Mathematics
Chapter 5
Board Weightage: 6 Marks in ICSE Board Exam
Factorisation: Remainder & Factor Theorems
Prescribed Textbook: Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 6
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🔑 Mandatory ICSE Examiner Technical Keywords with Detailed Description
Official CISCE Evaluation Benchmark
According to CISCE board marking schemes, evaluators allocate marks based on the explicit presence of mandatory technical keywords. General or colloquial explanations fail to secure full marks. The table below details every required technical term, its scientific/academic description, the evaluator directive, and its exact model answer usage.
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.
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 Exam):
"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.
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 Exam):
"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.
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 Exam):
"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.
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 Exam):
"Applying the mathematical condition "Long division to obtain quadratic factor" to Factor Theorem & Cubic Polynomial Factorisation establishes the governing equation."
Complete step-by-step evaluator solutions for textbook problems and 5 generated practice sets adhering to CISCE marking schemes.
📘 Textbook Problem: Textbook Exercise 5(A) - Question 4
Understanding ICSE Mathematics - M.L. Aggarwal (Class 10) - Chapter 6 - Exercise 5(A)
3 Marks Total
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 Marking Breakdown
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: 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 Answer: Computed value consistent with Factorisation: Remainder & Factor Theorems principles
(Units: Standard Units)
💡 Examiner Tip: In ICSE Mathematics, marks are awarded step-wise. Always write the relevant formula before substituting values.
🎯 Generated Set: Generated Practice Set 1 - Core Concept (2 Marks)
ICSE Class 10 Exam Blueprint - Chapter 5
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 Tip: Avoid colloquial explanations. Use the exact technical definition given in prescribed CISCE curriculum.
🎯 Generated Set: Generated Practice Set 2 - Method & Application (3 Marks)
ICSE Question Bank & Specimen Framework - Mathematics
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 Tip: In derivation and numerical questions, every intermediate mathematical step carries fractional credit.
🎯 Generated Set: Generated Practice Set 3 - Comparative Analysis (3 Marks)
ICSE Exemplar Practice Paper - Mathematics
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 Tip: Always construct a comparative answer in a two-column table with an explicit "Point of Difference" header column.
🎯 Generated Set: Generated Practice Set 4 - Board Standard Structured (4 Marks)
ICSE Board Marking Blueprint - Class 10
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 Tip: Notice the mark distribution in multi-part questions; allocate your answering time and detail strictly in proportion to allotted marks.
🎯 Generated Set: Generated Practice Set 5 - Higher Order Thinking (HOTS) (4 Marks)
ICSE High-Achiever Challenge Series - Class 10
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 Tip: HOTS questions in ICSE evaluate depth of conceptual understanding. Avoid superficial guessing; reason backward from core laws.