i want you to analyse this syllabus and tell me wh...

Here’s a definition + statement checklist from the syllabus you shared (Unit-wise). If you can write these cleanly in the exam, you’re covered for “define / state” type questions.

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Unit I

A) Contributions & biography of Indian mathematicians (short notes)

Be ready to write 2–5 lines each on:

• Bodhayan (Baudhayana) – key work (Śulba Sūtra), main mathematical idea(s)
• Apasthamb (Apastamba) – Śulba Sūtra, geometric constructions
• Katyayan (Kātyāyana) – Śulba Sūtra tradition
• Mahaveeracharya (Mahāvīra) – arithmetic/algebra contributions (Ganita-type work)
• Brahmagupta – rules for zero/negatives, important formulas
• Bhaskaracharya (Bhāskara II) – Leelavati (Lilavati), algebra/calculus-style ideas

Also know: What is “Leelavati (Lilavati)”? (a classical maths text, associated with Bhāskara II)

B) Sequences, Continuity & Differentiability (core definitions)

Sequences & series

• Sequence, general term $a_n$
• Bounded sequence
• Monotonic sequence (increasing/decreasing)
• Limit of a sequence
• Convergent / divergent sequence
• Series $\sum a_n$, partial sums $S_n$
• Convergence / divergence of a series (idea via partial sums)

Limits & continuity

• Real-valued function
• Limit of a function at a point
• Left-hand limit (LHL) and right-hand limit (RHL)
• Continuity at a point
• Continuity on an interval

Differentiability

• Derivative (definition)
• Differentiability at a point
• Relationship: differentiable ⇒ continuous (know this fact)
• Geometrical meaning of derivative (slope of tangent)
• Elementary differentiation rules (statements):
  • derivative of power, sum/difference
  • product rule, quotient rule, chain rule

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Unit II — Expansion of Functions (theorems + terms)

You should be able to state conditions + conclusion for each:

• Rolle’s Theorem (conditions: continuous, differentiable, equal end values)
• Lagrange’s Mean Value Theorem (LMVT)
• Cauchy’s Mean Value Theorem (CMVT)
• Geometrical interpretation (what it means on the graph)

Also definitions/statements:

• Successive differentiation ($f', f'', f^{(n)}$)
• Leibniz theorem (nth derivative of a product)

Taylor/Maclaurin

• Taylor’s theorem (expansion about $x=a$)
• Maclaurin’s theorem (special case $a=0$)
• “Remainder term” / error term meaning (at least conceptually)

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Unit III — Curvature, Asymptotes & Curve Tracing (definitions to memorize)

Curvature

• Curvature at a point
• Radius of curvature (if taught with it)

Asymptotes

• Asymptote (general definition)
• Vertical asymptote, horizontal asymptote, oblique (slant) asymptote
• Asymptotes parallel to axes
• Asymptotes of algebraic curves (idea: behavior as $x\to\infty$, etc.)

Shape & tracing vocabulary

• Symmetry (about x-axis, y-axis, origin)
• Concavity and convexity
• Point of inflection
• Tangent at origin
• Multiple point
• Double point (and “nature/position of double points” — node/cusp type classification if your teacher covered it)
• Cartesian curve, polar curve, parametric curve (definitions)

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Unit IV — Integration (definitions + key terms)

• Indefinite integral / antiderivative
• Constant of integration
• Definite integral (as limit of sums / area idea)
• Properties of definite integrals (common ones: reversing limits, additivity on intervals)
• Elementary integration (standard forms)
• Integration of transcendental functions (log, exp, trig, inverse trig forms—whatever your class includes)
• Reduction formula (what it means + how it’s used)

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High-priority “must write perfectly” list (most exam-friendly)

If you’re short on time, prioritize these definitions/statements first:

1. Limit (sequence + function)
2. Continuity
3. Derivative + geometrical meaning
4. Rolle’s theorem, LMVT, CMVT (full statement with conditions)
5. Taylor & Maclaurin series (form + meaning)
6. Asymptote types
7. Concavity/convexity + inflection point
8. Definite vs indefinite integral + reduction formula meaning

If you tell me your exam pattern (e.g., “2-mark definitions + 5-mark proofs” or “mostly numericals”), I can reorder this into a perfect 1-day / 1-week study checklist.

I can’t guarantee what your examiner will pick, but in this kind of syllabus 4-mark questions usually come from “define/state/short note” topics, and 10-mark questions usually come from “theorem + proof / big numerical / full curve tracing / reduction formula” topics.

Most likely 10-mark topics (long answers)

These are the highest probability from your syllabus:

Unit II (very high chance)

1. Rolle’s Theorem — statement + conditions + proof + one application problem
2. Lagrange’s Mean Value Theorem (LMVT) — statement + proof + application
3. Cauchy’s Mean Value Theorem (CMVT) — statement + proof + application
4. Taylor’s theorem / Maclaurin’s theorem — expansion + example (expand a function)

Unit III (very common for long problems)

5. Curve tracing of Cartesian/polar/parametric curves (full question: symmetry, intercepts, tangents, asymptotes, nature, sketch)
6. Asymptotes of algebraic curves (find asymptotes and use in tracing)
7. Curvature / radius of curvature (formula + compute at a point)

Unit IV (common long numerical)

8. Reduction formulae (derive + use to evaluate an integral)
9. Definite integrals (properties + evaluate using properties/substitution)

➡️ If you want the top 3 most “10-mark probable”:
LMVT/CMVT (Unit II), Taylor/Maclaurin (Unit II), Curve tracing + asymptotes (Unit III).

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Most likely 4-mark topics (short answers)

These are the topics that commonly come as define / state / explain briefly.

Unit I (definitions)

• Convergence of sequence/series (define convergent/divergent)
• Limit of a function (LHL, RHL idea)
• Continuity (at a point)
• Differentiability (definition)
• Geometrical interpretation of derivative (slope of tangent)
• Elementary differentiation (state basic rules)

Also sometimes:

• Short notes: Indian mathematicians + Leelavati context (4 marks style)

Unit II (short theorem statements)

• State Rolle / LMVT / CMVT (only statement + conditions)
• Geometrical interpretation of MVT
• Successive differentiation (meaning of $f''$, $f^{(n)}$)
• Leibniz theorem (statement)

Unit III (short definitions)

• Asymptote (definition + types)
• Symmetry (about x-axis/y-axis/origin)
• Concavity/convexity
• Point of inflection
• Tangent at origin (meaning)
• Multiple point / double point (definitions)
• Definitions of Cartesian / polar / parametric curves

Unit IV (short definitions/properties)

• Indefinite integral and constant of integration
• Definite integral (definition/meaning)
• Properties of definite integrals (any 2–3)
• Reduction formula (what it means)

➡️ If you want the top 6 most “4-mark probable”:
Continuity, Differentiability, Statement of LMVT, Taylor/Maclaurin form, Asymptote types, Concavity/Inflection point.

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Quick scoring strategy (based on marks)

• For 10 marks: prepare full format (statement → conditions → proof/derivation → 1 example).
• For 4 marks: prepare perfect textbook definitions + 1 small example/diagram (especially for MVT geometry/asymptotes/inflection).

If you tell me your university/board name + last year pattern (like “2+4+10 marks format”), I’ll rank the topics even more accurately in the exact order they usually ask.

Sure — from the syllabus in your image, these are the main “theorem + proof” items you should prepare (unit-wise). These are the ones examiners most often ask for 10 marks (or sometimes 6–8 marks) as “State and prove”.

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Unit II: Expansion of Functions (MOST important for proofs)

1) Rolle’s Theorem

• Statement + conditions + proof
• Often with 1 application problem

2) Lagrange’s Mean Value Theorem (LMVT)

• Statement + conditions + proof
• Often asked with geometrical meaning / application

3) Cauchy’s Mean Value Theorem (CMVT)

• Statement + conditions + proof
• Sometimes asked as “prove CMVT” or “derive LMVT from CMVT” (depends on teacher)

4) Leibniz Theorem (nth derivative of product)

• Usually asked as statement + proof by induction
• Then small example using it

5) Taylor’s Theorem

• Statement + proof idea (some colleges ask full proof, some ask only statement + expansion form)
• Use for expansion problems

6) Maclaurin’s Theorem

• Same as Taylor’s but at a = 0
• Sometimes asked as “State and prove Maclaurin’s theorem” (or state + derive from Taylor)

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Unit IV: Integration (proof/derivation style)

7) Reduction Formulae

• Very commonly asked as “Derive the reduction formula for …” Examples depend on your class, like:
  • $\int \sin^n x\,dx$
  • $\int \cos^n x\,dx$
  • $\int x^n e^x\,dx$
  • $\int (\ln x)^n dx$
• In exam: derivation + evaluation using it

8) Definite integral properties (sometimes as “prove”)

Not always full 10 marks, but can come as:

• Prove any property like:
  • $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
  • $\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
  • $\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$ Usually these are 4–6 marks, but some papers make them bigger.

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Unit I and Unit III

These units are mostly definitions + numericals/curve tracing, not classic “theorem proofs” (as per the text shown).

• Unit I: limits/continuity/differentiability → mainly definitions + results
• Unit III: asymptotes/curve tracing/curvature → mainly methods + problems (not long proofs typically)

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Final “must-read proof list” (most likely)

1. Rolle’s Theorem
2. LMVT
3. CMVT
4. Leibniz Theorem
5. Taylor’s Theorem
6. Maclaurin’s Theorem
7. Reduction formula derivations (at least 1–2 standard types)

If you tell me your textbook name (or send 1 more photo of the “detailed syllabus” page if it exists), I can list the exact reduction formula types your course expects.