13 September 2025 · Shift 3

Letter series — consecutive triplets Concept: watch how each cluster is built — here every cluster has three CONSECUTIVE letters of the alphabet. A-B-C, then D-E-F, then G-H-I — each new cluster simply starts from the very next letter. After I, the next three consecutive letters are J, K, L. Hence JKL — option (d).

Letter series (−2, +4, +7) Trick: track each position separately across the clusters. 1st letters: X, V, T, R — every time −2 ⇒ next is P. 2nd letters: I, M, Q, U — every time +4 ⇒ next is Y; 3rd letters: N, U, B, I — every time +7 (cyclic) ⇒ next is P. Hence PYP — option (c).

Series (n² + 1) Trick: when terms grow very fast (5 → 677), think of squaring. Rule here: next term = (previous term)² + 1. Check: 1²+1 = 2 ✓; 2²+1 = 5 ✓; so ? = 5²+1 = 26; and 26²+1 = 677 ✓ — the rule fits the whole series. Hence 26 — option (a).

Prime numbers First identify what is common: 59, 61, 71 and 79 are all PRIME numbers. Now test each option: 111 = 3 × 37 ✗; 147 = 3 × 49 ✗; 169 = 13 × 13 ✗. 101 has no factor other than 1 and itself — it is prime. Hence 101 — option (b).

Statement — assumptions Rule: an assumption is something the statement TAKES FOR GRANTED — without it, the action makes no sense. Why launch an anti-pollution campaign unless pollution is a serious problem? — I is implicit. Why promote EVs 'to reduce pollution' unless EVs can actually reduce it? — II is implicit. III (people are willing to switch) is a hoped-for OUTCOME, not a necessary assumption — campaigns are launched precisely to create willingness. Hence only I and II — option (a).

Alphabetical arrangement — BALANCE Write both forms one below the other and compare position by position. Original: B A L A N C E; Alphabetical: A A B C E L N. Position 1: B/A ✗; position 2: A/A ✓; positions 3–7: L/B, A/C, N/E, C/L, E/N — all change. Only ONE letter (the A at position 2) keeps its place — option (d).

Address matching Method: compare word-by-word — spellings, abbreviations and the PIN code. Option (a) shortens everything (Bldg., Sec-17, Rd.) and writes Gurugram; option (c) changes the PIN to 122011; option (d) writes Gurugram. Option (b) keeps every word and the PIN 122001 exactly as given. Hence option (b).

Word formation — MARKET Step 1 — pick the letters: M (1st), A (2nd), R (3rd), T (6th). Step 2 — try arrangements that make real words: MART (a market) ✓ and TRAM (a rail car) ✓. Other arrangements like RAMT, TARM, MATR are not meaningful English words. So 2 words — option (b).

Row positions — interchange Key idea: after the swap, Amit stands exactly where Sunil was standing. So Sunil's seat is 15th from the left AND (as given) 12th from the right. Formula: Total = (position from left) + (position from right) − 1 = 15 + 12 − 1. = 26 persons — option (c).

Coding — letter splits into ±1 Pattern: each letter of the word EXPANDS into two letters — (letter − 1) followed by (letter + 1). Check with TIP: T → S,U; I → H,J; P → O,Q — giving SUHJOQ ✓. Apply to CUT: C → B,D; U → T,V; T → S,U — giving BD TV SU. Hence BDTVSU — option (b).

Coding — +13 cipher Find the shift using the first pair: W(23) → J(10). Going forward, 23 + 13 = 36 → 36 − 26 = 10 — a +13 cyclic shift. Verify: R+13 = E ✓, O+13 = B ✓, N+13 = A ✓, G+13 = T ✓ — the rule holds for every letter. Apply +13 to WRITE: W→J, R→E, I→V, T→G, E→R. Hence JEVGR — option (a).

Perfect squares These are perfect squares of the counting numbers. 1² = 1, 2² = 4, 3² = 9, 4² = ?, 5² = 25. 4² = 16. Hence 16 — option (c).

Pattern — letter, number, symbol Common pattern in (a), (b), (c): the letters move +1 each time (A,B,C / X,Y,Z / P,Q,R) and the numbers also move +1. In option (d): M → O jumps +2, and the numbers 10, 11, 13 rise unevenly (+1, then +2). Both chains break in (d). Hence option (d).

Diagonals of a polygon Formula: number of diagonals of an n-sided polygon = $\frac{n\left(n-3\right)}{2}$. Why n − 3? From each vertex you cannot draw a diagonal to itself or to its 2 neighbours. For an octagon, n = 8: $\frac{8\times 5}{2}$ = $\frac{40}{2}$ = 20. Hence 20 — option (a).

Number puzzle (a² + b) Test a simple rule: a @ b = a² + b. Check: 4² + 1 = 17 ✓ and 5² + 2 = 27 ✓ — the rule works for both given pairs. Apply: 6 @ 3 = 6² + 3 = 36 + 3. = 39 — option (d).

Coding — position-wise rules Decode the first pair position by position: S→H (opposite letter), P→T (+4), E→W (+18), C→F (+3), T→Z (+6), O→L (−3), R→I (opposite letter). Apply the SAME position-wise changes to PHANTOM: P→K (opposite), H→L (+4), A→S (+18), N→Q (+3), T→Z (+6), O→L (−3), M→N (opposite). Reading the results together: K-L-S-Q-Z-L-N. Hence KLSQZLN — option (c).

Coding (+1,+1,+1,−1,+1) Compare letter by letter: S→T (+1), T→U (+1), O→P (+1), N→M (−1), E→F (+1). So the pattern is +1, +1, +1, −1, +1 by position. Apply to PEARL: P→Q, E→F, A→B, R→Q, L→M. Hence QFBQM — option (b).

Ratio — add water Step 1 — find actual quantities: ratio 4 : 6 means 10 units = 50 L, so 1 unit = 5 L ⇒ alcohol = 20 L, water = 30 L. Step 2 — alcohol stays the same (only water is added), so keep alcohol = 4 units = 20 L. For the new ratio 4 : 9, water must become 9 units = 9 × 5 = 45 L. Water to add = 45 − 30 = 15 L — option (a).

Number puzzle (a² + b²) Test the rule a & b = a² + b². Check: 4² + 3² = 16 + 9 = 25 ✓ and 2² + 5² = 4 + 25 = 29 ✓. Apply: 3 & 6 = 3² + 6² = 9 + 36. = 45 — option (c).

Blood relation Break the sentence from inside out: 'my father's only son' — since Ram's father has only one son, that son is Ram himself. So the sentence becomes: 'She is the only daughter of RAM.' The woman is therefore Ram’s daughter. Hence option (d).

Blood relation — gender unknown B is the father of C, and C is the sister of D — so C and D are siblings, both children of B. D is therefore B’s child. But is D a son or a daughter? Nothing in the question reveals D’s gender. Hence Cannot be determined — option (d).

Interchange check Apply BOTH swaps to each option: + becomes ×, × becomes +, 3 becomes 5, 5 becomes 3. Option (c): (3+6)×5 becomes (5×6)+3. = 30 + 3 = 33 — exactly the value written on the right side ✓ (the others give 23, 11 and 6). Hence option (c).

Symbol substitution Replace each letter with its sign: 20 ÷ 4 × 3 + 5 − 5. BODMAS: do division and multiplication first, left to right: 20 ÷ 4 = 5, then 5 × 3 = 15. Now the additions: 15 + 5 − 5 = 15. Hence 15 — option (d).

Symbol substitution Replace the signs as instructed: 15 ÷ 5 + 6 × 2. BODMAS: division and multiplication first: 15 ÷ 5 = 3 and 6 × 2 = 12. Then add: 3 + 12 = 15. Hence 15 — option (b).

Simple arithmetic First the sum: 55 + 22 = 77. Then multiply: 77 × 5. Shortcut: 77 × 5 = 77 × 10 ÷ 2 = 770 ÷ 2 = 385. Hence 385 — option (b).