Class 12 Mathematics Notes Chapter 14 (Design of the Question Paper , Set-I) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 14 from your NCERT Exemplar book for Class 12 Mathematics. Now, remember, this chapter isn't about learning a new mathematical concept like calculus or probability. Instead, it's titled 'Design of the Question Paper, Set-I', and it provides a blueprint, a sample model, of what your final examination paper might look like. Understanding this design is crucial for smart preparation, especially for competitive government exams where pattern familiarity is key.

Think of this chapter as a strategic guide. It helps you understand:

1. The Structure and Pattern: How the paper is divided into sections.
2. Marks Distribution: Weightage given to different types of questions (VSA, SA, LA).
3. Chapter Weightage (Indicative): Which chapters tend to carry more marks (though this can vary slightly year to year).
4. Difficulty Level: The mix of easy, average, and challenging questions.
5. Typology of Questions: The kinds of skills tested (Remembering, Understanding, Application, HOTS).

Let's break down the typical design presented in Set-I (based on the general pattern followed over the years, which this Exemplar set reflects):

Detailed Notes on Design of the Question Paper (Based on Set-I Principles)

1. General Structure:

   • Total Marks: Usually 80 or 100 (The exemplar might be based on 100, but board patterns can change, often to 80 marks theory + 20 internal assessment. Focus on the proportions).
   • Time Allowed: Typically 3 Hours.
   • Sections: The paper is generally divided into sections:
     • Section A: Very Short Answer (VSA) type questions or Multiple Choice Questions (MCQs). Often 1 mark each. Tests basic concepts and recall. (Number of questions varies, e.g., 10-20).
     • Section B: Short Answer (SA-I) type questions. Often 2 marks each. Require some calculation or a couple of steps. (e.g., 6-8 questions).
     • Section C: Long Answer (LA-I) type questions. Often 4 marks each. Require more detailed solutions, derivations, or multi-step problem-solving. (e.g., 6-8 questions).
     • Section D: Long Answer (LA-II) type questions. Often 6 marks each. These are typically comprehensive problems, often involving proofs, detailed applications, or combining concepts from different areas. (e.g., 4-6 questions).
   • Internal Choice: Choices are usually provided within certain questions, particularly in the longer answer sections (Sections C and D). You don't get a choice between all questions but within specific question numbers (e.g., "Solve Question 15 OR Solve the alternative Question 15").

2. Typology and Weightage (Approximate % based on typical patterns):

   • Remembering & Understanding (Knowledge-Based): ~25-35%. These test your recall of definitions, formulae, basic concepts, and properties. (Often dominate Section A & B).
   • Application: ~40-50%. These require you to apply concepts and formulae to solve problems. (Common in Sections B, C, and D).
   • Higher Order Thinking Skills (HOTS) / Analysis / Evaluation: ~15-25%. These are more challenging questions that might require combining multiple concepts, non-routine thinking, or deeper analysis. (Often found in Sections C and D, sometimes as tricky MCQs).

3. Indicative Chapter-wise Weightage (Observe this carefully in Set-I):

   • Relations and Functions (including ITF): Moderate weightage. Expect questions on types of relations/functions, composition, inverse, properties of ITF.
   • Algebra (Matrices & Determinants): High weightage. Properties of determinants, solving linear equations, matrix operations, inverse. Often includes a long answer question.
   • Calculus (Continuity, Differentiability, Applications of Derivatives, Integrals, Applications of Integrals, Differential Equations): Highest weightage (often 40-50% of the paper). This is the backbone. Expect a wide variety of questions from all sub-topics, including long answer questions on maxima/minima, area under curves, solving differential equations.
   • Vectors and 3-D Geometry: High weightage. Dot/cross products, lines, planes, shortest distance. Often includes long answer questions.
   • Linear Programming (LPP): Usually one dedicated question, often a long answer (4 or 6 marks) involving formulation and graphical solution. Relatively easy to score if practiced well.
   • Probability: Moderate to High weightage. Conditional probability, Bayes' theorem, probability distributions, binomial distribution. Can include application-based or case-study type questions.

4. Key Takeaways for Preparation:

   • No Skipping Chapters: Every chapter carries some weight. While focusing on high-weightage areas like Calculus and Vectors/3D is important, don't completely ignore others.
   • Concept Clarity: Rote learning isn't enough. Focus on understanding the underlying concepts, theorems, and formulae.
   • Practice Diverse Questions: Solve problems ranging from basic recall (for Section A) to complex applications and HOTS (for Sections C & D). The Exemplar book itself is excellent for this.
   • Time Management: Solving Set-I (and other sample papers) under timed conditions is crucial practice for the actual exam.
   • Presentation: Learn to present your answers clearly, especially for longer questions. Draw diagrams where needed (LPP, 3D Geometry, Area under curves). Write formulae used.
   • Internal Choice Strategy: During the exam, quickly assess both options in an internal choice question and attempt the one you are more confident about or can solve faster.

Important Note: The design in Set-I is a model. The exact number of questions per section or the precise marks distribution might slightly differ in your actual government exam or board exam, but the overall structure, typology, and relative chapter weightage usually remain consistent with the NCERT framework.

Multiple Choice Questions (MCQs)

Here are 10 MCQs representative of the type you might find, covering various topics as reflected in the sample paper's design:

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 value of sin⁻¹(cos(3π/5)) is:
   (A) π/10
   (B) 3π/5
   (C) -π/10
   (D) -3π/5

3. If y = log(cos(eˣ)), then dy/dx is:
   (A) -eˣ tan(eˣ)
   (B) eˣ sin(eˣ)
   (C) -eˣ sin(eˣ)
   (D) eˣ tan(eˣ)

4. The interval in which the function f(x) = 2x³ + 9x² + 12x – 1 is decreasing is:
   (A) [-1, ∞)
   (B) [-2, -1]
   (C) (-∞, -2]
   (D) [-1, 1]

5. The value of the integral ∫(from 0 to π/2) [sin(x) / (sin(x) + cos(x))] dx is:
   (A) π/2
   (B) π/4
   (C) 0
   (D) π

6. The order and degree (if defined) of the differential equation d²y/dx² + (dy/dx)¹ᐟ³ + x¹ᐟ⁴ = 0 are respectively:
   (A) 2, 3
   (B) 2, 1
   (C) 2, not defined
   (D) 1, 3

7. The projection of the vector a = 2i + 3j + 2k on the vector b = i + 2j + k is:
   (A) 10/√6
   (B) 10/6
   (C) √6/10
   (D) 8/√6

8. The distance of the plane 2x – 3y + 6z + 7 = 0 from the point (2, -3, -1) is:
   (A) 1
   (B) 2
   (C) 7
   (D) 4

9. Two events A and B will be independent if:
   (A) A and B are mutually exclusive
   (B) P(A'B') = [1 – P(A)][1 – P(B)]
   (C) P(A) = P(B)
   (D) P(A) + P(B) = 1

10. In a Linear Programming Problem, the objective function is always:
    (A) Quadratic
    (B) Cubic
    (C) Linear
    (D) Constant

Answers to MCQs:

1. (C) I
2. (C) -π/10
3. (A) -eˣ tan(eˣ)
4. (B) [-2, -1]
5. (B) π/4
6. (C) 2, not defined (because of the fractional power on the derivative)
7. (A) 10/√6
8. (A) 1 (Use formula |Ax1+By1+Cz1+D|/√(A²+B²+C²))
9. (B) P(A'B') = [1 – P(A)][1 – P(B)] (This is a condition for independence)
10. (C) Linear

Study this design analysis carefully and use Set-I as a realistic practice tool. Good luck with your preparation!