COMPLETE MATHS in ONE SHOT || All Chapters || Class 12 Boards || PW

Main ideas / lessons conveyed

• “One-shot” style complete revision for Class 12 Maths boards (PW/Hacker series) The speaker presents the video as a final revision plan: revise only the remaining topics/chapters not covered fully earlier, and focus on board-likely questions—especially:

  • Objective (MCQ) speed
  • Subjective (long) writing strategy

• Speed + accuracy strategy for boards

  • Objective questions: “finish in a minute.” Don’t rush subjective steps, don’t write wrongly, and don’t miss sign/± handling.
  • Subjective questions: write only the steps needed to score marks, but don’t over-write “story” and waste time.
  • Always revise by practice: attempt questions after pausing, then check how you prepared.

• Heavy emphasis on “formulas/properties + board-ready procedures” Repeated teaching pattern:

  1) the relevant theorem/property, and 2) how to apply it in exam-writing steps, plus 3) common mistakes (especially sign mistakes like forgetting ±, and matrix formula order/transposition errors).

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Methodology / instruction bullet points (as presented)

A) General board-attempt strategy

• Attempt questions after a brief watch; pause and try yourself.
• For objective questions:
  • Use quick reasoning and eliminate options.
  • Apply speed with correct logic.
• For subjective questions:
  • Don’t rush—prioritize correctness.
  • Write only what earns marks:
    • key steps,
    • the final substitution,
    • correct simplification.
  • Avoid unnecessary “full theorem whole story” unless required.
• After solving long problems (especially matrix/derivative/integration):
  • Check by substituting back where applicable—particularly for matrix-linear equation solutions.

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B) Determinants + Adjoint/Avers (inverse) workflow

Determinant questions

• Determinants are equal when the corresponding values match (often by setting two determinant expressions equal).
• Use minors/cofactors:
  • Cofactor = ((-1)^{i+j}) × minor
  • Minor is found by deleting row i and column j

Adjoint of a matrix

• Form the cofactor matrix, then transpose it to get the adjoint.

Key properties (repeated throughout)

• Determinant relations involving adjoint/determinant properties, including an explanation using the matrix order (n).

• A key relationship: [ \det(\text{adj}(A)) = (\det(A))^{n-1} ] where (n) is the order of the square matrix.

• Adjoint and determinant relationships used to derive the inverse.

• If (\det(A)\neq 0), adjoint is used to compute (A^{-1}).

Inverse of a matrix

• For a 3×3 matrix (A) with (\det(A)\neq 0): [ A^{-1}=\frac{\text{adj}(A)}{\det(A)} ]

• To solve (AX=B): [ X=A^{-1}B ] (i.e., multiply the adjoint/determinant factor with (B)).

Common mistake explicitly warned

• Forgetting ± sign or making an early wrong tick can break the entire downstream solution.

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C) Linear equations in 3 variables using matrices

Board-focused steps:

1. Convert the word problem into three linear equations in (x, y, z).

2. Write the matrix form: [ A\begin{bmatrix}x\y\z\end{bmatrix}=B ]

3. Compute (\det(A)).

4. Find the adjoint of (A).
5. Use the inverse factor and multiply (A^{-1}B) to get ((x,y,z)).
6. Verify by substituting into the original equations.

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D) Trigonometry (inverse trig) range/domain discipline

• Always respect domain/range of inverse trigonometric functions.
• For (\cos^{-1}), the answer must lie in the permitted range (quadrant/range handling stressed).
• Use standard squared identities:
  • (\csc^2\theta = 1+\cot^2\theta)
  • (\sec^2\theta = 1+\tan^2\theta)
• For inverse half-angle-style mapping:
  • If (\theta=\cos^{-1}(\cdots)), substitute correctly and keep the valid range.

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E) Applications of derivatives: rate of change + maxima/minima

Related rates method

• Build a relation (often using geometry such as Pythagoras with a ladder setup).
• Differentiate implicitly w.r.t. time.
• Substitute the given instantaneous value to find the rate.
• Interpret sign carefully:
  • “height decreasing” ⇒ report decreasing rate; apply/remove minus based on wording.

Traffic/camera case study

• Convert geometry into a (\tan^{-1}) / trigonometric expression.
• Use: [ \frac{d}{dx}\left(\tan^{-1}u\right)=\frac{u’}{1+u^2} ] and apply chain rule for “rate w.r.t. time.”

Monotonicity clarification

• “Strictly increasing” means the derivative sign does not change from left to right.
  • The derivative may be zero at a point and still be increasing if the sign doesn’t change.
• Maxima/minima occur where the derivative changes sign.

Optimization via calculus (ellipse rectangle / soccer field)

• Use the ellipse equation as a constraint.
• Express area as a single-variable function.
• Find critical points via (A’(x)=0).
• Use sign change and/or second derivative test.

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F) Integrals: area under curve / between curves / symmetry

• Interpret definite integrals as area:
  • Between curve and x-axis: integrate upper and lower parts carefully.
  • If the region covers negative (y), the integral may be negative—split intervals to compute the correct absolute area.

• Between two curves: [ \text{Area}=\int(\text{upper}-\text{lower})\,dx ]

• Use symmetry for ellipse-type regions:

  • Compute one quadrant and multiply by 4.

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G) Vectors: angle, magnitude, dot/cross, projection

Angle between vectors

[ \cos\theta=\frac{A\cdot B}{|A||B|} ]

Magnitude/dot-based questions

• Compute norms and simplify.

Projection

• Projection of A on B: [ \text{proj}_B(A)=\frac{A\cdot B}{|B|} ] (direction along (B)).

• Ratio-based forms may rely on unit vector ideas.

Cross product

• Magnitude: [ |A\times B|=|A||B|\sin\theta ]

• Parallelogram area: [ |A\times B| ]

• Triangle area: [ \frac{1}{2}|A\times B| ]

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H) Linear Programming (LP): feasible region + corner-point method

• Maximize/minimize objective with constraints:
  • Graph constraints.
  • Identify the feasible region (intersection).
• If the feasible region is closed:
  • Maximum/minimum occurs at a corner point.
• If feasible region is unbounded/open-ended:
  • Maximum may not exist; for minimum, boundary checking and comparison along feasible boundaries is needed.

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I) Relations & functions + continuity/differentiability basics

Continuity

• Must satisfy:
  • left limit,
  • right limit,
  • and the function value.

Differentiability

• If the derivative exists ⇒ the function is continuous.
• At corners/cusps, it becomes non-differentiable.

Function types

• One-one (injective), onto (surjective), bijective.
• Domain/codomain vs range emphasis.

Relations

• Reflexive: ((a,a)) always included.
• Symmetric: if ((a,b)) then ((b,a)).
• Transitive: if ((a,b)) and ((b,c)) then ((a,c)).
• Equivalence relation = reflexive + symmetric + transitive.
  • Also emphasized in explanation flow: reflexive + transitive often appear in “equivalence context.”

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J) Probability

• Conditional probability: [ P(A|B)=\frac{P(A\cap B)}{P(B)} ]

• Disjoint events:

  • If (A\cap B=0) (probability sense), events are mutually exclusive.
• Independence:
  • Standard theme: independent events allow multiplying probabilities.
• Conditional “truth/contradiction” logic:
  • Uses complement cases and conditional probability reasoning.

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Speakers / sources featured (as present in the subtitles)

• “Brother” / main instructor (primary speaker) — dominant voice throughout, teaching solutions and strategies.
• Sachin Sir — mentioned as teaching certain parts/chapters earlier in the series.
• Tarun Sir — mentioned as covering the remaining portion, especially in the later “marathon” part.
• NCERT — referenced as a curriculum source for definitions/properties (not a person).