Chapter 1: A Square and A Cube

Chapter 1: A Square and A Cube

Study Material

• Square of a number: The square of a number n is n × n, written as n². Example: 5² = 25. (Competency: Understanding and computing squares)
• Cube of a number: The cube of a number n is n × n × n, written as n³. Example: 3³ = 27. (Competency: Understanding and computing cubes)
• Properties of squares:
  • All squares are positive numbers.
  • Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
• Properties of cubes:
  • Cubes can be positive or negative depending on the number.
  • Cube numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ...
• Perfect square: A number whose square root is a whole number. Example: 81 is a perfect square because √81 = 9. (Competency: Identifying perfect squares)
• Perfect cube: A number whose cube root is a whole number. Example: 27 is a perfect cube because ∛27 = 3. (Competency: Identifying perfect cubes)
• Patterns: The difference between consecutive square numbers increases by 2 each time: 1, 4, 9, 16 → differences 3, 5, 7, ...

Insert NCERT diagram of squares and cubes from textbook page 2.

Worksheet

I. Multiple Choice Questions (MCQs)

1. The square of 12 is:
   1. 124
   2. 144
   3. 122
   4. None of these
   (Competency: Calculating squares)
2. The cube of -4 is:
   1. -64
   2. 64
   3. -124
   4. None of these
   (Competency: Calculating cubes with negative numbers)
3. Which of the following is NOT a perfect square?
   1. 49
   2. 36
   3. 50
   4. 64
   (Competency: Identifying perfect squares)

II. Assertion–Reasoning

1. Assertion (A): The square of any odd number is odd. Reason (R): Odd × Odd = Even.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false and R is true. (Competency: Logical reasoning with properties of numbers)
2. Assertion (A): The cube of any even number is even. Reason (R): Even × Even × Even = Even.
   (a) Both A and R are true and R is the correct explanation of A.
   (b) Both A and R are true but R is not the correct explanation of A.
   (c) A is true but R is false.
   (d) A is false and R is true. (Competency: Logical reasoning with cubes)

III. True or False

1. All prime numbers are perfect squares. (Competency: Understanding perfect squares)
2. 125 is a perfect cube. (Competency: Identifying perfect cubes)
3. The square root of 81 is 9. (Competency: Computing square roots)

IV. Short Answer (1 mark)

1. Find the square of 25. (Competency: Calculating squares)
2. Find the cube of 6. (Competency: Calculating cubes)

V. 2-Mark Questions

1. Write the first five square numbers. (Competency: Listing square numbers)
2. Write the first five cube numbers. (Competency: Listing cube numbers)

VI. 3-Mark Questions

1. Find the difference between the square of 15 and the cube of 5. (Competency: Application of squares and cubes)
2. If x² = 196, find x. (Competency: Finding square roots)

VII. Long Answer (5 Marks)

1. A box contains small cubes of side 2 cm. How many such cubes can be placed in a cubical box of side 10 cm? (Competency: Application of cube and volume concepts)

VIII. Case-Based Questions (CBQ – Competency Based Questions)

Read the following and answer the questions:

A builder is making a square floor of side 12 m and wants to decorate it with cubic tiles of side 1 m. He also needs to paint a cubic water tank of side 4 m.

1. Area of the floor is:
   1. 144 m²
   2. 120 m²
   3. 100 m²
   4. None of these
   (Competency: Calculating area of a square)
2. Number of tiles needed:
   1. 144
   2. 120
   3. 100
   4. 96
   (Competency: Application of square measurement in real life)
3. Volume of the water tank is:
   1. 64 m³
   2. 48 m³
   3. 32 m³
   4. None of these
   (Competency: Calculating volume of a cube)
4. If 1 m² requires 2 liters of paint, total paint needed for the tank is:
   1. 192 liters
   2. 160 liters
   3. 128 liters
   4. None of these
   (Competency: Application of surface area of a cube)

Insert NCERT image of cube and square real-life examples from textbook page 6.