = 20 (remember "EJOT").
Answer as per the official UPSC answer key.
If in a certain code, 'POT' is written as ATOP and 'TRAP' is written as APART, then how is 'ARENA' written in that code?
Answer (d) & explanation
ARENA is written as AANERA (d). Step 1: look at the first example: POT reversed is TOP; adding A in front gives ATOP. Step 2: check the second: TRAP reversed is PART; with A in front: APART. So the rule is: reverse the word, then put A in front. Step 3: ARENA reversed is ANERA. With A in front: AANERA. Options (a), (b) and (c) do not follow the rule.
Tip: in coding questions, test your rule on every example before applying it. Reversal plus a fixed added letter is a common pattern.
Answer as per the official UPSC answer key.