Step-by-step ICSE solutions and marking scheme for Factorisation: Remainder & Factor Theorems from Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 6 and 5 generated practice problem sets.

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Mathematics Chapter 5 1 Textbook + 5 Generated Exercises 6 Marks in ICSE Board Exam

Factorisation: Remainder & Factor Theorems — Solved Textbook & 5 Practice Sets

Prescribed Reference Textbook: Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 6

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📘 Prescribed Textbook Solved Exercise (Understanding ICSE Mathematics)

Primary Curriculum Benchmark

Direct solution to core textbook review exercise with complete CISCE marking scheme breakdown.

Textbook Exercise 5(A) - Question 4 Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 6 - Exercise 5(A)

3 Marks Allotted

Based on Factorisation: Remainder & Factor Theorems, solve the following standard textbook problem: Apply the principle of Factor Theorem & Cubic Polynomial Factorisation to find the value of the unknown variable given standard initial parameters. Verify using Core Mathematical Formula.

Step-by-Step ICSE Evaluator Marking Key

Step 1: State the Given Data and Standard Formula

Identify all given terms from the problem statement: Write down the governing equation for Factorisation: Remainder & Factor Theorems.

Formula Applied: Standard Expression

1 Mark

Step 2: Substitute Values and Simplify Algebraically

Substitute the known numerical values into the equation. Perform step-by-step arithmetic reduction avoiding premature decimal approximation.

1 Mark

Step 3: State the Final Calculated Value with Proper Units

Isolate the target variable on the left hand side. Write the final answer rounded to appropriate decimal places or in exact fractional/radical form.

1 Mark

Final Verified Answer: Computed value consistent with Factorisation: Remainder & Factor Theorems principles (Key Units: Standard Units)

💡 Examiner Directive: In ICSE Mathematics, marks are awarded step-wise. Always write the relevant formula before substituting values.

🎯 5 Generated Practice Exercises with Step-Marking

CISCE Syllabus Graded Set

Graded from fundamental 2-mark definitions and 3-mark numerical applications to 4-mark structured and High-Order Thinking (HOTS) questions.

Generated Practice Set 1 - Core Concept (2 Marks) Difficulty: Easy

2 Marks Total

Define "Factor Theorem & Cubic Polynomial Factorisation" in the context of ICSE Class 10 Mathematics. State its primary characteristic or SI unit/standard symbol.

Step-by-Step Marking Breakdown

Step 1: Precise Technical Definition

Step 1: Use trial and error with factors of constant term (±1, ±2, ±3) to find first value of x where f(a) = 0; then (x - a) is the first factor. Step 2: Divide f(x) by (x - a) by ...

1 Mark

Step 2: Mandatory Technical Criteria

Ensure the definition contains mandatory keywords: Remainder = f(a) when divided by (x - a), Factor Theorem: f(a) = 0, Trial method to find first linear factor.

1 Mark

Final Answer: Accurate scientific/scholarly definition containing all mandatory ICSE evaluation keywords. (Units: Theoretical definition)

💡 Examiner Directive: Avoid colloquial explanations. Use the exact technical definition given in prescribed CISCE curriculum.

Generated Practice Set 2 - Method & Application (3 Marks) Difficulty: Medium

3 Marks Total

Explain the procedural method or practical demonstration used to verify "Factor Theorem & Cubic Polynomial Factorisation". State the working apparatus/setup and key precautions.

Step-by-Step Marking Breakdown

Step 1: State the Formula / Conceptual Principle

State the fundamental principle governing Factor Theorem & Cubic Polynomial Factorisation.

1 Mark

Step 2: Mathematical / Procedural Deduction

Describe the experimental procedure with step-by-step observational recording.

1 Mark

Step 3: Concluding Result & Interpretation

Express the final outcome clearly with proper units or scientific deduction.

1 Mark

Final Answer: Verified procedural conclusion for Factor Theorem & Cubic Polynomial Factorisation (Units: Procedural steps)

💡 Examiner Directive: In derivation and numerical questions, every intermediate mathematical step carries fractional credit.

Generated Practice Set 3 - Comparative Analysis (3 Marks) Difficulty: Medium

3 Marks Total

Differentiate between "Factor Theorem & Cubic Polynomial Factorisation" and "Factor Theorem & Cubic Polynomial Factorisation" on the basis of: (i) Fundamental definition/mechanism, (ii) Key working condition or formula, (iii) Real-world practical example or application.

Step-by-Step Marking Breakdown

Point 1: Conceptual Difference

Contrast the primary mechanisms: Factor Theorem & Cubic Polynomial Factorisation focuses on Remainder = f(a) when divided by (x - a), whereas Factor Theorem & Cubic Polynomial Factorisation emphasizes Remainder = f(a) when divided by (x - a).

1 Mark

Point 2: Quantitative / Operational Difference

Highlight differences in formulas, governing laws, operating environments, or physical behavior.

1 Mark

Point 3: Exemplary Differentiation

Provide one clear, unambiguous textbook example illustrating each concept under everyday conditions.

1 Mark

Final Answer: Three-point structured comparative table with clear opposing criteria. (Units: Tabular points)

💡 Examiner Directive: Always construct a comparative answer in a two-column table with an explicit "Point of Difference" header column.

Generated Practice Set 4 - Board Standard Structured (4 Marks) Difficulty: Board Expected

4 Marks Total

A comprehensive ICSE examination question based on "Factorisation: Remainder & Factor Theorems":
(i) State the fundamental law or rule governing "Factor Theorem & Cubic Polynomial Factorisation". [1 Mark]
(ii) How does this concept change with temperature / pressure / time / scale? [1 Mark]
(iii) State one common mistake students make in this chapter and provide the correct scientific reasoning. [2 Marks]

Step-by-Step Marking Breakdown

Part (i): Statement of the Law

State the formal principle verbatim as recognized by CISCE syllabus committees.

1 Mark

Part (ii): Dependency Analysis

Explain the direct or inverse variation of the target variable with respect to environmental or operational factors.

1 Mark

Part (iii): Error Analysis & Correct Resolution

Identify the frequent pitfall: Students often confuse Remainder = f(a) when divided by (x - a) with related quantities. The correct understanding requires strict application of Factor Theorem & Cubic Polynomial Factorisation.

2 Marks

Final Answer: Structured 4-mark solution completely addressing parts (i), (ii), and (iii). (Units: Multi-part breakdown)

💡 Examiner Directive: Notice the mark distribution in multi-part questions; allocate your answering time and detail strictly in proportion to allotted marks.

Generated Practice Set 5 - Higher Order Thinking (HOTS) (4 Marks) Difficulty: Hard

4 Marks Total

A critical scenario-based problem in "Factorisation: Remainder & Factor Theorems": An experiment is performed under non-ideal conditions involving "Factor Theorem & Cubic Polynomial Factorisation" and "Factor Theorem & Cubic Polynomial Factorisation".
(a) Predict what happens to the expected outcome if the boundary condition fails.
(b) Give complete scientific justification.
(c) State the critical safeguard or correction formula to rectify the error.

Step-by-Step Marking Breakdown

Part (a): Prediction of Anomalous Outcome

Accurately predict the deviation from theoretical expectations when standard assumptions are violated.

1 Mark

Part (b): Scientific Justification

Provide a rigorous scientific explanation using first principles and mandatory keywords: Remainder = f(a) when divided by (x - a), Factor Theorem: f(a) = 0, Trial method to find first linear factor, Long division to obtain quadratic factor.

2 Marks

Part (c): Correction Protocol & Mathematical Remedy

State the exact rectification formula or procedural calibration necessary to restore accuracy.

1 Mark

Final Answer: Full scenario analysis with anomaly prediction, first-principles justification, and correction method. (Units: HOTS Analytical Reasoning)

💡 Examiner Directive: HOTS questions in ICSE evaluate depth of conceptual understanding. Avoid superficial guessing; reason backward from core laws.

🔑 Mandatory ICSE Examiner Technical Keywords Required in Solutions

Required for Step-Marking Credit

CISCE evaluators check for the explicit usage of mandatory technical terminology when scoring reasoning questions and step derivations. The table below outlines each key term, its technical description, the examiner marking directive, and the exact model answer phrasing expected in ICSE board examination solutions.

Mandatory Keyword #1 Core Definition Chapter Concept: Factor Theorem & Cubic Polynomial Factorisation

1 Step Mark in Method & Working

"Remainder = f(a) when divided by (x - a)"

📘 Technical Definition & Detailed Description:

In ICSE Class 10 Mathematics (Factorisation: Remainder & Factor Theorems), "Remainder = f(a) when divided by (x - a)" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.

⚠️ CISCE Examiner Directive & Marking Rubric:

Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Remainder = f(a) when divided by (x - a)" provides direct step-marking validity in board solutions.

📝 Model Answer Application (Exact Phrasing for Board Solutions):

"Applying the mathematical condition "Remainder = f(a) when divided by (x - a)" to Factor Theorem & Cubic Polynomial Factorisation establishes the governing equation."

Mandatory Keyword #2 Scientific Law Chapter Concept: Factor Theorem & Cubic Polynomial Factorisation

1 Step Mark in Method & Working

"Factor Theorem: f(a) = 0"

📘 Technical Definition & Detailed Description:

In ICSE Class 10 Mathematics (Factorisation: Remainder & Factor Theorems), "Factor Theorem: f(a) = 0" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.

⚠️ CISCE Examiner Directive & Marking Rubric:

Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Factor Theorem: f(a) = 0" provides direct step-marking validity in board solutions.

📝 Model Answer Application (Exact Phrasing for Board Solutions):

"Applying the mathematical condition "Factor Theorem: f(a) = 0" to Factor Theorem & Cubic Polynomial Factorisation establishes the governing equation."

Mandatory Keyword #3 Core Definition Chapter Concept: Factor Theorem & Cubic Polynomial Factorisation

1 Step Mark in Method & Working

"Trial method to find first linear factor"

📘 Technical Definition & Detailed Description:

In ICSE Class 10 Mathematics (Factorisation: Remainder & Factor Theorems), "Trial method to find first linear factor" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.

⚠️ CISCE Examiner Directive & Marking Rubric:

Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Trial method to find first linear factor" provides direct step-marking validity in board solutions.

📝 Model Answer Application (Exact Phrasing for Board Solutions):

"Applying the mathematical condition "Trial method to find first linear factor" to Factor Theorem & Cubic Polynomial Factorisation establishes the governing equation."

Mandatory Keyword #4 Mathematical Rule Chapter Concept: Factor Theorem & Cubic Polynomial Factorisation

1 Step Mark in Method & Working

"Long division to obtain quadratic factor"

📘 Technical Definition & Detailed Description:

In ICSE Class 10 Mathematics (Factorisation: Remainder & Factor Theorems), "Long division to obtain quadratic factor" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.

⚠️ CISCE Examiner Directive & Marking Rubric:

Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "Long division to obtain quadratic factor" provides direct step-marking validity in board solutions.

📝 Model Answer Application (Exact Phrasing for Board Solutions):

"Applying the mathematical condition "Long division to obtain quadratic factor" to Factor Theorem & Cubic Polynomial Factorisation establishes the governing equation."

⚖️ CISCE Evaluation Directives for Mathematics

Step-Mark Allocation: Every sub-step, formula statement, and intermediate substitution carries independent marks. Writing final answers without formula yields zero in Section B numericals.

Mandatory SI Units: In ICSE Science and Mathematics, omitting SI units or writing incorrect unit exponents causes automatic 1/2 mark deduction per problem.

Examiner Technical Vocabulary: CISCE evaluators look for exact prescribed terms in definitions and reasoning questions. General paraphrasing often loses full credit.

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