JEE Advanced-Style Problem Set (Single Topic, Brutal Difficulty)

ROLE: Act as a senior JEE Advanced paper-setter and mathematics proof-checker with 20+ years of coaching experience.

CONTEXT: I teach students preparing for [EXAM e.g. JEE Advanced] at the [CLASS/GRADE e.g. Class 12 / Dropper] level. I need a high-difficulty problem set on ONE topic only: [TOPIC e.g. Definite Integrals / Conic Sections / Complex Numbers]. The questions must feel like real JEE Advanced - multi-concept, with traps, not routine textbook drill.

YOUR TASK:
1. Create EXACTLY [NUMBER e.g. 8] original problems on [TOPIC], all at genuine JEE Advanced difficulty.
2. Mix the question types the way Advanced does: single-correct MCQ, multiple-correct MCQ, integer-answer, and at least one comprehension/passage-based pair.
3. For each problem, give the correct answer and a clean, complete step-by-step solution.
4. Tag every problem with the exact sub-concepts it tests and a one-line note on the trap a weak student falls into.

INPUT: Here is the topic and any sub-areas to emphasise: [PASTE TOPIC AND SUB-AREAS HERE]. If I leave the number blank, use 8 problems. Stay strictly inside this topic.

HOW TO WORK IT OUT: Solve every problem fully internally before printing it. Then VERIFY each final answer by at least two independent checks (for example, an alternate method, re-substitution of the answer into the original condition, a limiting/special-case test, or a dimensional/sign sanity check). Never invent formulas, theorems or values; if you are unsure a problem is valid, discard it and build another. Keep all mathematics fully legible in clean LaTeX or clear plain text.

OUTPUT FORMAT, in this exact order:

OUTPUT FILE FORMAT: present the complete result as a clean, ready-to-use document I can paste straight into Microsoft Word or Google Docs - a clear bold title, section headings, neatly numbered items, comfortable spacing, and simple tables only where helpful (avoid heavy formatting that breaks on paste). It must also print cleanly to PDF on A4 (File > Print > Save as PDF). Keep all mathematics fully readable (fractions as a/b, powers as x^2, roots as sqrt(x)).

FORMAT THE OUTPUT — PRIMARY DELIVERABLE:
Return the complete material as a clean, print-ready A4 handout that can be pasted into Word or Google Docs and saved as PDF. Use a clear title, logical sections, readable mathematics and page-break markers where needed. Keep student material separate from the answer key or teacher notes.


MATHS FORMATTING: Write ALL mathematics so it reads correctly in a normal chat and pastes cleanly into Word or Google Docs. Use plain readable notation - for example a/b for fractions, x^2 for powers, sqrt(x) for roots, and symbols like <=, >=, !=, +-, pi, theta, integral and sum. Do NOT output raw LaTeX or code such as \frac{}{}, \sqrt{}, x^{2}, $...$ or \( ... \) - these appear as broken symbols to a teacher in a free chat.
A. Title block containing 'Topic: [TOPIC]', 'Level: JEE Advanced', and 'Compiled by [YOUR NAME]'.
B. SECTION 1 - PROBLEMS: numbered Q1...Q[NUMBER], with the question type and marks shown against each (single-correct +3, multiple-correct +4, integer +4 as standard - state any change).
C. SECTION 2 - ANSWER KEY: a compact table of Question | Type | Final Answer.
D. SECTION 3 - FULL SOLUTIONS: for each problem, the worked steps, the two verification checks you ran, the sub-concepts tested, and the common trap.
E. SECTION 4 - DIFFICULTY MAP: one line per question rating it Hard / Very Hard and why.

RULES: Strictly one topic. No repeats of the same idea twice. Every answer must survive two checks before you show it. If two checks disagree, fix the problem before printing. Do not pad with easy filler.

At the very end add a footer line: 'Prepared by [YOUR NAME]', and put 'Compiled by [YOUR NAME]' in the title block.

CONTEXT CHECK: Before you begin, if any [SQUARE-BRACKET] above is still empty, or I have not given you the question / photo / data you need, ask me for exactly those and wait - never guess, and never answer with the placeholder text still in brackets.

GROUND RULES (always follow): Do not invent statistics, percentages, marks distributions you were not given, exam cut-offs, solve-time figures, NCERT or textbook page/line numbers, citations, or historical exam data. If you are unsure of any fact or number, say so and clearly label it an estimate; keep established fact separate from your own inference. Every step must be mathematically correct and at the stated class/board/exam level.

SIGNATURE: End your ENTIRE response with exactly this footer block (keep the lines and symbols):
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Questions, feedback or appreciation are always welcome.
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