Grade 12 Relations and Functions Exercise 1.2

Grade 12 Relations and Functions Exercise 1.2 (Solutions)

Grade 12 Relations and Functions Exercise 1.2 (All Questions & Solutions)

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Focus: Exercise 1.2 — roster, domain, range, inverse relations

Grade 12 Relations and Functions Exercise 1.2

Q1. If A = {1,2,3}, write the relation R = {(a,b) : a ≤ b} in roster form.

All ordered pairs (a,b) with a ≤ b and a,b ∈ {1,2,3} are:

R = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}

Q2. Write in roster form: R = {(x,y): y = x+1, x ∈ ℤ, 0 < x < 3}.

Values of x satisfying 0 < x < 3 are x = 1,2. So

R = {(1,2), (2,3)}

Q3. Let A = {2,4,6}, B = {1,3}. Write R = {(x,y): x ∈ A, y ∈ B, x > y} in roster form.

Compare each x ∈ A with each y ∈ B:

R = {(2,1), (4,1), (4,3), (6,1), (6,3)}

Q4. Write the domain and range of R = {(1,2),(2,3),(3,4),(4,5)}.

Domain (set of first coordinates) = {1,2,3,4}.

Range (set of second coordinates) = {2,3,4,5}.

Q5. Let R = {(x,y) : x,y ∈ ℤ, x - y = 3}. Write domain and range when 0 < x < 6.

Possible x: 1,2,3,4,5. For each such x, y = x – 3.

Pairs: (1,-2), (2,-1), (3,0), (4,1), (5,2)

Domain = {1,2,3,4,5}. Range = {-2,-1,0,1,2}.

Q6. Let A = {1,2,3,4}. Write R = {(x,y): x is prime and y is composite} in roster form.

Primes in A: 2,3. Composites in A: 4 (1 is neither prime nor composite).

R = {(2,4), (3,4)}

Q7. Write inverse relation R⁻¹ for R = {(a,b): a = b + 2}.

By definition, R⁻¹ = {(b,a): (a,b) ∈ R}. Since a = b+2 in R, swapping gives:

R⁻¹ = {(x,y): y = x + 2}

(Equivalently: if (a,b) ∈ R then (b,a) ∈ R⁻¹.)

Q8. If R = {(1,2),(2,3),(3,1)}, find R⁻¹.

Swap each ordered pair:

R⁻¹ = {(2,1), (3,2), (1,3)}

Q9. Let A={1,2,3}, B={4,5}, and R = A × B. Write domain and codomain.

Cartesian product A×B = {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}.

Domain = A = {1,2,3}. Codomain = B = {4,5}. (Actual range = {4,5}.)

Q10. Let R = {(x,y): y = x², x ∈ {-2,-1,0,1,2}}. Write the range of R.

Compute y for each x: (-2)²=4, (-1)²=1, 0²=0, 1²=1, 2²=4.

Range = {0,1,4}

Q11. If A = {2,4,6,8}, write R = {(x,y): x divides y} in roster form.

List all pairs (x,y) in A×A where x | y:

R = {(2,2),(2,4),(2,6),(2,8), (4,4),(4,8), (6,6), (8,8)}

Q12. If R = {(1,1),(2,4),(3,9)}, find domain and range.

Domain = {1,2,3}. Range = {1,4,9}.

Q13. Let A = {-3,-2,-1,0,1,2,3}. Write R = {(x,y): y = |x|} in roster form.

Compute y = |x| for each x:

R = {(-3,3), (-2,2), (-1,1), (0,0), (1,1), (2,2), (3,3)}

Q14. If A = {1,2,3}, write all possible relations on A containing exactly 3 ordered pairs.

There are C(9,3) = 84 distinct relations on A that contain exactly 3 ordered pairs (since A×A has 9 pairs and a relation with 3 pairs is just any 3-element subset).

Listing all 84 is lengthy. A few examples of 3-pair relations:

  • {(1,1),(1,2),(1,3)}
  • {(1,2),(2,1),(3,3)}
  • {(2,2),(3,1),(3,2)}

If you want, I can generate and show the full list of 84 relations (I can paste them or give a downloadable file).

Q15. Let R = {(x,y): x = y - 3}. Find R⁻¹.

If x = y – 3 then y = x + 3. Swapping coordinates gives the inverse:

R⁻¹ = {(x,y): y = x - 3} (equivalently, if (a,b) ∈ R then (b,a) ∈ R⁻¹).

Check: Original (1,4) (since 1 = 4-3) ⇒ inverse includes (4,1) which satisfies 1 = 4-3.

Q16. If R = {(1,2),(2,1),(2,3)}, write all ordered pairs of R⁻¹.

Swap each pair:

R⁻¹ = {(2,1),(1,2),(3,2)}

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Unit I: Diversity of Living Organisms

  1. The Living World
  2. Biological Classification
  3. Plant Kingdom
  4. Animal Kingdom

Unit II: Structural Organisation in Animals and Plants
5. Morphology of Flowering Plants
6. Anatomy of Flowering Plants
7. Structural Organisation in Animals

Unit III: Cell: Structure and Function
8. Cell – The Unit of Life
9. Biomolecules
10. Cell Cycle and Cell Division

Unit IV: Plant Physiology
11. Transport in Plants
12. Mineral Nutrition
13. Photosynthesis in Higher Plants
14. Respiration in Plants
15. Plant Growth and Development

Unit V: Human Physiology
16. Digestion and Absorption
17. Breathing and Exchange of Gases
18. Body Fluids and Circulation
19. Excretory Products and their Elimination
20. Locomotion and Movement
21. Neural Control and Coordination
22. Chemical Coordination and Integration

Class 11 Biology NCERT Solution – New NCERT Syllabus (2024–25)

These 14 chapters are included in the official CBSE syllabus for the 2024–25 board exam.

Unit I: Diversity of Living Organisms

  1. The Living World
  2. Biological Classification
  3. Plant Kingdom
  4. Animal Kingdom

Unit II: Structural Organisation in Animals and Plants

  1. Morphology of Flowering Plants

Note: “Anatomy of Flowering Plants” and “Structural Organisation in Animals” have been removed from the CBSE syllabus.

Unit III: Cell Structure and Function

  1. Cell: The Unit of Life
  2. Biomolecules
  3. Cell Cycle and Cell Division

Unit IV: Plant Physiology

  1. Photosynthesis in Higher Plants
  2. Respiration in Plants
  3. Plant Growth and Development

Unit V: Human Physiology

  1. Breathing and Exchange of Gases
  2. Body Fluids and Circulation
  3. Excretory Products and Their Elimination

Chapters like “Locomotion and Movement”, “Neural Control and Coordination”, and “Chemical Coordination and Integration” are excluded from the new syllabus.

Total Chapters in New Syllabus: 14

These are the only chapters needed for the 2024–25 CBSE board exam.

Chapters in NCERT Textbook But Excluded from NEW CBSE Syllabus (2024–25)

These chapters are part of the NCERT book but are not included in the CBSE 2024–25 exam syllabus. Still important for NEET.

Chapters Excluded from CBSE Class 11 Biology Syllabus (2024–25)

  1. Anatomy of Flowering Plants
  2. Structural Organisation in Animals
  3. Locomotion and Movement
  4. Neural Control and Coordination
  5. Chemical Coordination and Integration

Important Notes:

  • These chapters still exist in the NCERT textbook, but are not required for the CBSE 2024–25 exam.
  • Do not prepare or publish these chapters as part of the main syllabus for CBSE board content.

Important Notes:

  • Only 14 chapters are to be studied for the Class 11 CBSE exam (2024–25).
  • The remaining 5 chapters are not for board exams, but are useful for NEET and concept clarity.
NEET BIOLOGY

Class-wise Solutions

Class 12:

Class 12 Physics – NCERT Solutions

Class 11:

Class 10:

Class 9:

Class 8:

Class 7:

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Subject-wise Solutions

Physics:

Chemistry:

Biology:

Math:

Science:

NEET BIOLOGY

For additional reference and to access the official NCERT Class 10 Maths textbook, visit the NCERT website. This will help you understand the concepts covered in Class 10 Math Ch 1 Real Numbers Ex 1.1 more effectively.

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Grade 12 Relations and Functions Exercise 1.2