Class 12 Mathematics Notes Chapter 15 (Design of the Question Paper , Set-II) – Examplar Problems (English) Book

Alright class, let's delve into Chapter 15 of your NCERT Exemplar book for Mathematics, which focuses on the 'Design of the Question Paper, Set-II'. Now, this chapter isn't about learning new mathematical concepts. Instead, it provides a blueprint, a sample question paper (Set-II), designed according to the CBSE guidelines for your Class 12 board examination. Understanding this design is crucial for effective preparation, not just for boards but also as a foundation for various government exams that test Class 12 level mathematics.

Understanding the Purpose and Design (Based on Typical Class 12 Patterns)

The primary goal of providing a sample question paper like Set-II in the Exemplar is to familiarize you with:

1. The Structure and Pattern: How the final examination paper will look.
2. Marking Scheme: How marks are allocated to different types of questions and steps.
3. Distribution of Questions: How questions are spread across different units and topics.
4. Difficulty Level: The expected mix of easy, average, and challenging questions.
5. Types of Questions: The nature of questions asked (knowledge, understanding, application, higher-order thinking skills - HOTS).

Key Components of the Question Paper Design (General Pattern):

(Note: The exact number of questions and marks per section might vary slightly year to year based on CBSE guidelines, but the overall structure remains similar. Refer to the specific Set-II in your Exemplar for the exact details of that particular set.)

1. General Instructions: Always read these carefully. They provide information about the number of sections, compulsory questions, internal choices, and specific instructions for certain questions (like graph paper for LPP).
2. Sections: The paper is typically divided into sections (e.g., Section A, B, C, D, E or similar).
   • Very Short Answer (VSA) / Multiple Choice Questions (MCQ) / Assertion-Reasoning: Often carry 1 or 2 marks each. These test fundamental concepts, definitions, formulae, and quick calculation skills. MCQs require selecting the correct option, sometimes involving calculations. Assertion-Reasoning questions test deep conceptual understanding.
   • Short Answer Type I (SA-I): Usually carry 2 or 3 marks. These require brief explanations, calculations, or derivations.
   • Short Answer Type II (SA-II) / Long Answer Type I (LA-I): Often carry 3 or 4 marks. These involve more detailed solutions, proofs, or problem-solving steps.
   • Long Answer Type II (LA-II) / Case-Study Based: Typically carry 5 or 6 marks (Case studies might be 4 marks each). These demand comprehensive understanding, detailed derivations, solving complex problems (like LPP graphically, matrix method, integration, 3D geometry lines/planes), or analyzing a given scenario (case studies).
3. Internal Choices: To provide some flexibility, internal choices are usually provided within certain questions, especially in the higher-mark sections. You have to attempt only one of the alternatives. Choose wisely based on which concept you are more confident in.
4. Syllabus Coverage: The paper is designed to cover the entire Class 12 syllabus prescribed by NCERT/CBSE. Weightage is assigned to different units:
   • Relations and Functions
   • Algebra (Matrices, Determinants)
   • Calculus (Continuity & Differentiability, Applications of Derivatives, Integrals, Applications of Integrals, Differential Equations)
   • Vectors and Three-Dimensional Geometry
   • Linear Programming
   • Probability
     (Check the latest CBSE syllabus blueprint for exact unit-wise weightage)
5. Weightage to Difficulty Level: A balanced paper includes:
   • Easy Questions: (~15-20%) - Direct application of formulas or definitions.
   • Average Questions: (~60-70%) - Require understanding and standard problem-solving techniques.
   • Difficult Questions (HOTS): (~10-15%) - Test deeper understanding, analytical skills, and application in unfamiliar situations.
6. Time Allocation: The total time (usually 3 hours) needs to be managed effectively across different sections. The design implicitly guides time allocation based on marks.

How to Use Set-II for Preparation:

1. Analyze the Structure: Note the number of questions in each section and the marks allotted.
2. Identify Topic Weightage: Solve the paper to see which topics are frequently asked in which section (e.g., LPP is often a Long Answer question).
3. Practice Time Management: Solve Set-II (and other sample papers) under timed conditions (3 hours) to simulate the exam environment.
4. Self-Assessment: After solving, evaluate your performance. Identify weak areas, conceptual gaps, and calculation errors.
5. Understand Step-Marking: Look at the solutions (if provided) or discuss with your teacher to understand how marks are awarded for intermediate steps, formulas, diagrams, and the final answer. This is crucial for maximizing your score.
6. Familiarize with Question Types: Pay attention to the phrasing and requirements of different question types (e.g., 'Find', 'Prove', 'Show', 'Solve graphically', 'Evaluate').

Key Takeaway: Chapter 15 Set-II is not just a list of questions; it's a tool. Use it strategically to understand the exam's demands and fine-tune your preparation strategy for government exams requiring Class 12 Mathematics proficiency.

Multiple Choice Questions (MCQs based on Class 12 Syllabus):

Here are 10 MCQs covering various topics, similar in style to what you might encounter:

1. If A is a square matrix such that A² = A, then (I + A)³ - 7A is equal to:
   (a) A
   (b) I - A
   (c) I
   (d) 3A

2. The function f(x) = |x - 1| is:
   (a) Differentiable at x = 1
   (b) Continuous but not differentiable at x = 1
   (c) Neither continuous nor differentiable at x = 1
   (d) Continuous and differentiable everywhere

3. The value of the integral ∫ (dx / (x² + 2x + 2)) is:
   (a) tan⁻¹(x + 1) + C
   (b) log|x² + 2x + 2| + C
   (c) ½ log|x² + 2x + 2| + C
   (d) x tan⁻¹(x + 1) + C

4. The order and degree (if defined) of the differential equation (d²y/dx²)² + (dy/dx)³ = x sin(dy/dx) are respectively:
   (a) 2, 2
   (b) 2, 3
   (c) 2, degree not defined
   (d) 1, 3

5. If a and b are unit vectors and θ is the angle between them, then |a - b| is equal to:
   (a) sin(θ/2)
   (b) 2 sin(θ/2)
   (c) cos(θ/2)
   (d) 2 cos(θ/2)

6. The distance of the point (1, 2, 3) from the plane x + y + z = 1 is:
   (a) 5 / √3
   (b) 6 / √3
   (c) 7 / √3
   (d) 4 / √3

7. For an LPP, the objective function is Z = 4x + 3y, and the feasible region determined by a set of constraints (linear inequalities) is bounded with corner points (0, 0), (25, 0), (16, 16), and (0, 24). The maximum value of Z occurs at:
   (a) (0, 0)
   (b) (25, 0)
   (c) (16, 16)
   (d) (0, 24)

8. If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) is:
   (a) 0.24
   (b) 0.3
   (c) 0.48
   (d) 0.96

9. Let R be the relation in the set N given by R = {(a, b) : a = b - 2, b > 6}. Choose the correct answer.
   (a) (2, 4) ∈ R
   (b) (3, 8) ∈ R
   (c) (6, 8) ∈ R
   (d) (8, 7) ∈ R

10. The slope of the tangent to the curve y = x³ - x at x = 2 is:
    (a) 10
    (b) 11
    (c) 12
    (d) 6

Answer Key for MCQs:

1. (c) I
2. (b) Continuous but not differentiable at x = 1
3. (a) tan⁻¹(x + 1) + C
4. (c) 2, degree not defined (because of sin(dy/dx))
5. (b) 2 sin(θ/2)
6. (a) 5 / √3 (|1+2+3-1|/√(1²+1²+1²))
7. (c) (16, 16) (Z(0,0)=0, Z(25,0)=100, Z(16,16)=64+48=112, Z(0,24)=72)
8. (d) 0.96 (P(A∩B) = P(B|A)P(A) = 0.6 * 0.4 = 0.24. P(A∪B) = P(A)+P(B)-P(A∩B) = 0.4+0.8-0.24 = 1.2-0.24 = 0.96)
9. (c) (6, 8) (Since a = b-2 => 6 = 8-2, and b=8 > 6)
10. (b) 11 (dy/dx = 3x² - 1. At x=2, dy/dx = 3(2)² - 1 = 12 - 1 = 11)

Remember to practice solving the entire Set-II paper from your Exemplar book to get the best feel for the examination pattern. Good luck!