Coal India 2025 Previous Year Questions (PYQs) with Solutions

Mission Coking Coal was launched in 2021 to support India's domestic coking-coal roadmap.

After the Right to Property was removed from the list of Fundamental Rights, it continued as a constitutional/legal right under Article 300A.

A separate annual library grant under Samagra Shiksha was provisioned from 2018-19 to help develop reading habits among children.

Afonso de Albuquerque captured Goa in 1510. This marked a major beginning of Portuguese dominance in the Indian Ocean region.

The Central University of Andhra Pradesh was established through the Central Universities (Amendment) Act, 2019.

Madam Bhikaji Cama was associated with the revolutionary journal Bande Mataram published from Paris. Therefore, Bande Mataram is correct.

India's industrial-sector growth in FY23 was lower than in FY22. Hence the statement saying that the sector witnessed lower growth in FY23 compared to FY22 is correct.

The 2025 Padma Vibhushan award in Literature and Education (posthumous) was conferred on M.T. Vasudevan Nair.

Cachar is a region of Assam, and Assam's Cachar area is known for tea cultivation. Therefore, Tea is the correct answer.

Tropical humid climates remain hot and humid with a low annual temperature range. They receive high annual rainfall.

An e-way bill is an electronic permit used for the movement of goods under the GST framework.

Reports for November 2024 state that businesses generated 101.8 million e-way bills, which is 10.18 crore. Therefore, the figure signifies the number of electronic permits generated for transporting goods in that month, not GST registrations, GST revenue, or tax returns filed.

Fundamental Duties were inserted into the Indian Constitution by the Constitution (Forty-second Amendment) Act, 1976.

The amendment added Part IVA and Article 51A, which list the Fundamental Duties of citizens.

Therefore, the correct answer is 1976.

EEPROM stands for Electrically Erasable Programmable Read-Only Memory.

It is a non-volatile memory, so it retains data even when power is removed.

It can also be erased and reprogrammed electrically. Therefore, the correct answer is EEPROM.

Fundamental Duties in the Indian Constitution were inspired by the Soviet Constitution.

They were added later through the 42nd Constitutional Amendment Act, 1976, under Part IVA.

Therefore, the correct answer is Fundamental Duties.

Kudankulam Nuclear Power Station Unit-1 is a pressurised water reactor with an installed capacity of 1000 MWe.

So, the correct option is 1000 MW.

The Indian Ports Association was constituted in 1966 under the Societies Registration Act.

Therefore, the correct answer is 1966.

Among the listed countries, India shares its longest international land border with Bangladesh. The India-Bangladesh border is about 4096.7 km long.

Concept: In India an enterprise is classified as micro, small or medium by a composite criterion: its investment in plant, machinery or equipment and its annual turnover must both stay within the ceiling fixed for that category. Crossing either ceiling pushes the unit into the next higher category, so the ceilings are revised periodically to keep growing firms from losing MSME benefits merely by expanding.

Application: The Union Budget 2025-26, presented on 1 February 2025, revised these ceilings: investment limits were raised to 2.5 times and turnover limits to 2 times their existing values. So the micro-enterprise turnover ceiling is obtained by doubling the ceiling then in force.

1. Micro-enterprise limits before the revision: investment up to ₹1 crore and annual turnover up to ₹5 crore.

2. Turnover limits raised to 2 times: ₹5 crore × 2 = ₹10 crore.

3. Investment limits raised to 2.5 times: ₹1 crore × 2.5 = ₹2.5 crore.

4. Revised micro-enterprise definition: investment up to ₹2.5 crore and annual turnover up to ₹10 crore.

Cross-check: Applying the same two multipliers to the other categories reproduces the full revised classification, which confirms the arithmetic:

Result: Therefore the turnover threshold for micro enterprises has been increased to ₹10 crore. The revised limits were notified through Gazette Notification S.O. 1364(E) dated 21 March 2025 and apply from 1 April 2025.

Concept — where the venue of a treaty Meeting of States Parties comes from. A multilateral disarmament treaty usually does not name any city in its own text. The Treaty on the Prohibition of Nuclear Weapons (TPNW) follows this pattern: Article 8 makes the United Nations Secretary-General, as depositary, responsible for convening the Meetings of States Parties on a biennial rhythm unless the states parties agree otherwise, while the states parties themselves settle the place and dates of the next session in the decisions of the current one. So the venue of any particular meeting is never deduced from the treaty text — it is read off the decision of the previous meeting and the Secretary-General’s convening notification, and it may be a United Nations complex or an ordinary conference centre.

Applying it here. The TPNW opened for signature in 2017 and entered into force on 22 January 2021. Its second session had already decided that the next one would be convened at United Nations Headquarters, and the Secretary-General’s notification duly set the third Meeting of States Parties for United Nations Headquarters, which stands in New York, from 3 to 7 March 2025 — the exact window named in the question.

Cross-check against the series of meetings.

Contrasting the other cities offered.

• Vienna did host the first Meeting of States Parties in June 2022, at the Austria Center Vienna, but the 2025 session was not convened there.

• Geneva hosts the United Nations Office at Geneva and the Conference on Disarmament, which is a separate standing negotiating body rather than this treaty’s own Meeting of States Parties; no TPNW Meeting of States Parties has been convened in that city.

• Tokyo has hosted no TPNW Meeting of States Parties; Japan has not joined the treaty and takes part in nuclear-disarmament diplomacy through other channels.

Result: the third Meeting of States Parties to the Treaty on the Prohibition of Nuclear Weapons was held at United Nations Headquarters in New York from 3 to 7 March 2025.

Concept — A national foundation-model mission is a government-funded programme that builds base generative-AI models — for language (text), speech and vision — trained on indigenous data, so that many downstream applications, including public services, can be built on top of them. An initiative of this kind is identified by three markers: the sponsoring ministry or mission that funds it, the technical scope of the models it builds, and its launch event.

Application — BharatGen is the initiative that matches all three markers:

• Sponsor: funded by the Department of Science & Technology under the National Mission on Interdisciplinary Cyber-Physical Systems (NM-ICPS), and executed through the TIH Foundation for IoT & IoE at IIT Bombay.

• Scope: foundation models for language, speech and computer vision across Indian languages, built to improve public service delivery and citizen engagement.

• Launch: inaugurated in New Delhi on 30 September 2024 by Dr. Jitendra Singh, Union Minister of State for Science & Technology, and described as the first government-funded multimodal large language model initiative.

Cross-check — The other names denote programmes of a different kind, so none of them fits a multimodal foundation-model mission:

Result — BharatGen.

Note on the date — the primary Department of Science & Technology record places the inauguration on 30 September 2024; several current-affairs digests carried the news in early October 2024, which is why the month is sometimes quoted as October.

Concept — A river-valley project is defined by the river it impounds: the dam sits at a chosen gorge or narrow point on one river, and that river's drainage basin fixes the project's geography. So to name the river behind a project, locate the dam site, identify the valley it stands in, and match that valley to the river that drains it.

Application — The Indira Sagar Hydroelectric Power Project stands at Narmada Nagar (Punasa), in Khandwa district of Madhya Pradesh. That dam site lies inside the rift valley that runs east–west between the Vindhya range to the north and the Satpura range to the south — the valley drained by the Narmada. Impounding the Narmada there creates India's largest reservoir by storage volume (about 12.22 billion cubic metres) and drives an installed capacity of 1,000 MW (8 units of 125 MW). On the same river it lies upstream of the Omkareshwar and Sardar Sarovar projects, while Bargi (Rani Avanti Bai Sagar) lies further upstream of it.

Cross-check — The other rivers named here drain different valleys, so a dam in the Vindhya–Satpura rift cannot belong to any of them:

Hence the Indira Sagar Hydroelectric Power Project is located on the Narmada.

Concept

A register is a very small, very fast storage cell built directly into the processor chip. It sits at the top of the storage hierarchy — registers, then cache, then main memory, then secondary storage — where capacity grows and access speed falls at every step outward.

Because a register can be read or written within a single clock cycle, it acts as the processor's working scratchpad: it holds the operand being used at this instant, the partial result just produced, and the instruction currently being executed. Its contents are volatile and change continuously as execution proceeds.

Application

1. Fetch: the program counter holds the address of the next instruction, and the instruction brought in from memory is placed in the instruction register.

2. Decode: the control unit reads the instruction out of that register and works out which operation is required and which operands it needs.

3. Execute: the operands are loaded into general-purpose registers, the arithmetic and logic circuitry works on those values, and the result is placed back into a register such as the accumulator.

4. Write-back: only when a value is no longer needed for the immediate computation is it copied out of the register into memory.

Every one of these steps is a short-lived hold of data or of an instruction while processing is under way, which is precisely why the primary role of registers is to hold data and instructions temporarily during processing.

Cross-check

• Storing large amounts of data permanently needs non-volatile media of gigabyte-to-terabyte capacity — hard disks, SSDs, optical discs — not a handful of volatile cells inside the processor.

• Managing communication with peripheral devices is done by I/O interfaces and device controllers using status flags, buffers, interrupt lines and DMA channels.

• Performing complex mathematical calculations is the work of the arithmetic logic unit and the floating-point unit; registers only supply the operands to that circuitry and receive the results from it.

Registers therefore support computation by holding its inputs and outputs briefly; they neither carry out the computation themselves nor archive data for the long term.

Concept — A wind always moves from an area of higher air pressure towards an area of lower air pressure, and the moisture it carries is decided by the surface it has just travelled across. Air that has spent its journey over open sea reaches its destination laden with water vapour; air that has spent its journey over a continent reaches it with very little.

Application — In the cold weather season the landmass of northern India and Central Asia loses heat rapidly and builds up a high pressure cell over the interior, while the neighbouring ocean cools far more slowly and remains comparatively low in pressure. The pressure gradient therefore points outward, from the interior towards the ocean, and the northeast trade winds of this season move off the continent and out over the water. Because the stretch of their track that crosses India runs over land and not over water, they carry almost no water vapour while they are over the country, and the greater part of it stays rainless.

Cross-check — Test the rule where it appears to break. The same northeast winds that travel over the Bay of Bengal do acquire a sea leg, arrive moist, and give the Coromandel coast of Tamil Nadu its October to December rainfall. The exception shows up precisely where a stretch of open water is added to the track, which confirms that it is the outward land to sea direction that keeps the rest of the country dry.

Result — The northeast trade winds of the winter season blow from land to sea.

Concept: Under Article 164 of the Constitution, the Governor of a State appoints as Chief Minister the person who commands the confidence of a majority in the Legislative Assembly. After a general election to the Assembly, that person is whoever the winning party or coalition elects as its legislature-party leader, so the office follows the post-poll numbers and the alliance's own choice rather than seniority or incumbency.

Application: Polling for all 288 seats of the Maharashtra Legislative Assembly was held on 20 November 2024 and the results were declared on 23 November 2024. The Mahayuti alliance won about 230 seats, of which the BJP alone took 132, the single largest tally in the House. On 4 December 2024 the BJP legislature party elected Devendra Fadnavis as its leader and the alliance backed him for the top post. He took oath as Chief Minister of Maharashtra on 5 December 2024 at Azad Maidan in Mumbai, with Eknath Shinde and Ajit Pawar sworn in as Deputy Chief Ministers. This was Fadnavis's third term as Chief Minister, after 2014-2019 and a brief term in November 2019.

Cross-check: where each of the four leaders stood around the November-December 2024 change of government.

Hence the leader who became Chief Minister of Maharashtra in December 2024, after the November 2024 Assembly elections, was Devendra Fadnavis.

Concept — how a letter-group analogy is fixed. Two independent rules decide such an analogy. The first is the internal shape of a group: the gaps between the alphabet positions of its letters, written in the form (X, X + k, …). The second is the step that carries the first group onto the second, applied letter by letter at matching places; that step counts only when it is the same at every place. A candidate reproduces the analogy only when both rules hold exactly.

Applying it to the given pairs.

1. Take HLH and write the alphabet positions: H = 8, L = 12, H = 8. The second letter stands 4 places after the first and the third repeats the first, so the shape is (X, X + 4, X) with X = 8.

2. Take JNJ: J = 10, N = 14, J = 10. The same shape (X, X + 4, X) appears, now with X = 10.

3. Compare the two groups place by place: H → J, L → N, H → J. Each letter moves forward by 2, so the step is a uniform +2.

4. Repeat on the second given pair. AEA: A = 1, E = 5, A = 1, again (X, X + 4, X). CGC: C = 3, G = 7, C = 3, again (X, X + 4, X). Place by place A → C, E → G, A → C, again a uniform +2.

5. So the signature to reproduce is: both groups carry the shape (X, X + 4, X), and the second group is the first group moved forward by exactly 2 at every place.

Cross-check against every offered pair.

Result. Only MQM-OSO satisfies both rules at once: each of its groups carries the shape (X, X + 4, X), and the second group is the first moved forward by 2 at every place. Reading it back, M + 4 = Q with the third letter repeating M, then M + 2 = O, Q + 2 = S and M + 2 = O — precisely the relation seen in HLH-JNJ and in AEA-CGC. The pair that follows the same pattern is therefore MQM-OSO.

Concept

In a constant-difference (arithmetic) series, every consecutive pair of terms differs by the same fixed step d, so the list of successive differences is itself constant. A single mistyped term therefore disturbs exactly two neighbouring differences: the one entering that term falls short of d and the one leaving it overshoots d by the same amount.

Application

Take the successive differences of the printed series 109, 135, 161, 177, 213, 239, 265:

1. 135 − 109 = 26

2. 161 − 135 = 26

3. 177 − 161 = 16

4. 213 − 177 = 36

5. 239 − 213 = 26

6. 265 − 239 = 26

Four of the six differences equal 26, so the intended step is d = 26. The two irregular differences, 16 and 36, sit next to each other, and the only term they share is the fourth printed term, 177. Their sum is 16 + 36 = 52 = 2 x 26, which is the signature of exactly one mistyped term: replacing that single term restores both differences at once. The value that makes both differences 26 is 161 + 26 = 187.

Cross-check

With 187 in place the series reads 109, 135, 161, 187, 213, 239, 265, and every successive difference is 26. Equivalently the nth term is 109 + 26(n - 1): that formula reproduces 213 = 109 + 26 x 4, 239 = 109 + 26 x 5 and 265 = 109 + 26 x 6 exactly as printed, so no term other than the fourth needs changing.

Hence the number that does not belong in the printed series is 177; it should be 187.

Concept

In a floor-arrangement puzzle, place the absolute clues before the relative ones. In a building of n floors, a clue of the form "exactly k people live above X" pins X to floor n - k, and a clue naming exactly who lives above someone pins both of them at once.

Only after every absolute position is fixed do you fit the relative blocks: a clue of the form "P lives immediately above Q" needs a pair of consecutive floors that are both still free.

Applying it to this building

1. The six floors are numbered 1 at the bottom up to 6 at the top, and R, E, M, I, N and S occupy one floor each.

2. Four people live above I, so four floors lie above I. That pins I to floor 6 - 4 = 2.

3. Only S lives above M, so exactly one floor lies above M and S occupies it. That pins M to floor 5 and S to floor 6.

4. Floors 1, 3 and 4 are still free, and they must be taken by R, N and E.

5. N lives immediately above R, which needs two consecutive free floors. Among the free floors 1, 3 and 4 the consecutive pair is 3 and 4, so R is on floor 3 and N is on floor 4. The pair 1 and 2 is unusable because floor 2 already holds I.

6. E takes the single floor left over, floor 1.

7. The sum of the floor numbers of E and N is therefore 1 + 4 = 5.

Cross-check

• Above I on floor 2 live R, N, M and S, which is exactly four people.

• Above M on floor 5 lives S and nobody else.

• N on floor 4 sits immediately above R on floor 3.

• Every clue is satisfied and no other seating survives all three clues, so the sum 1 + 4 = 5 is forced.

A letter-cluster classification is decided on alphabet positions, not on how the letters look. Replace every letter by its position in the English alphabet (A = 1, B = 2, ... Z = 26); the group is then defined by two positional properties that every member must satisfy: a fixed step between the two letters inside each half of the pair, and a fixed shift that carries the first half onto the second, with the count continuing cyclically from Z back to A. The member that fails those properties is the one outside the group.

Written in alphabet positions, the four pairs are:

Comparing the rows:

• VX-BD, GI-MO and UW-AC each keep a +2 step inside both halves and a uniform +6 shift from the first half onto the second.

• JL-OT keeps the +2 step only in its first half; O to T is a +5 step, and its two shifts, J to O (+5) and L to T (+8), are unequal, so it matches neither property.

• The cyclic count is what lets the wrapping pairs follow the same rule: U(21) + 6 = 27 -> 1 = A and W(23) + 6 = 29 -> 3 = C, exactly as G(7) + 6 = 13 = M.

• This also agrees with the note in the question: the three grouped pairs do not share a common vowel or consonant count (VX-BD has no vowel, GI-MO has two, UW-AC has two), so the rule has to be positional.

Therefore the pair that does not belong to the group is JL-OT.

Concept: a letter-pair pattern is fixed by two independent things — the internal shape of each three-letter group, meaning the alphabetical gaps between its own letters, and the constant shift that carries the first group onto the second. A candidate pair belongs to the same family only when it reproduces both of them.

1. Write the alphabet positions of CAC: C = 3, A = 1, C = 3. The outer letters are identical and the middle letter sits 2 places earlier, so the shape is X, X−2, X.

2. FDF gives F = 6, D = 4, F = 6 — the same shape X, X−2, X.

3. Now measure the shift from CAC to FDF: C (3) → F (6), A (1) → D (4), C (3) → F (6). Every position advances by exactly +3.

4. Test the rule on the second example: OMO = 15, 13, 15 and RPR = 18, 16, 18 are both X, X−2, X, and O (15) → R (18), M (13) → P (16), O (15) → R (18) is again a uniform +3.

5. So the governing rule has two parts: each group must read X, X−2, X, and the second group must be the first group shifted forward by 3.

Only 20, 18, 20 → 23, 21, 23 keeps the X, X−2, X shape in both groups and moves every position by the same +3. TRT-WUW is therefore the pair that follows the pattern of CAC-FDF and OMO-RPR.

Concept: In a parallel-row (double-row) arrangement the two rows face each other, so a direction word does not carry the same meaning in both rows: whichever way "left" runs for a person in one row, it runs the opposite way for a person in the row facing it. The dependable method is to fix one drawing frame, restate every left/right clue inside that single frame, and start from the clue that admits the fewest placements. Because the rows face each other, the person in seat n of one row always faces the person in seat n of the other row.

Application:

1. Set the frame. Draw four seats numbered 1 to 4 from left to right for both rows, so that seat n of Row 1 faces seat n of Row 2. Row 1 (R, S, T, U) faces south, so for a Row 1 person "left" runs toward seat 4 and "right" runs toward seat 1. Row 2 (L, M, N, O) faces north, so for a Row 2 person "left" runs toward seat 1 and "right" toward seat 4.

2. Apply "T sits third to the left of U" in Row 1. Left for this row means moving toward seat 4, so T's seat is three greater than U's seat. In a four-seat row the only fit is U at seat 1 and T at seat 4.

3. Apply "R sits to the immediate right of S" in Row 1. Right for this row means one seat toward seat 1, so R sits one seat before S. Only seats 2 and 3 remain, which gives R at seat 2 and S at seat 3.

4. Apply "L sits second to the left of O" in Row 2. Left for this row means moving toward seat 1, so L's seat is two less than O's seat. That leaves two candidates: O at seat 3 with L at seat 1, or O at seat 4 with L at seat 2.

5. Apply "the person facing T is an immediate neighbour of O". T is at seat 4, so the Row 2 person in seat 4 must sit next to O, and seat 4 has only one neighbour, seat 3. Hence O is at seat 3, which forces L to seat 1, and M and N are left for seats 2 and 4.

6. Apply "the person facing M is an immediate neighbour of U". U is at seat 1. If M were at seat 4, the person facing M would be T at seat 4, whose only neighbour is S at seat 3, and that is not U. So M is at seat 2, faced by R at seat 2, and R does sit next to U. N therefore takes seat 4.

7. Read off the two end seats of each row. Row 1 has U at seat 1 and T at seat 4; Row 2 has L at seat 1 and N at seat 4. The people at the extreme ends are therefore U, T, L and N, so the pair whose two names both belong to that set is L and T.

Cross-check:

• For a south-facing person, T at seat 4 is three seats to the left of U at seat 1, and R at seat 2 is immediately to the right of S at seat 3.

• For a north-facing person, L at seat 1 is two seats to the left of O at seat 3; N at seat 4 faces T and sits next to O; R at seat 2 faces M and sits next to U. All five clues hold.

• Mirroring the whole diagram, that is numbering the seats the other way round, changes every seat number but leaves the same four people on the end seats, so the result does not depend on which way the rows were drawn.

Concept

In a two-row facing arrangement the two rows have opposite left–right senses: a person facing north has west on the left and east on the right, while a person facing south has east on the left and west on the right. Persons occupying the same seat number in the two rows form a facing pair, and in a four-seat row the statement that exactly two people sit between two persons forces those two persons onto the two end seats.

Applying it here

Row 1 (A, B, C, D) faces south and row 2 (L, M, N, O) faces north, so row 1 lies to the north of row 2. Number the seats of each row 1 to 4 from west to east; seat i of row 1 faces seat i of row 2.

1. For a south-facing row-1 person the immediate left is one seat to the east, that is seat number + 1; for a north-facing row-2 person the immediate left is one seat to the west, that is seat number − 1.

2. Only two people sit between N and M, so in a four-seat row N and M take seats 1 and 4 in some order.

3. The person facing N is the row-1 occupant of N’s seat number, and D sits immediately to that person’s left, so seat(D) = seat(N) + 1.

4. The person facing O is the row-1 occupant of O’s seat number, and that person sits immediately to D’s left, so seat(O) = seat(D) + 1 = seat(N) + 2.

5. If seat(N) = 4 then seat(D) = 5, a seat that does not exist. Hence seat(N) = 1, seat(M) = 4, seat(D) = 2 and seat(O) = 3.

6. Seat 2 of row 2 is the one row-2 seat still free, so it belongs to L.

7. A sits to the immediate right of C, and for a south-facing person the right is one seat to the west, so seat(A) = seat(C) − 1. Seat 2 is already taken by D, so the fit among the remaining seats 1, 3 and 4 is seat(C) = 4 and seat(A) = 3, leaving seat(B) = 1.

Cross-check

• L and O are the two people sitting between N at seat 1 and M at seat 4.

• B faces N, and the seat immediately to B’s left — one step east — is seat 2, held by D.

• A faces O, and A at seat 3 lies immediately to the left of D at seat 2.

• C at seat 4 has A at seat 3 immediately to its right.

D occupies seat 2 of row 1, so the person sitting opposite D is the occupant of seat 2 of row 2, namely L.

Concept

On the cyclic alphabet every letter carries a position — A = 1, B = 2, … Z = 26 — and a count that passes 26 wraps back to A. A pair of letter groups is then fixed by two independent quantities: the inner gaps between consecutive letters within a group, and the positional shift that carries each letter of the first group to the letter standing in the same place in the second group.

A candidate reproduces the pattern only when BOTH quantities repeat the values the given pairs establish — matching one of them is not enough.

Applying the concept to the two given pairs

1. First group of the first pair: K = 11, N = 14, R = 18, so the inner gaps are 14 − 11 = +3 and 18 − 14 = +4.

2. Second group of the first pair: U = 21, X = 24, B = 2; since 24 + 4 = 28 wraps to 2, its inner gaps are again +3 and +4.

3. Compare the two groups letter by letter: 11 + 10 = 21 (K → U), 14 + 10 = 24 (N → X), 18 + 10 = 28 → 2 (R → B). The shift is a constant +10.

4. Repeat on the second pair: N = 14, Q = 17, U = 21 give gaps +3 and +4; X = 24, A = 1, E = 5 give 24 + 3 = 27 → 1 and 27 + 4 = 31 → 5, so gaps +3 and +4 again.

5. Its shift: 14 + 10 = 24 (N → X), 17 + 10 = 27 → 1 (Q → A), 21 + 10 = 31 → 5 (U → E) — a constant +10 once more.

6. So the required pair must show inner gaps of +3 and +4 in both groups and a constant +10 between them.

7. Build it from V = 22: 22 + 3 = 25 = Y and 25 + 4 = 29 → 3 = C, giving the first group VYC.

8. Now add 10 to each of its letters: 22 + 10 = 32 → 6 = F, 25 + 10 = 35 → 9 = I, 3 + 10 = 13 = M, giving the second group FIM.

Cross-check against the other candidate values

• VXA – EHL: inside VXA the gaps are +2 and +3, and the group-to-group shifts read +9, +10, +11, so neither quantity is constant.

• VYC – EIL: inside EIL the gaps run +4 then +3 rather than +3 then +4, and the shifts read +9, +10, +9.

• VXA – FIM: inside VXA the gaps are +2 and +3, and the shifts read +10, +11, +12.

Both requirements — inner gaps of +3 and +4 in each group and a constant +10 between the groups — hold together only for VYC – FIM.

Concept: In a statement and course of action item, a suggested step follows only when it is practical, ethical and aimed directly at the problem the statement describes. A step whose purpose is to protect the decision maker's own interest — image, revenue or share price — while the stated harm continues does not follow. A step that removes the stated harm and repairs the damage it has caused does follow.

Application: the statement fixes three facts — the dysfunction is conclusively established, it is severe, and it can make a smartphone built with that component suddenly swell or catch fire. So every unit carrying the faulty component is a live safety risk to whoever ends up using it, and each suggested step must be judged against that risk.

1. Course of Action I asks the company to keep the finding confidential because disclosure would move the share price. This protects the company's market valuation, not the user: the defective handsets stay in the market, so the hazard the statement describes is left entirely untouched, and concealing a known safety defect exposes the company to far larger legal and reputational damage later. It is therefore not a step that follows from the statement.

2. Course of Action II asks the company to recall the defective batch immediately and compensate its customers. The recall withdraws the hazardous units from circulation before the risk the statement establishes can materialise, and the compensation answers for the defective, unsafe handsets the company has supplied. It is practical, immediate and entirely within the company's power, so it is a step that follows.

Cross-check: apply the standard test — does the step end the problem stated? Concealment leaves the hazard live and adds liability, while recall with compensation ends it. Hence only Course of Action II follows, and the answer is “Only II follows”.

Concept — odd days The days of the week repeat in a cycle of 7, so when a stretch of days is added to a known weekday, only the remainder of that stretch on division by 7 matters. That remainder is the number of odd days: a common year of 365 days carries 1 odd day, a leap year of 366 days carries 2, and adding n odd days moves the weekday forward by n places.

Applying it to these two dates Split the stretch from 21 January 1998 to 31 December 2000 into two whole year-to-year legs and one tail inside 2000.

1. 21 January 1998 to 21 January 1999 spans a common year, 365 days. Since 365 = 52 × 7 + 1, this leg carries 1 odd day.

2. 21 January 1999 to 21 January 2000 spans February 1999, again a common year, 365 days. So this leg also carries 1 odd day.

3. 2000 is divisible by 400, so it is a leap year of 366 days. 21 January is its 21st day, so the tail from 21 January 2000 to 31 December 2000 is 366 − 21 = 345 days. Since 345 = 49 × 7 + 2, this tail carries 2 odd days.

4. Total odd days = 1 + 1 + 2 = 4.

5. Move the anchor weekday forward by 4 places from Wednesday: Thursday, Friday, Saturday, Sunday.

Cross-check Re-anchor the count at 1 January 1998 instead. 21 January is the 21st day of 1998, so 1 January 1998 sits 20 days earlier, and 20 = 2 × 7 + 6, which places it 6 steps behind the given Wednesday, on a Thursday. From 1 January 1998 to 31 December 2000 is 365 + 365 + 365 = 1095 days, because 31 December is the 366th day of 2000 and so 365 days have elapsed within that year. Since 1095 = 156 × 7 + 3, the target date lands 3 places after Thursday, giving the same weekday as the first route.

Where the count usually slips Two whole years from 21 January 1998 reach 21 January 2000 after 730 days, and 730 = 104 × 7 + 2, which is only 2 odd days. The remaining 345 days of 2000 add 2 more and carry the total to 4, so stopping at the two-year mark leaves the count short.

So 31 December 2000 falls on a Sunday.

Concept. In a "find the wrong term" number series, the rule is a relation the terms are built on rather than a property of any single term — most commonly the successive differences between consecutive terms, though ratios, powers, or digit operations can carry it instead. Whenever those first differences themselves form an arithmetic progression, the whole series is fixed by its opening term together with that difference rule, so every later term is forced and can be rebuilt independently of what is printed.

Application. Here the differences are meant to be consecutive even numbers rising by 2 each time: +4, +6, +8, +10, +12, +14, +16. Rebuilding the series from the opening term 56 gives:

1. 56 + 4 = 60

2. 60 + 6 = 66

3. 66 + 8 = 74, but the printed series carries 70 at this place

4. 74 + 10 = 84

5. 84 + 12 = 96

6. 96 + 14 = 110

7. 110 + 16 = 126

Cross-check. Read the differences off the printed series directly: 4, 6, 4, 14, 12, 14, 16. Two of them break the +2 rise, and both breaks straddle the same single entry. Replacing that entry with 74 turns the difference list into the clean 4, 6, 8, 10, 12, 14, 16, so one substitution repairs both breaks at once — exactly what the stem's promise that only one number is wrong demands. Changing any other single term repairs at most one break and leaves the other standing.

Result. The number that does not belong is 70; the term at that place should be 74.

Concept

In a substitution code language every whole word is replaced by a fixed code group, and the groups are not written in the same order as the words they stand for. Such a code is broken by intersection: when two coded statements have one word in common, the one code group those two statements also have in common must stand for that word. Once the shared words are pinned down, the leftover group inside each statement is settled by elimination, and a word that never appears in any given statement can be represented only by a code group that has not been used anywhere.

Application

1. 'time wait none' = 'li mo zo' and 'tide wait some' = 'mo gy pu' have the word wait in common and the code group mo in common, so wait = mo.

2. 'time wait none' = 'li mo zo' and 'time and tide' = 'gy zo bh' have the word time in common and the code group zo in common, so time = zo.

3. 'tide wait some' = 'mo gy pu' and 'time and tide' = 'gy zo bh' have the word tide in common and the code group gy in common, so tide = gy.

4. In 'time wait none' = 'li mo zo', time = zo and wait = mo are already placed, so the group left over gives none = li.

5. The same elimination on the other two statements gives some = pu from 'tide wait some', and and = bh from 'time and tide'.

6. The phrase to be coded is 'eternal none'. Step 4 has already fixed none = li.

7. The word eternal never appears in the three given statements, so no code group has been assigned to it. Every group that has appeared — li, mo, zo, gy, pu and bh — is already taken by another word, so eternal must be carried by a fresh two-letter group.

8. A fresh two-letter group written with li is therefore the probable code, that is 'kz li'.

Cross-check

The complete mapping recovered from the three statements is:

Substituting this mapping back reproduces all three coded statements exactly, which confirms the deduction. Note also that a group's position does not track the word's position — in 'time wait none' = 'li mo zo' the group for none is written first — so the two groups of the coded phrase may be written in either order.

Concept: In a letter-cluster series, every cluster is generated from one anchor letter (its first letter) by a fixed pair of alphabetical offsets, while the anchors themselves advance through the alphabet by a fixed step, counted cyclically with A = 1 and Z = 26. The cluster that disobeys the internal offset rule is the defective one, so the method is to read the rule off the regular clusters and test each cluster against it.

Application: Number the letters and test the given series against that idea.

1. Write the anchors with their positions: D (4), Z (26), V (22), R (18), N (14). Each anchor is 4 letters before the previous one, where 4 - 4 = 0 is counted cyclically as 26, that is Z, so the anchor chain is regular across all five clusters.

2. Read the internal rule off the first cluster DAC: the second letter is anchor - 3, because 4 - 3 = 1 = A, and the third letter is anchor - 1, because 4 - 1 = 3 = C.

3. Apply that rule to every anchor and compare the letters it predicts with the letters actually written.

Cross-check: Four clusters match the rule exactly. VTW does not: its second letter T sits 2 letters before the anchor V and its third letter W sits 1 letter after it, whereas the rule requires 3 letters before and 1 letter before. Rewriting that cluster as VSU, with positions 22, 19 and 21, makes all five clusters obey the single rule while leaving the -4 anchor step untouched, which confirms that exactly one cluster is defective.

Hence the incorrect letter-cluster is VTW.

Concept: A conclusion follows from a set of categorical statements only when it is true in every arrangement of the classes that satisfies all the statements; one consistent arrangement in which it fails is enough to reject it. Two rules settle such items. First, a universal chain: if the whole of class A lies inside class B and the whole of B lies inside C, then the whole of A lies inside C, and because a named class is taken to be non-empty, this also guarantees that some C are A. Second, a particular statement of the form “Some X are Y” guarantees only that an overlap exists somewhere, so on its own it can never force a universal claim about a third class.

Application to the given statements:

1. “All Turquoise are Beige” places the whole Turquoise class inside Beige.

2. “All Beige are Pink” places the whole Beige class inside Pink, so chaining the two universals puts the whole Turquoise class inside Pink.

3. Turquoise is a non-empty class, so at least one Pink item is Turquoise. “Some Pink are Turquoise” therefore holds in every arrangement that satisfies the statements, so it follows.

4. “Some Pink are Maroon” fixes only that Pink and Maroon overlap somewhere; it says nothing about where that overlap sits, so it cannot force the universal claim “All Beige are Maroon”.

Cross-check with a counter-arrangement:

• Let Turquoise = {t}, Beige = {t, b}, Pink = {t, b, m} and Maroon = {m}. Every Turquoise item is Beige, every Beige item is Pink, and m is both Pink and Maroon, so all three statements hold.

• In this arrangement b is Beige but not Maroon, so “All Beige are Maroon” is false; t is both Pink and Turquoise, so “Some Pink are Turquoise” is true. One such arrangement is enough to reject the first conclusion, and no arrangement can defeat the second because the chain already proves it.

Hence only conclusion (II), “Some Pink are Turquoise”, logically follows from the statements.

Concept — In sentence-based coding, each distinct word of the original sentence is replaced by one fixed code that keeps the same meaning in every statement, and the codes of a statement are written in a scrambled order. A code is therefore identified by intersection: the code that two coded statements have in common must stand for the word that those two statements have in common, and once a statement's shared codes are settled, the code left over must stand for the word left over.

Application — The three coded statements are:

• 'tenant pays rent' → 'mh si pa'

• 'owner receive rent' → 'zy pa cu'

• 'owner pays tax' → 'si zy ke'

Intersecting them two at a time:

1. 'tenant pays rent' and 'owner receive rent' have the word 'rent' in common, and their codes 'mh si pa' and 'zy pa cu' have 'pa' in common, so rent = pa.

2. 'owner receive rent' and 'owner pays tax' have the word 'owner' in common, and their codes 'zy pa cu' and 'si zy ke' have 'zy' in common, so owner = zy.

3. 'tenant pays rent' and 'owner pays tax' have the word 'pays' in common, and their codes 'mh si pa' and 'si zy ke' have 'si' in common, so pays = si.

4. In 'tenant pays rent' coded as 'mh si pa', taking away pays = si and rent = pa leaves the word 'tenant' against the code 'mh', so tenant = mh.

5. Therefore 'tenant owner' = 'mh' + 'zy', and because the codes of a group are written in a scrambled order this pair is presented as 'zy mh'.

Cross-check — The two codes not used above fall out of the remaining statements: 'cu' is what is left in 'owner receive rent' coded as 'zy pa cu' after owner = zy and rent = pa, so receive = cu; 'ke' is what is left in 'owner pays tax' coded as 'si zy ke' after owner = zy and pays = si, so tax = ke. The full dictionary is:

Re-encoding with this dictionary reproduces every given line: 'tenant pays rent' gives mh si pa, 'owner receive rent' gives zy pa cu, and 'owner pays tax' gives zy si ke, which is the given 'si zy ke' in scrambled order. Each word carries one code and each code one word, so the mapping is consistent.

Result — the probable code for 'tenant owner' is 'zy mh'.

Concept

A letter-cluster series is governed by two rules that must both hold. The internal rule links the letters inside a single cluster, and the step rule fixes how one cluster moves to the next through the alphabet. To test such a series, convert every letter to its alphabet position (A = 1, B = 2, ..., Z = 26) and then check both rules on every cluster.

Application

Written with alphabet positions, the printed series reads:

1. Convert every printed cluster to alphabet positions, as in the table above.

2. Read the internal rule from PLP, MIM, JFJ and GCG: in each of them the last letter repeats the first letter, and the middle letter stands 4 places before the first letter.

3. Read the step rule from the first letters of those clusters: 16, 13, 10, 7 - a constant fall of 3. The fifth cluster must therefore open at position 7 - 3 = 4, the letter D.

4. Test the printed fifth cluster, CYF: it opens at position 3 (C) instead of 4 (D), and its last letter F (6) does not repeat its first letter C (3). Both rules break, so CYF is the incorrect letter-cluster.

5. Continuing both rules gives the cluster that belongs in that place: D as the outer letters, and a middle letter 4 places before D, which runs past A and comes back to Z - that is, DZD.

Cross-check

Apply both rules to each remaining printed cluster: PLP (16, 12, 16), MIM (13, 9, 13), JFJ (10, 6, 10) and GCG (7, 3, 7) all repeat their first letter at the end, all place their middle letter 4 places before their first letter, and their first letters fall by exactly 3 each time. All four are consistent with both rules, so exactly one cluster of the series is incorrect, as the question states.

Concept: A direction-sense path is solved by fixing one origin and two perpendicular axes, then writing every leg as a change in exactly one coordinate — take east as the positive x-direction and north as the positive y-direction.

A left turn rotates the walker 90 degrees anticlockwise, so the facing cycles from north to west to south to east and back to north. Once both endpoints are written as coordinates, the shortest distance between them is the straight line joining them.

Application: track each leg of the two walks as a change in one coordinate.

1. Place Point C at the origin (0, 0), with east as +x and north as +y.

2. Tara walks 23 m south from C, reaching (0, -23), and is now facing south.

3. Facing south, her left turn points her east; walking 10 m east takes her to (10, -23).

4. Facing east, her next left turn points her north; walking 17 m north takes her to (10, -23 + 17) = (10, -6), which is Point D.

5. Point P lies 15 m east of D, so P is at (10 + 15, -6) = (25, -6).

6. Bruce walks 6 m north from P, reaching (25, -6 + 6) = (25, 0), and is now facing north.

7. Facing north, his left turn points him west; walking 16 m west takes him to (25 - 16, 0) = (9, 0), which is Point Q.

8. C is at (0, 0) and Q is at (9, 0), so the east-west change is 9 and the north-south change is 0. The straight-line distance is the square root of the sum of the squares of these two changes.

   √(92 + 02) = √81 = 9

Cross-check: Tara ends 10 m east and 6 m south of C, while Bruce ends 16 m west and 6 m north of P. Bruce’s 6 m northward leg exactly cancels Tara’s 6 m southward net displacement, so Q sits on the same east-west line as C and no north-south gap survives.

The whole east-west account is therefore 10 m east from Tara, plus 15 m east from D to P, minus 16 m west from Bruce, giving 9 m east. With the north-south gap zero, the straight-line distance equals that east-west gap.

The shortest distance between Point Q and Point C is 9 m.

Concept. A floor-arrangement puzzle is solved by working outward from the most restrictive clue, not by guessing an arrangement. Three clue shapes do almost all the work: "only n people live above P" pins P exactly, because with F floors in all P must stand on floor F - n; "only X lives between P and Q" forces P, X and Q onto three consecutive floors with X in the middle, so they behave as one rigid block; and "R lives immediately above T" forces a rigid pair of adjacent floors. Place the pinned person first, fit the rigid block into whatever consecutive floors remain, then drop the rigid pair into what is left, and use any remaining above/below clue purely as a consistency check.

Application to this question.

1. The building has six floors, numbered 1 at the bottom up to 6 at the top, and six residents: S, O, R, B, E and T.

2. Only two people live above B, so exactly two floors lie above B. Therefore B stands on floor 6 - 2 = 4.

3. Only E lives between S and O, so exactly one resident separates them and that resident is E. S, E and O must therefore occupy three consecutive floors with E in the middle.

4. Floor 4 is already taken by B, so the runs 2-3-4, 3-4-5 and 4-5-6 are all blocked. The only free run of three consecutive floors is 1-2-3. Hence E stands on floor 2, and S and O take floors 1 and 3 in some order.

5. Floors 5 and 6 are the only floors still empty, so R and T must fill them. R lives immediately above T, which forces T onto floor 5 and R onto floor 6.

6. The last clue, S lives on one of the floors below R, is satisfied automatically: S is on floor 1 or 3, and R is on floor 6.

7. Floor number 5 is therefore occupied by T.

Cross-check.

• Above B on floor 4 there are floors 5 and 6, that is exactly two residents, as required.

• S and O stand on floors 1 and 3, and the single floor between them is floor 2, held by E, as required.

• R on floor 6 is immediately above T on floor 5, as required.

• S on floor 1 or 3 is below R on floor 6, as required.

• S and O may swap floors 1 and 3, so the full arrangement is not unique, but floor 5 holds T in both arrangements. The answer to this question is therefore uniquely determined: T.

Concept. In a floor puzzle the word "only" makes a clue an exact count, not a mere comparison. "Only X lives above Y" says the number of people above Y is exactly one, which pins Y absolutely: in a building of n floors numbered 1 at the bottom to n at the top, a person with exactly k people above them lives on floor n − k, and a person with exactly k people below them lives on floor k + 1.

Concept (cont.). Once two floors are pinned, the number of people living strictly between floor a and floor b (with a < b) is b − a − 1. That count depends only on the two pinned floors, never on how the still-unplaced people are distributed among the remaining floors.

Application to this arrangement.

1. Six people occupy six floors, one person per floor, numbered 1 (lowest) up to 6 (topmost).

2. "Only R lives above L" → exactly one person lives above L, so L is on floor 6 − 1 = 5, and the single person above, R, is on floor 6.

3. "Only E lives below P" → exactly one person lives below P, so P is on floor 1 + 1 = 2, and the single person below, E, is on floor 1.

4. Floors 1, 2, 5 and 6 are now occupied by E, P, L and R, so the two people left, F and Q, take the two floors left, 3 and 4, in some order.

5. Apply the between-count: with P on floor 2 and L on floor 5, the number of people strictly between them is 5 − 2 − 1 = 2 — the residents of floors 3 and 4.

Final arrangement.

Cross-check and contrast.

• The two clues fix E, P, L and R uniquely and leave F and Q interchangeable on floors 3 and 4. The arrangement is therefore only partly determined, yet the quantity asked for is fully determined: whichever way F and Q sit, floors 3 and 4 are still the only floors strictly between floor 2 and floor 5, so the count is 2 either way.

• Common trap: reading "only R lives above L" as the weaker "R lives above L" would allow L on any floor below R and make the count indeterminate. The word "only" is what converts a comparison into an exact count and pins the floors.

• Sanity check on the endpoints: no one lives above R (floor 6) and no one lives below E (floor 1), which is consistent with R being the single person above L and E the single person below P.

Concept

Replace every letter by its position in the English alphabet (A = 1, B = 2, ..., Z = 26). A set of letter-cluster pairs belongs to one group when the same sequence of position differences runs through every member, read from the first letter to the last. The member whose difference sequence breaks that common pattern is the odd one out.

Application

Write the four letters of each pair as position numbers, then take the difference between each letter and the next:

Cross-check

• HG-ED, QP-NM and NM-KJ all show the step pattern -1, -2, -1: two letters running backwards through the alphabet, one letter skipped (F, O and L respectively), then two more letters running backwards.

• SR-OQ begins the same way, with S and R running backwards, but its second block moves forward from O to Q instead of backwards, giving the steps -1, -3, +2. Following the group's rule, S and R would have been paired with PO.

Therefore SR-OQ is the letter-cluster pair that does not belong to the group.

Concept — In a coded-relationship chain every symbol relates only the two letters standing immediately on either side of it, and the chain is read from left to right. Calling one person the father of another needs two independent facts to hold at the same time: that person has to be male, and the other has to be his child. Maleness comes from a male-specific link such as husband or son. Parenthood can be reached directly through a parent link, or indirectly through marriage, because the husband of a woman who is the mother of a child is that child's father; and since siblings share the same parents, a parent of one sibling is a parent of the other.

So two things have to be located inside a chain before F can be called the father of T:

• a link that marks F as male, which here can only be the husband link or the son link;

• a route from F to T that ends in parenthood, either because F is given directly as a parent of T, or because F is the husband of a woman who is the mother of T or of a sister of T.

Applying it — Each statement is decoded link by link, and the last column records what the decoded chain makes F to T.

Cross-check — Statements II and III carry the same operative sub-chain, F & S % N # T: F is the husband of S, S is the mother of N, and N is the sister of T. That sub-chain is precisely the indirect route to fatherhood set out above, and the point where those two statements differ, R as the mother of F against R as the sister of F, lies outside it and changes nothing. The remaining two break at an identifiable link instead: in I the final link is a husband link, so T arrives as a spouse rather than as a child, and in IV the second link marks F himself as a son.

The statements that mean F is the father of T are therefore II and III.

Concept: in a symbol-coded blood-relation chain every operator fixes two things at once — the relationship it names and the gender of the person written on its left. '&' makes that person a husband (male), '%' makes that person a mother (female), '#' makes that person a sister (female), and '+' makes that person a son (male).

A chain is read link by link from left to right; each link binds only the pair it stands between, and the links together assemble one family tree. By definition X is the father-in-law of Y when X is male and Y is married to a child of X, so a chain qualifies only when it makes F male, gives F a child, and marries that child to M.

Applying the definition to each chain:

• Chain I — F & T % S # U & M

  1. F & T: F is the husband of T, so F is male and T is his wife.

  2. T % S: T is the mother of S, so S is a child of T and of her husband F.

  3. S # U: S is the sister of U, so U is a sibling of S and therefore another child of F and T.

  4. U & M: U is the husband of M, so M is the wife of U.

  M is married to U, who is a child of F, and F is male — every part of the definition is met.

• Chain II — F # T % S # U & M

  1. F # T: F is the sister of T, so F is female.

  2. T % S: T is the mother of S.

  3. S # U: S is the sister of U, so U is another child of T.

  4. U & M: U is the husband of M.

  F is the sister of the mother of M's husband, so F is an aunt of M's husband, and a female relative cannot be a father-in-law.

• Chain III — F & T % S # U % M

  1. F & T: F is the husband of T.

  2. T % S: T is the mother of S, so S is a child of F and T.

  3. S # U: S is the sister of U, so U is another child of F and T.

  4. U % M: U is the mother of M, so M is the child of U.

  M is the child of a child of F, so F stands as M's maternal grandfather rather than as the father of M's spouse.

• Chain IV — F & T # S # U % M

  1. F & T: F is the husband of T.

  2. T # S: T is the sister of S, so S is T's sibling and no parent-child link is created here.

  3. S # U: S is the sister of U, so U is a sibling of T as well.

  4. U % M: U is the mother of M.

  F is the husband of the sister of M's mother, so F stands as M's uncle by marriage.

Cross-check — the relation each chain assigns to F with respect to M:

Exactly one chain, I, satisfies the definition of father-in-law, so the intended answer is 'Only I'.

Step-by-Step Solution

In a partnership business, profit is distributed between partners in proportion to their respective investments and the time period for which they invested. Since both invested for the same duration (one year), we distribute the profit based on their initial investment ratio.

1. Find the Investment Ratio:

   • Ramu's investment = 1300

   • Somu's investment = 2200

   • Ratio = 1300 : 2200 = 13 : 22

2. Determine Total Ratio Parts:

   • Total parts = 13 + 22 = 35

3. Calculate Ramu's Share:

   • Ramu's share = (Ramu's ratio part / Total ratio parts) * Total profit

   • Ramu's share = (13 / 35) * 7350

   • Ramu's share = 13 * (7350 / 35)

   • Ramu's share = 13 * 210 = 2730

Step-by-Step Solution

1. Understand the Concept: When two items are sold at the same selling price, and one is sold at a gain of x% while the other is sold at a loss of x%, there is always a net loss in the overall transaction.

2. Use the Shortcut Formula: For cases with an equal percentage gain and loss (x%), the net loss percentage is calculated as: Net Loss % = (common percentage / 10)^2 OR (x^2 / 100)

3. Apply the Values:

   • Here, x = 15.

   • Net Loss % = (15^2) / 100

   • Net Loss % = 225 / 100

   • Net Loss % = 2.25%.

Step-by-Step Solution

To find the mean of grouped data, use the formula: Mean = (Σfx) / Σf, where 'f' is the frequency (number of customers) and 'x' is the data value (consumption in units).

1. Calculate the product of each consumption unit (x) and its frequency (f):

   • 20 units * 4 customers = 80

   • 25 units * 4 customers = 100

   • 30 units * 7 customers = 210

   • 35 units * 3 customers = 105

   • 40 units * 2 customers = 80

2. Find the sum of these products (Σfx):

   • 80 + 100 + 210 + 105 + 80 = 575

3. Find the sum of frequencies (Σf):

   • 4 + 4 + 7 + 3 + 2 = 20

4. Calculate the Mean:

   • Mean = 575 / 20 = 28.75 units.

Step-by-Step Solution

1. Understand the Relationship: Since the distance between point A and point B is constant, the product of speed and time must be equal for different scenarios (Distance = Speed × Time). Let the distance be D and the time taken at 16 km/hr be t hours.

2. Set up the Equations:

   • Scenario 1: At 16 km/hr, time = t. So, D = 16 × t.

   • Scenario 2: At 20 km/hr, he reaches 1 hour earlier (3 p.m. to 2 p.m.), so time = t - 1. So, D = 20 × (t - 1).

3. Solve for t:

   • 16t = 20(t - 1)

   • 16t = 20t - 20

   • 4t = 20

   • t = 5 hours.

4. Find the Distance (D):

   • D = 16 km/hr × 5 hours = 80 km.

5. Calculate Required Speed:

   • To reach at 1 p.m. (which is 2 hours earlier than 3 p.m.), the time taken should be t - 2 = 5 - 2 = 3 hours.

   • Required Speed = Distance / Required Time = 80 km / 3 hours = 26 and 2/3 km/hr.

Concept

The median is the middle observation of a data set once the values are arranged in order of size. For n observations written in ascending order the median is the ((n + 1)/2)-th term when n is odd, and the average of the (n/2)-th and (n/2 + 1)-th terms when n is even. Because the median is defined by rank, sorting the data is the first and essential step; the order in which the values happen to be printed carries no information.

Application

1. Count the observations. The list 1/2, 4/5, 7/12, 4/3, 5/6, 6/7, 2/3 contains n = 7 values, and 7 is odd, so the median is the (7 + 1)/2 = 4th term of the sorted list.

2. Pick a common denominator so that the fractions can be compared exactly. The denominators are 2, 5, 12, 3, 6 and 7, and their LCM is 420, since 420 = 22 × 3 × 5 × 7.

3. Rewrite each value over 420: 1/2 = 210/420, 4/5 = 336/420, 7/12 = 245/420, 4/3 = 560/420, 5/6 = 350/420, 6/7 = 360/420 and 2/3 = 280/420.

4. With equal denominators, comparing the fractions is just comparing the numerators: 210 < 245 < 280 < 336 < 350 < 360 < 560, so the data in ascending order is 1/2, 7/12, 2/3, 4/5, 5/6, 6/7, 4/3.

5. Read off the 4th term of this ordered list: 336/420 = 4/5. Therefore the median of the data is 4/5.

Cross-check

Converting each value to a decimal reproduces exactly the same ranking, independently of the common-denominator method.

Exactly three values (1/2, 7/12 and 2/3) are smaller than 4/5 and exactly three (5/6, 6/7 and 4/3) are larger, which is the defining property of the median of an odd-sized data set. A frequent slip is to take the middle entry of the list as it is printed, which would give 4/3; the median is always read from the sorted list, never from the printed order.

Median = 4/5.

Concept

A markup and a discount are percentages of different bases: the markup is taken on the cost price, while each discount is taken on the price standing just before it. So the chain is multiplicative — SP = CP × (1 + markup) × (1 − d1) × (1 − d2) — and the loss is simply CP − SP. Since every percentage factor is known, one unknown cost price can be recovered from the given loss amount.

Applying it here

1. Let the cost price be ₹x. A markup of 40% above cost gives the marked price MP = 1.40x.

2. The first discount of 20% leaves 80% of the marked price: 1.40x × 0.80 = 1.12x.

3. The second discount of 25% is taken on that reduced price and leaves 75% of it: 1.12x × 0.75 = 0.84x. So SP = 0.84x.

4. Loss = CP − SP = x − 0.84x = 0.16x.

5. The loss is given as ₹720, so 0.16x = 720, which gives x = 720 ÷ 0.16 = ₹4,500 as the cost price.

6. Therefore the selling price is SP = 0.84 × 4,500 = ₹3,780.

Cross-check

Rebuild the whole chain from a cost price of ₹4,500 and confirm that the loss comes back to ₹720:

Note the trap: discounts of 20% and 25% are not a single 45% discount. Their combined effect is 1 − (0.80 × 0.75) = 0.40, i.e. 40% off. Even so, a 40% markup followed by 40% off ends in a loss, because the 40% is taken off the larger marked price while the 40% was added to the smaller cost price.

Concept — For a random experiment in which every outcome is equally likely, the probability of an event E is a ratio of counts: P(E) = (number of outcomes favourable to E) / (total number of possible outcomes).

When an event is described as "A or B" and the two cases can never happen on the same trial, they are mutually exclusive, so their favourable counts simply add: n(A or B) = n(A) + n(B). A divisibility condition on a sum must first be converted into the list of multiples that actually lie inside the range of possible sums.

Applying it here

1. Two dice thrown together give ordered outcomes of the form (first die, second die). Each die has 6 faces, so the total number of equally likely outcomes is 6 x 6 = 36.

2. The smallest possible sum is 1 + 1 = 2 and the largest is 6 + 6 = 12, so every sum lies between 2 and 12.

3. Multiples of 7 inside that range: only 7, because 14 is already greater than 12. Multiples of 9 inside that range: only 9, because 18 is greater than 12. So the condition "divisible by 7 or 9" means the sum equals 7 or the sum equals 9.

4. Outcomes whose sum is 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That is 6 outcomes.

5. Outcomes whose sum is 9: (3,6), (4,5), (5,4), (6,3). That is 4 outcomes.

6. A single roll cannot total 7 and 9 at the same time, so the two cases are mutually exclusive and their counts add: 6 + 4 = 10 favourable outcomes.

7. Apply the ratio: P = 10 / 36. Dividing numerator and denominator by 2 gives P = 5/18.

Cross-check — Cross-check with the standard tally for two dice: the number of ordered outcomes with sum s is (s - 1) for sums from 2 to 7, and (13 - s) for sums from 7 to 12. This gives 7 - 1 = 6 outcomes for sum 7 and 13 - 9 = 4 outcomes for sum 9, confirming 10 favourable outcomes out of 36.

Result — Probability that the sum of the two faces is divisible by 7 or 9 = 5/18.

Step-by-Step Solution

To achieve a "neither gain nor loss" (break-even) position, the total profit from the remaining goods must exactly equal the total loss incurred from the first part.

1. Analyze the first part:

   • Portion of goods = 1/3

   • Loss percentage = 33%

   • Loss amount = (1/3) * 33% = 11% of the total cost price (CP).

2. Analyze the remaining part:

   • Portion of goods = 1 - 1/3 = 2/3

   • We need this part to generate a profit percentage (P%) that covers the 11% loss.

3. Set up the equation:

   • Profit from remaining = Loss from first part

   • (2/3) * P% = 11%

4. Solve for P%:

   • P% = 11% * (3 / 2)

   • P% = 33 / 2

   • P% = 16.5%

Concept — Average speed is never the plain average of the separate speeds used on a journey. It is defined as total distance divided by total time. So whenever a trip is broken into stretches covered at different speeds, each stretch's time must be found on its own from time = distance ÷ speed, all those times added up, and only then is the total distance divided by that total time.

Application — Fix the three speeds from the given ratios, then time each of the four stretches of the round trip.

1. Walking speed = 5 km/hr. Jogging is three times walking, so jogging = 3 × 5 = 15 km/hr. Running is twice jogging, so running = 2 × 15 = 30 km/hr.

2. Home to office, walking stretch: one-third of 10 km = 10/3 km at 5 km/hr, so time = (10/3) ÷ 5 = 2/3 hr.

3. Home to office, jogging stretch: the remaining 20/3 km at 15 km/hr, so time = (20/3) ÷ 15 = 4/9 hr.

4. Office to home, jogging stretch: half of 10 km = 5 km at 15 km/hr, so time = 5 ÷ 15 = 1/3 hr.

5. Office to home, running stretch: the remaining 5 km at 30 km/hr, so time = 5 ÷ 30 = 1/6 hr.

6. Total time = 2/3 + 4/9 + 1/3 + 1/6 = 12/18 + 8/18 + 6/18 + 3/18 = 29/18 hr.

7. Total distance = 10 + 10 = 20 km, so average speed = 20 ÷ (29/18) = 20 × 18/29 = 360/29 = 12.4137… ≈ 12.41 km/hr.

Cross-check — Collapse each 10 km leg into a single leg average speed and recombine the two legs independently.

• Onward leg: 10 km in 2/3 + 4/9 = 10/9 hr, so the onward leg averages 10 ÷ (10/9) = 9 km/hr.

• Return leg: 10 km in 1/3 + 1/6 = 1/2 hr, so the return leg averages 10 ÷ (1/2) = 20 km/hr.

• The two legs are equal in distance, so the round-trip average is the harmonic mean of the leg speeds: (2 × 9 × 20) ÷ (9 + 20) = 360/29 ≈ 12.41 km/hr — the same value, reached by an independent route.

• Note that the arithmetic mean of 9 and 20 is 14.5 km/hr, which is not the average speed of the trip; that gap is exactly what the concept above warns against.

Average speed for the complete round trip = 12.41 km/hr.

Concept: a number is divisible by 9 exactly when the sum of its digits is divisible by 9. The reason is that 10 leaves a remainder of 1 when divided by 9, and so does every power of 10; a number therefore leaves the same remainder on division by 9 as the plain sum of its digits.

Application: the digit in the starred place is unknown, so write the digit sum with that digit as x and force the total to be a multiple of 9.

1. The six known digits are 4, 8, 1, 8, 6 and 5, and they add to 4 + 8 + 1 + 8 + 6 + 5 = 32.

2. Writing x for the digit in the starred place, the digit sum of 481x865 is 32 + x, where x is a single digit from 0 to 9.

3. Divisibility by 9 requires 32 + x to be a multiple of 9. Since 32 = 9 × 3 + 5, the known digits already carry a remainder of 5, so x must supply the missing 4.

4. x = 4 gives 32 + 4 = 36 = 9 × 4, a multiple of 9. The next value that would work, x = 13, is not a single digit, so 4 is the only admissible digit and hence the smallest.

Cross-check: every single-digit choice can be tested against the rule directly.

Substituting the digit gives 4814865, and 4814865 divided by 9 is exactly 534985 with no remainder. So the smallest whole number that can replace * is 4.

Concept: A percentage change always describes a movement measured against the base it acts on, and that base is whatever the quantity is worth at the moment the change is applied. So when a quantity first falls and then has to be brought back to its starting value, the fall is measured on the original base while the restoring rise is measured on the smaller value that survived the fall. The restoring rise is therefore (shortfall ÷ value after the fall) × 100, and a fall of x% needs a rise of 100x/(100 − x)% to return to the start.

Applying it here:

1. Let the original salary be 100 units. Any base works, because the answer is a ratio and the units cancel.

2. The salary falls by 40%, and that 40% is taken on the original base of 100, so the fall is 40 units.

3. Salary after the fall = 100 − 40 = 60 units. This 60 is now the base for anything applied to the reduced salary.

4. Shortfall still to be made up = 100 − 60 = 40 units.

5. Express that shortfall on the new base: required rise = (40 ÷ 60) × 100 = (2/3) × 100 = 66 2/3%.

Cross-check: Raise 60 by 66 2/3%: the rise is (2/3) × 60 = 40, and 60 + 40 = 100, which is the original salary. The general formula agrees: 100x/(100 − x) with x = 40 gives 4000/60 = 66 2/3%.

Common slip: Adding the same 40% back on the reduced salary gives 60 + 24 = 84 units, short of 100, because 40% of 60 is smaller than 40% of 100. The percentage that comes off and the percentage that must go back on are never equal once the base has changed.

Concept: The surface area of a sphere is fixed by its radius alone, through the identity S = 4πr2. The radius enters only as a square, so a radius written as a surd is squared exactly using (a√b)2 = a2b, and no decimal approximation of the surd is needed at any stage.

Application: here the radius is 14√5 cm and the question fixes π = 22/7.

1. Square the radius exactly: r2 = (14√5)2 = 142 × 5 = 196 × 5 = 980 cm2.

2. Substitute into the identity: S = 4 × (22/7) × 980 cm2.

3. Cancel the 7 against 980 before multiplying: 980 ÷ 7 = 140, which leaves S = 4 × 22 × 140.

4. Multiply the remaining factors: 4 × 22 = 88, and 88 × 140 = 12320.

5. Therefore S = 12320 cm2.

Cross-check: reverse the arithmetic. 12320 ÷ 4 = 3080, and 3080 × 7/22 = 980, which is exactly r2; since √980 = √(196 × 5) = 14√5, the radius recovered is the one the question gave, so the value is consistent.

Contrast: the factor in front of πr2 decides which figure is being measured — 2πr2 is the curved surface of a hemisphere (6160 cm2 here) and 3πr2 is the total surface of a hemisphere (9240 cm2 here). A full sphere always carries the factor 4.

Concept: when two ratios share a common term — here Q appears in both P : Q and Q : R — they can be merged into a single three-term ratio by scaling each ratio so that the shared term carries the same value in both.

Once one chained ratio a : b : c exists, a total T is split into (a + b + c) equal parts, one part is worth T ÷ (a + b + c), and any member's share is its own part-count multiplied by that part value.

Applying this to the match:

1. Write the two given ratios: P : Q = 5 : 4 and Q : R = 1 : 2. The shared term Q is 4 in the first ratio but 1 in the second.

2. Scale the second ratio so that Q agrees: multiply Q : R = 1 : 2 throughout by 4, which gives Q : R = 4 : 8.

3. Merge the two into one chain: P : Q : R = 5 : 4 : 8.

4. Add the parts: 5 + 4 + 8 = 17 parts in all.

5. Find the value of one part: 884 ÷ 17 = 52 runs per part.

6. P holds 5 parts, so P scored 5 × 52 = 260 runs.

Cross-check by distributing all three shares:

The three shares add to 260 + 208 + 416 = 884, and 260 : 208 = 5 : 4 while 208 : 416 = 1 : 2, so both given ratios hold. P scored 260 runs.

Step-by-Step Solution

To solve this, we determine the status of the tank step-by-step based on which pipes are active.

1. Define Pipe Rates:

   • Pipe A (filling): 1/10 tank/hour

   • Pipe B (filling): 1/12 tank/hour

   • Pipe C (emptying): 1/4 tank/hour

2. Calculate Cumulative Fill Status by 9 a.m.:

   • From 6 a.m. to 7 a.m. (1 hour): Only A is open.

     • Work done = 1/10

   • From 7 a.m. to 9 a.m. (2 hours): A and B are open.

     • Combined rate = 1/10 + 1/12 = (6+5)/60 = 11/60 tank/hour.

     • Work done = (11/60) * 2 = 11/30

   • Total filled by 9 a.m.: 1/10 + 11/30 = 3/30 + 11/30 = 14/30 = 7/15.

3. Calculate Net Rate from 9 a.m. onwards:

   • When A, B, and C are all open:

   • Net rate = (1/10 + 1/12) - 1/4

   • Net rate = (11/60) - 15/60 = -4/60 = -1/15.

   • Since the rate is negative, the tank is emptying at a rate of 1/15 tank/hour.

4. Find Time to Empty:

   • Tank to be emptied = 7/15.

   • Time required = (7/15) / (1/15) = 7 hours.

5. Determine Final Time:

   • Starting from 9 a.m., adding 7 hours gives 4 p.m.

Concept

When a task is carried out in stages performed one after another, the total number of ways is the product of the number of choices available at each stage — the fundamental counting principle. If both stages draw from the same pool of k distinct objects and the second stage may not repeat the object chosen at the first, then the first stage offers k choices and the second offers only k − 1, so the number of ordered pairs of distinct objects is k × (k − 1).

This count is exactly the number of ordered selections of 2 distinct objects out of k. Order is part of the arrangement, so a pair taken in one sequence is counted separately from the same pair taken in the opposite sequence.

Applying it here

1. The pool here is the hall's 14 gates, and the man performs two ordered stages: he picks the gate he enters by, and then the gate he leaves by.

2. Entry stage: any gate of the hall may be used, so there are 14 choices.

3. Exit stage: the gate already used to enter is barred, so 13 of the 14 gates remain and there are 13 choices.

4. By the counting principle the two stages multiply: 14 × 13 = 182 ways.

Cross-check

Count it the other way round as a check. Ignoring the restriction there are 14 × 14 = 196 ordered (entry gate, exit gate) pairs. Exactly 14 of these use the same gate twice, one for each gate, so the admissible count is 196 − 14 = 182, which matches the direct count.

In permutation notation the same figure is 14P2 = 14! / 12! = 14 × 13 = 182.

Step-by-Step Solution

To solve this, we must ensure the amount of sugar represents the desired percentage of the total weight of the solution.

1. Analyze the Initial State:

   • Total initial solution = 300 grams

   • Sugar concentration = 40%

   • Initial sugar amount = 0.40 * 300 = 120 grams

   • Initial water/other components amount = 300 - 120 = 180 grams

2. Analyze the Target State:

   • Let 'x' be the grams of sugar to be added.

   • New sugar amount = 120 + x

   • New total solution weight = 300 + x

   • Target sugar concentration = 55%

3. Set up the Equation:

   • (New sugar amount) / (New total solution weight) = 0.55

   • (120 + x) / (300 + x) = 0.55

4. Solve for x:

   • 120 + x = 0.55 * (300 + x)

   • 120 + x = 165 + 0.55x

   • x - 0.55x = 165 - 120

   • 0.45x = 45

   • x = 45 / 0.45

   • x = 100 grams.

Step-by-Step Solution

1. Understand the Principle: The Addition Principle states that if you have several mutually exclusive categories of items, the total number of ways to choose one item is the sum of the number of items in each category.

2. List the Items:

   • Cupcakes: 17

   • Donuts: 25

   • Cookies: 48

   • Breads: 19

   • Chocolates: 36

   • Muffins: 33

3. Calculate the Total:

   • Total = 17 + 25 + 48 + 19 + 36 + 33 = 178

Step-by-Step Solution

1. Determine A's Individual Rate:

   • A completes 1/3 of the work in 5 days.

   • A's daily rate = (1/3) / 5 = 1/15 of the work per day.

2. Calculate the Remaining Work:

   • Remaining work = Total work - Work already done = 1 - 1/3 = 2/3.

3. Determine the Combined Rate of A and B:

   • A and B together complete 2/3 of the work in 6 days.

   • Combined rate = (2/3) / 6 = 2/18 = 1/9 of the work per day.

4. Calculate Total Days for A and B to Finish the Work:

   • Total days = 1 / (Combined rate)

   • Total days = 1 / (1/9) = 9 days.

Concept: For any two numbers a and b, the difference of cubes factorises as a3 − b3 = (a − b)(a2 + ab + b2). So whenever a2 + ab + b2 is not zero, the quotient (a3 − b3) ÷ (a2 + ab + b2) reduces to the single difference a − b, and the cubes never have to be evaluated.

Application

1. Match the expression to the identity: here a = 0.867 and b = 0.321, the numerator is a3 − b3, and the denominator is exactly a2 + ab + b2.

2. Factorise the numerator using the identity: a3 − b3 = (a − b)(a2 + ab + b2).

3. Cancel the common factor a2 + ab + b2, which is positive here, so the cancellation is valid; the expression becomes simply a − b.

4. Substitute the given values: a − b = 0.867 − 0.321 = 0.546.

Cross-check by direct computation

• Numerator: 0.8673 − 0.3213 = 0.651714363 − 0.033076161 = 0.618638202.

• Denominator: 0.8672 + 0.867 × 0.321 + 0.3212 = 0.751689 + 0.278307 + 0.103041 = 1.133037.

• Quotient: 0.618638202 ÷ 1.133037 = 0.546, which matches the factorised result exactly.

Watch out: If the numerator carried a plus sign, the sum-of-cubes identity a3 + b3 = (a + b)(a2 − ab + b2) would apply instead, and the middle term of the quadratic factor would be −ab, giving a + b. Mixing the two forms breaks the cancellation, so always check the sign pattern of the middle term before cancelling.

Concept: In an isosceles triangle, the altitude drawn to the base bisects that base and splits the triangle into two congruent right triangles. In each of them the legs are the altitude and half the base, and the hypotenuse is one of the two equal sides. Two relations therefore govern every such triangle: the Pythagorean relation (equal side)2 = (half of base)2 + (altitude)2, and the area formula Area = 1/2 × base × altitude.

Application: Let each equal side be a cm and the base be b cm, with the altitude to the base equal to 12 cm.

1. Perimeter: 2a + b = 36, so a = 18 − b/2.

2. The altitude bisects the base, so one right triangle has legs b/2 and 12 with hypotenuse a: a2 = (b/2)2 + 122.

3. Substitute a = 18 − b/2: (18 − b/2)2 = (b/2)2 + 144.

4. Expand the left-hand side: 324 − 18b + b2/4 = b2/4 + 144.

5. The b2/4 terms cancel, leaving 324 − 18b = 144, so 18b = 180 and b = 10.

6. Then a = 18 − 10/2 = 13, so the base is 10 cm and each equal side is 13 cm.

7. Area = 1/2 × base × altitude = 1/2 × 10 × 12 = 60 cm2.

Cross-check: the right triangle formed by the altitude has legs 5 cm and 12 cm and hypotenuse 13 cm, the 5-12-13 Pythagorean triple, and the three sides give a perimeter of 13 + 13 + 10 = 36 cm, exactly the value stated in the question.

Hence the area of the triangle is 60 cm2.

Step-by-Step Solution

To calculate the average of a set of numbers, you sum all the values and then divide by the total count of numbers. The formula is: Average = (Sum of all values) / (Total count of values)

1. Sum all the numbers:

   • 308 + 135 + 94 + 45 + 102 + 99 = 783

2. Count the numbers:

   • There are 6 numbers in the given set.

3. Divide the sum by the count:

   • Average = 783 / 6 = 130.5

Concept: fractions can be added only after they are written over one common denominator. Reduce each fraction to lowest terms using a/b = (a ÷ k)/(b ÷ k), rewrite every term over the LCM of the denominators, and then add the numerators. A whole number written beside a fraction denotes their sum; the standard mixed-number form additionally requires that fraction to be proper, so a fractional part whose numerator is at least its denominator must be normalised by carrying the extra whole units into the whole-number part.

Applying this to the given expression 3 + 1/2 + 1/4 + 1/14 + 2/28:

1. Reduce the last term to lowest terms: 2/28 = (2 ÷ 2)/(28 ÷ 2) = 1/14, so the expression becomes 3 + 1/2 + 1/4 + 1/14 + 1/14.

2. Find the LCM of the denominators 2, 4 and 14. Their prime factorisations are 2 = 2, 4 = 22 and 14 = 2 × 7, so the LCM is 22 × 7 = 28.

3. Rewrite every fraction over 28: 1/2 = 14/28, 1/4 = 7/28, 1/14 = 2/28 and 1/14 = 2/28.

4. Add the numerators over the common denominator: (14 + 7 + 2 + 2)/28 = 25/28.

5. Since 25 is less than 28, the fractional total 25/28 is already a proper fraction, so it stays beside the whole number 3. The expression equals 3 25/28.

Cross-check with decimals: 1/2 = 0.5, 1/4 = 0.25, 1/14 ≈ 0.0714 and 2/28 ≈ 0.0714, and these add to about 0.8929. Since 25/28 ≈ 0.8929, the whole expression is about 3.893, which agrees with the fractional working.

Step-by-Step Solution

To calculate the Simple Interest (SI), you can use the standard formula: SI = (P * R * T) / 100

Where:

• P = Principal amount (the initial sum of money)

• R = Rate of interest per annum

• T = Time period in years

1. Identify the given values:

   • Principal (P) = 247

   • Rate (R) = 6%

   • Time (T) = 5 years

2. Apply the formula:

   • SI = (247 * 6 * 5) / 100

3. Calculate the result:

   • First, multiply the rate and time: 6 * 5 = 30

   • Next, multiply by the principal: 247 * 30 = 7410

   • Finally, divide by 100: 7410 / 100 = 74.10

Step-by-Step Solution

1. Define the Number: Let the number be x.

2. Determine the Values:

   • Correct value = (11/15) * x

   • Incorrect value = (15/11) * x

3. Calculate the Absolute Error: The error is the difference between the incorrect value and the correct value.

   • Error = (15/11)x - (11/15)x

   • To subtract, find a common denominator (11 * 15 = 165).

   • Error = (225x / 165) - (121x / 165) = (104/165)x.

4. Calculate Percentage Error: The formula for percentage error is (Error / Correct Value) * 100.

   • Percentage Error = [((104/165)x) / ((11/15)x)] * 100

   • Percentage Error = (104 / 165) * (15 / 11) * 100

   • Simplify: 104 / 11 * 1/11 * 100 (since 165/15 = 11)

   • Percentage Error = (104 / 121) * 100 = 10400 / 121%.

5. Convert to Mixed Fraction:

   • 10400 divided by 121 equals 85 with a remainder of 115.

   • Thus, the result is 85 and 115/121%.

Step-by-Step Solution

1. Identify the Variables:

   • Principal (P) = 6455

   • Rate (r) = 20%

   • Time (n) = 2 years

2. Use the Amount Formula for Compound Interest: The formula to find the total amount (A) is: A = P * (1 + r/100)^n

3. Perform the Calculation:

   • A = 6455 * (1 + 20/100)^2

   • A = 6455 * (1.20)^2

   • A = 6455 * 1.44

   • A = 9295.20

4. Find the Compound Interest: The compound interest (CI) is the total amount minus the principal:

   • CI = A - P

   • CI = 9295.20 - 6455 = 2840.20

The word 'morose' means gloomy, sullen, ill-humoured, or depressed in mood. We need its antonym (opposite).

'happy' (cheerful, light-hearted) directly opposes the gloomy sense of 'morose', so it is the correct antonym.

'sympathetic' (compassionate toward others) and 'diseased' (ill) describe attitude and health, not mood; 'halting' (hesitant, faltering) describes speech or movement. None of these is the opposite of 'morose'.

A question tag repeats the auxiliary (helping) verb of the main statement and reverses its polarity. A negative statement therefore takes a positive tag, and a positive statement takes a negative tag; the subject pronoun in the tag stays the same.

The statement "He has not been here before" is built on the present-perfect auxiliary "has" (in "has ... been") and is negative. So the tag uses the same auxiliary "has," keeps the pronoun "he," and becomes positive: "has he?"

Full sentence: "He has not been here before, has he?"

A question tag mirrors the auxiliary verb of the main clause and reverses its polarity: a negative statement takes a positive tag, and a positive statement takes a negative tag.

Here the main clause is "You have not been here before" — present perfect ("have ... been") and negative ("not"). The tag must therefore reuse the auxiliary "have" in the positive form and keep the subject pronoun "you."

The complete sentence is: "You have not been here before, have you?" Tags such as "will you," "would you," or "had you" do not match the present-perfect auxiliary of the statement.

The sentence contains the misspelt word "benvolence", which is missing the second "e". The correct spelling is "benevolence" (meaning kindness or goodwill).

The remaining words in the sentence — "magnanimity", "philanthropic", and "alleviating" — are all spelt correctly, so the option that rectifies the error is "benevolence".

The sentence describes Bengaluru’s climate as “salubrious.” The word salubrious (from Latin salubris, “healthful, wholesome”) means favourable to health or well-being, and it is used especially of climate or air.

Therefore the closest synonym is “healthful” — a healthful climate is one that is good for your health, exactly the idea conveyed by salubrious.

The other options do not fit: “luxurious” relates to comfort and wealth, “hectic” means very busy or frantic, and “unpredictable” means changeable or hard to forecast — none of these carry the sense of being good for one’s health.

The blank needs a word describing a manner of speaking that made it hard for people to grasp her true feelings — i.e. a manner that is unclear or open to several interpretations.

“Ambiguous” means open to more than one interpretation, vague or unclear; an ambiguous manner of speaking naturally leaves listeners uncertain of the real meaning, matching the clue “difficult to understand her true feelings.”

The other words point the wrong way: “forthright” (direct, frank), “candid” (honest, open) and “effusive” (gushing, openly emotional) all describe speech that reveals feelings clearly, so they contradict the context. Hence “ambiguous” is the most appropriate choice.

The correct one-word substitute is Protagonist.

A protagonist is the principal or central character of a literary work. This character faces the main conflict and whose decisions and actions drive the narrative forward, exactly as the phrase describes.

Distinguishing the other terms: an antagonist is the opposing character who creates conflict for the protagonist; a narrator is the voice that tells the story; and a confidant is a trusted character to whom secrets are confided. None of these denote the main, story-driving character.

The underlined word "deceitful" means dishonest, misleading, or intending to cheat. We need its antonym (opposite).

"Honest" means truthful, sincere, and free from deceit, which is the exact opposite of "deceitful." Hence, "honest" is the correct answer.

The other options do not fit: "fortunate" means lucky, "tasteful" means showing good taste, and "diseased" means affected by illness — none of these relate to truthfulness.

The task is to find the word that is spelt wrongly in the sentence. Each option is one of the words used in the sentence.

The word "maintanance" is incorrect. The correct spelling is "maintenance" (main-ten-ance). Although it derives from the verb "maintain", the "ai" is dropped in the noun, which is why it is a very commonly misspelt word.

The other three words match their dictionary spellings: "perceived" (from perceive: c followed by ei), "gorgeousness" (from gorgeous: the -eous ending plus -ness), and "accommodate" (double c, double m).

A question tag mirrors the main clause and flips its polarity: a negative statement takes a positive tag, and a positive statement takes a negative tag. The tag also repeats the auxiliary or be-verb of the statement together with its subject pronoun.

Here the statement 'She isn't your sister' is negative and uses the present be-verb 'is' (contracted in 'isn't'), with the subject 'she'. So the tag must be positive, use 'is', and keep the pronoun 'she' — giving 'is she?'.

Complete sentence: 'She isn't your sister, is she?' The other auxiliaries ('did', 'will', 'can') do not appear in the statement and so cannot form the tag.

Concept

A research-report paragraph follows a fixed narrative spine: it first states the purpose/aim of the study, then describes the method (who was studied and how), then reports the results obtained, and finally draws a conclusion or implication from those results. Sequencing by this Aim -> Method -> Result -> Conclusion logic, and by cohesion cues (a sentence that begins with a back-reference such as "These findings" must come AFTER what it refers to), fixes the order uniquely.

Application

Classify each numbered sentence by its role, then chain them in narrative order:

1. Sentence 2 states the AIM ("The study aimed to evaluate the impact of digital learning tools") - this introduces the topic, so it opens the paragraph.

2. Sentence 3 gives the METHOD ("Researchers conducted a year-long study involving over 500 students") - it describes how the aim was pursued, so it follows the aim.

3. Sentence 1 reports the RESULT ("The results showed a significant improvement in academic performance") - results can only be stated after the study was conducted.

4. Sentence 4 is the CONCLUSION ("These findings suggest that integrating technology can be beneficial") - the pronoun phrase "These findings" refers back to the results, so it must come last.

Chaining these roles gives the order 2, 3, 1, 4.

Cross-check

Cohesion test: "These findings" in sentence 4 has no antecedent unless the results (sentence 1) precede it - so 4 cannot come before 1. Likewise "The results" in sentence 1 presupposes a study has already been run (sentence 3). Both anchors confirm the chain 2 -> 3 -> 1 -> 4.

One might instead argue that sentence 3 ("Researchers conducted a year-long study involving over 500 students") should open the paragraph, since it introduces "a study" before sentence 2 refers back to "the study" - testing definite/indefinite article cohesion. This does not overturn the chosen order for two reasons: first, aim must causally precede method - a study's purpose explains why a particular method (a year-long, 500-student design) was chosen, so the aim sentence has to come before the method sentence, not after it; opening with the method before its motivating purpose is known leaves the method unmotivated rather than fixing any cohesion gap. Second, an opening sentence stating "The study aimed to..." is the standard topic-sentence convention for a research-summary paragraph (the same way an abstract opens with "This study examines...") - the definite article here names the paragraph's own subject rather than requiring an antecedent from an earlier sentence.

A question tag is a short question added to the end of a statement. Two rules decide it:

1. Polarity reverses: a negative statement takes a positive tag (and a positive statement takes a negative tag).

2. The tag repeats the auxiliary or main verb of the statement in the same tense, and replaces the subject with a pronoun.

The statement "The weather is not nice today" is present tense, uses the verb "is," and is negative ("is not"). So the tag must be present tense, use "is," be positive, and refer to "the weather" as "it." This gives "is it?"

Complete sentence: "The weather is not nice today, is it?"

A question tag must be formed using the same auxiliary verb as the main clause, and its polarity must be the opposite of that clause.

Here the statement is "You are not coming to the party" — it is negative and uses the auxiliary verb "are" (the be-verb in the present continuous "are coming"). Since the statement is negative, the tag must be positive, and it must reuse the same auxiliary "are" with the pronoun "you".

This gives the positive tag "are you?". The other options use the wrong auxiliaries — "will you" (future/request), "do you" (simple present) and "did you" (simple past) — none of which match the auxiliary "are" in the statement, so the completed sentence is: "You are not coming to the party, are you?"

The opening sentence S1 introduces 'machinery'. Sentence B continues this idea directly: 'Where there is machinery, there are large cities.' This makes B the natural first link after S1.

B ends with 'large cities', and D opens with 'Where there are large cities, there are cars, buses, and electric lights.' This repetition of 'large cities' forms a mandatory B-D link.

D mentions 'cars, buses, and electric lights' (means of artificial locomotion). A picks this up: 'Honest physicians will tell you that where means of artificial locomotion have increased, the health of the people has suffered.' So A follows D.

C then concludes the body: 'English villages do not boast of any of these things' - 'these things' refers to the machinery, cities, cars and lights mentioned in B, D and A. C leads naturally into the final sentence S6.

Therefore the correct sequence is S1-B-D-A-C-S6, i.e. BDAC.

The passage is a sustained explanation of 'critical appreciation' in literature. It opens by stating that critical appreciation is central to literary analysis and involves a detailed examination of structure, language, characters and themes to understand the author's intentions, then illustrates each of these elements with examples (Faulkner's structure, Woolf's language, Dostoevsky's characters, Orwell's themes), and concludes that this discerning approach lets readers analyse and interpret a text's complexities.

The central theme is therefore the process of analysing, evaluating and interpreting literary works. The other choices either address a sub-detail or a setting that is too narrow (literary criticism only in academic settings), or introduce ideas the passage never raises (historical context; the impact of technology). Hence the correct answer is the option describing the process of analysing, evaluating and interpreting literary works.

The passage explains that critical appreciation is a method of literary analysis that requires examining several distinct elements of a text — its structure, language, characters, and themes — in order to understand the author's intentions and messages.

It supports this by giving concrete examples for each element: Faulkner's fragmented structure, Woolf's stream-of-consciousness language, Dostoevsky's morally conflicted characters, and Orwell's anti-totalitarian themes. The opening and the conclusion both frame critical appreciation as a discerning, detailed process built on analysing these multiple components.

Therefore, the inference that best captures the passage as a whole is that the process of critical appreciation involves analysing various elements of literary works.

Concept: In a reading-comprehension real-life-connection question, the correct option must restate the passage's own stated position in application terms, not merely offer a real-world claim that sounds plausible on its own. An option that the passage does not support, or that contradicts its stated conclusion, is wrong regardless of how sensible it seems outside the passage.

Application: This passage's central argument is that critical appreciation trains a reader to closely examine a text's structure, language, characters and themes, and to interpret its complexities through a discerning, analytical approach. The most appropriate real-life connection must carry these exact habits — close examination, interpretation of meaning, discerning judgment — into everyday reasoning and expression. The option stating that understanding literary criticism enhances communication skills and critical thinking abilities directly maps the passage's described skills onto real-world reasoning and communication, which is exactly the transfer the passage supports.

Why the other options don't fit:

• Technology has replaced the need for critical appreciation in modern society — the passage never discusses technology at all; nothing in the text supports this claim.

• The process of literary analysis is irrelevant in everyday life — this directly contradicts the passage's conclusion, which insists that a discerning, analytical approach remains valuable.

• Historical context has no impact on the interpretation of literary works in contemporary times — the passage never raises historical context as a factor, so asserting it has no impact is an unsupported claim invented outside the passage.

The correct real-life connection is therefore the option linking literary criticism to stronger communication and critical-thinking skills.

The question asks which statement is a fact actually stated in the passage. Test each option against the text.

Option about historical context: the passage never mentions historical context, so nothing supports it.

Option about technology and modern literature: the passage is about critical appreciation through structure, language, characters and themes; technology is never discussed.

Option about critical appreciation examining structure, language, characters and themes: the passage states directly that critical appreciation 'involves a detailed examination of elements such as structure, language, characters and themes to understand the author's intentions.' This is a near-verbatim restatement of the text, so it is the fact given in the passage.

Option about personal opinions over objective analysis: the passage describes a careful, discerning, detailed approach, not an opinion-driven one, so this is not stated.

Hence the correct choice is the statement that critical appreciation involves examining the structure, language, characters and themes of literary works.

The passage moves through three clear stages. It opens by explaining what critical appreciation is and the process it involves — examining structure, language, characters and themes to grasp the author's intentions.

It then applies this process through worked examples: structure (Faulkner's The Sound and the Fury), language (Woolf's Mrs. Dalloway), character (Dostoevsky's Raskolnikov) and themes (Orwell's 1984).

It closes with a conclusion restating that critical appreciation lets readers appreciate a work's artistry and depth. So the organisation is: explanation of the critical appreciation process, examples of its application, conclusion.

Blank 3 sits in the sentence: "Despite these challenges, the advantages of AI in enhancing the efficacy and effectiveness of healthcare are ___3___."

The phrase "Despite these challenges" is a concession marker. Having just listed the drawbacks (bias, cost, resource limits), the writer pivots to affirm that the benefits still stand out clearly. The required word must therefore be strongly positive and emphatic about the advantages.

undeniable = impossible to deny; clearly true. This matches the emphatic, affirmative tone and the strong claim about improved efficacy and effectiveness. CORRECT.

negligible = so small it can be ignored — contradicts "advantages". Wrong. questionable = open to doubt — clashes with the concessive, pro-benefit framing. Wrong. minimal = very small in degree — undercuts the praise of the advantages. Wrong.

Hence the most appropriate word for blank 3 is "undeniable".

Blank 4 sits in the clause: “… establish ethical standards and best practices to guarantee AI’s responsible and ___4___ use.” The word must pair with “responsible” to describe a positive, desirable quality of how AI is used.

‘Haphazard’ (disorderly/random), ‘negligent’ (failing in care/duty) and ‘careless’ (without attention) are all negative traits, so each contradicts the sentence’s positive intent of using AI responsibly.

‘Equitable’ means fair and impartial. It complements “responsible” and reflects the passage’s earlier worry about bias in AI algorithms harming underrepresented populations — calling for treatment that is just and unbiased. Hence ‘equitable’ best fills blank 4.

Blank 1 needs a word describing how the integration of AI into various sectors has changed in today's fast-evolving technological landscape.

The passage's overall tone is that AI is being adopted more and more widely. The word that fits is "prevalent" (widespread, common): AI integration has become increasingly prevalent.

The other choices are eliminated because they carry negative or opposite meanings: "obsolete" (outdated), "redundant" (unnecessary) and "ineffective" (not working) all contradict a passage that highlights AI's growing and beneficial role.

Blank 2 sits in the clause ‘the potential for bias in AI algorithms, which can result in ___ in treatment outcomes, particularly for underrepresented populations.’ Bias is introduced as a negative, problematic force, so the missing word must name an unwanted, uneven effect on certain groups.

‘Disparities’ means differences or inequalities between groups. Biased algorithms produce unequal treatment outcomes for underrepresented populations, which matches both the meaning and the negative tone of the clause, so it is the correct choice.

The remaining choices fail the context: ‘equality’ and ‘uniformity’ describe sameness or fairness, the opposite of what bias causes, and ‘improvements’ is a positive result that contradicts the problem being described.

The sentence reads: "Balancing innovation with ethical considerations and ___5___ remains essential for the successful assimilation of AI into healthcare."

Blank 5 needs a positive value that is set alongside "ethical considerations" and weighed against "innovation." The passage has already flagged that bias in AI algorithms can produce unfair treatment outcomes "particularly for underrepresented populations," so the closing call is for AI that serves everyone fairly. "Inclusivity" — making sure AI works well for all groups, including those underrepresented — is exactly the value that addresses that concern, so it is the correct answer.

"Exclusivity" (leaving groups out) and "inequity" (unfairness) are the opposite of what should be balanced into responsible AI use, and "apathy" (a lack of interest or concern) is a negative quality, not a value to uphold. Therefore inclusivity is the most appropriate choice.