CLASS 8 NCERT WORKSHEET CH-1 squares and square roots WITH ANSWERKEY

CLASS 8 WORKSHEET 
CH-1 squares and square roots

Worksheet: Exploring Squares and Square Roots

Instructions: Read each question carefully and use the information from the provided text to answer. Show your work where applicable.

Part A: Understanding Squares

1. Define a "square number" or "perfect square" in your own words.

   
   

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2. The area of a square is found by multiplying its side length by itself.

• If a square has a side length of 5 units, what is its area? Express this using square notation. Area = _______________________
• What notation do we use for any number 'n' multiplied by itself? n × n = __________

3. List all the perfect squares between 1 and 100.

   
   

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Part B: Properties of Perfect Squares

1. What are the only possible digits that a perfect square can end with (its units place)?

   
   

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2. Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.

   • (i) 2032
   • (ii) 1027
   • (iii) 345
   • (iv) 576
   • (v) 2048
   • (vi) 1089

   Not Perfect Squares: ____________________________________________________  Reasoning:____________________________________________________________________

   
   

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3. Observe the pattern for the number of zeros at the end of a number and its square.

   • If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end?

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   • What can you conclude about the number of zeros at the end of any perfect square?

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4. Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:

   • 1 = 1 = 1²
   • 1 + 3 = 4 = 2²
   • 1 + 3 + 5 = 9 = 3²
   • 1 + 3 + 5 + 7 = __________ = __________
   • 1 + 3 + 5 + 7 + 9 = __________ = __________
   • 1 + 3 + 5 + 7 + 9 + 11 = __________ = __________

5. Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².

   • What is the 36th odd number? (Hint: The nth odd number is 2n-1)

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   • Calculate 36²: 36² = 35² + (36th odd number) = _______________________

6. How many numbers lie between the squares of 16 and 17?

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Part C: Finding Square Roots

1. Define "square root" and state how it is denoted.

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2. Every perfect square has two integer square roots. What are they for the number 64?

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3. Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps)

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   Is 1156 a perfect square? __________ If yes, √1156 = __________

4. Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = _______________________
   
   
   

5. Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares.

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6. Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut?

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Part D: Challenge Question

1. Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.

   • Person 1 opens every locker.
   • Person 2 toggles every 2nd locker.
   • Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.

   Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors).

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   Which specific locker numbers (up to 100) would remain open?

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Answer Key: Exploring Squares and Square Roots

Part A: Understanding Squares

1. Define a "square number" or "perfect square" in your own words. A square number, or perfect square, is a number obtained by multiplying a number by itself. It can also be described as the area of a square figure. Example definition: A perfect square is a number that results from multiplying an integer by itself. For instance, 4 is a perfect square because 2 × 2 = 4.

2. The area of a square is found by multiplying its side length by itself.

   • If a square has a side length of 5 units, what is its area? Express this using square notation. Area = 5 × 5 = 25 square units, or 5² = 25.
   • What notation do we use for any number 'n' multiplied by itself? n × n = n².

3. List all the perfect squares between 1 and 100. The perfect squares between 1 and 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Part B: Properties of Perfect Squares

1. What are the only possible digits that a perfect square can end with (its units place)? Perfect squares can only end with the digits 0, 1, 4, 5, 6, or 9.

2. Which of the following numbers are not perfect squares? Explain your reasoning based on their unit digits.

   • (i) 2032
   • (ii) 1027
   • (iii) 345
   • (iv) 576
   • (v) 2048
   • (vi) 1089

   Not Perfect Squares: (i) 2032, (ii) 1027, (v) 2048. Reasoning: If a number ends with 2, 3, 7, or 8, then it is definitely not a square.

3. Observe the pattern for the number of zeros at the end of a number and its square.

   • If a number ends with three zeros (e.g., 1000), how many zeros will its square have at the end? Six zeros (e.g., 1000² = 1,000,000).
   • What can you conclude about the number of zeros at the end of any perfect square? Perfect squares can only have an even number of zeros at the end.

4. Complete the following pattern showing the sum of consecutive odd numbers resulting in square numbers:

   • 1 = 1 = 1²
   • 1 + 3 = 4 = 2²
   • 1 + 3 + 5 = 9 = 3²
   • 1 + 3 + 5 + 7 = 16 = 4²
   • 1 + 3 + 5 + 7 + 9 = 25 = 5²
   • 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6²

5. Given that 35² = 1225, use the pattern of adding consecutive odd numbers to find 36².

   • What is the 36th odd number? (Hint: The nth odd number is 2n-1) The nth odd number is 2n–1. So, the 36th odd number is (2 × 36) - 1 = 71.
   • Calculate 36²: 36² = 35² + (36th odd number) = 1225 + 71 = 1296.

6. How many numbers lie between the squares of 16 and 17? This information is not directly provided in the source material in an explicit statement. However, based on the pattern n² and (n+1)², the numbers between them are (n+1)² - n² - 1 = (n² + 2n + 1) - n² - 1 = 2n. So, between 16² and 17² (where n=16), there are 2 × 16 = 32 numbers [16 (Implied by the question type)].

Part C: Finding Square Roots

1. Define "square root" and state how it is denoted. If y = x², then x is the square root of y. It is the inverse operation of squaring a number. The square root of a number is denoted by the symbol √.

2. Every perfect square has two integer square roots. What are they for the number 64? The square roots of 64 are +8 and –8.

3. Determine if 1156 is a perfect square using prime factorization. If it is, find its square root. (Show your prime factorization steps) Prime factorization of 1156: 1156 = 2 × 578 578 = 2 × 289 289 = 17 × 17 So, 1156 = 2 × 2 × 17 × 17.

   • Can factors be grouped into pairs? Yes, (2 × 2) × (17 × 17). This can be grouped as (2 × 17) × (2 × 17).
   • Is 1156 a perfect square? Yes.
   • If yes, √1156 = 2 × 17 = 34.

4. Find the square root of 441 m² (area of a square) to determine the length of its side. Length of side = √441 = 21 m.

5. Estimate the square root of 250 without calculating it exactly. Explain your reasoning by identifying the closest perfect squares. We know that 15² = 225 and 16² = 256. Therefore, the square root of 250 is between 15 and 16. Since 256 is much closer to 250 than 225, the square root of 250 is approximately 16 (but less than 16).

6. Akhil has a square piece of cloth with an area of 125 cm². He wants to cut out the largest possible square handkerchief with an integer side length. What is the maximum side length he can cut? The nearest perfect square less than 125 is 121 (which is 11²). Therefore, the maximum side length he can cut with an integer side length is 11 cm.

Part D: Challenge Question

1. Recall the puzzle given by Queen Ratnamanjuri involving 100 lockers.

   • Person 1 opens every locker.
   • Person 2 toggles every 2nd locker.
   • Person 3 toggles every 3rd locker, and so on, until all 100 people have taken their turn.

   Khoisnam immediately knew which lockers would remain open at the end. How did he know? (Hint: Think about how many times each locker is toggled and what type of numbers have an odd number of factors). Khoisnam knew that a locker would remain open only if it was toggled an odd number of times. The number of times a locker is toggled is equal to the number of its factors. Most numbers have an even number of factors because factors usually come in pairs (e.g., for 6, factors are 1&6, 2&3). However, numbers that are perfect squares have an odd number of factors because one factor is paired with itself (e.g., for 36, 6 × 6). Thus, every locker whose number is a perfect square will remain open.

   Which specific locker numbers (up to 100) would remain open? The locker numbers that remain open are the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

CH2 Lines and Angles WORKSHEET CLASS 6

Subject: Maths                        WORKSHEET   - CH-2 LINES AND ANGLES                  Class-VI            

Q1. How many lines can be drawn that pass through the point?
Q2.How many different lines can be drawn that pass through both of the points?

Q3. Draw and label an angle with arms ST and SR. 

Q4. Write names of two points and 2 line segments in the picture.

Q5. A line segment has ___ end points.

Only 1    (b) no end point (c) 2    (d)  3

Q6. If the measure of an angle is 50°, what type of angle is it?

(a) Obtuse angle   (b) Acute angle (c) Right angle (d) Straight angle
Q7. How many degrees are there in a straight angle?

90°     (b)180°    (c) 45°     (d) 360°

Q8. The measure of a Right angle is-

60°    (b) 120°       (c) 270°     (d) 90°

Q9. What  will be the angle between the hands of clock at  (a) 3 o’clock (b) 4 o’clock
Q10.Classify the angles as Acute , Obtuse , Right , Straight or Reflex angle-

Q11. Draw the following angles using a protractor

1. 80°                      (b) 135°    c)  60◦          d) 140◦

Q12. 

Q13. Name the line segments in Fig. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments?

Q14. Lines that never intersect are called:
a) Intersecting lines     b) Parallel lines     c) Perpendicular lines
Q15. Lines that intersect and form a right angle are called:
a) Parallel lines        b) Perpendicular lines      c) Skew lines
Q16. Draw the following pairs of lines:a)  Two parallel lines b) Two intersecting lines c) Two perpendicular lines
Q17. Choose the correct option:
a)  Both Assertion and reason are correct and reason is correct explanation for Assertion.

b)  Both Assertion and reason are correct but reason is not correct explanation for Assertion

c)  Assertion is true but reason is false.

d) Both Assertion and reason are false.

i)Assertion (A): A line has two end points.
Reason (R): A line segment has a fixed length.
ii) Assertion (A): An angle formed by two rays with a common endpoint is called a straight angle.
Reason (R): A straight angle measures 180 degrees.
iii)  Assertion (A): Parallel lines meet at a point.
Reason (R): Perpendicular lines meet at right angles.
iv) Assertion (A): An acute angle is smaller than a right angle.
Reason (R): A right angle measures 90 degrees.
v) Assertion (A): A ray has a starting point but no end point.
Reason (R): A line goes endlessly in both directions.
vi) Assertion (A): Two lines that do not intersect are perpendicular lines.
Reason (R): Perpendicular lines form four right angles.
vii) Assertion (A): The hands of a clock at 3 o’clock form a right angle.
Reason (R): A right angle measures 90°.
viii) Assertion (A): An obtuse angle is less than 90°.
Reason (R): An acute angle is greater than 90°.
ix) Assertion (A): Intersecting lines meet at exactly one point.
Reason (R): Parallel lines are always the same distance apart.
x) Assertion (A): A straight angle looks like a straight line.
Reason (R): A straight angle measures 180°.
Assertion: 82° is an acute angle. Reason: Acute angle lies between  0° and 90°

Q18. Name the angles marked in the given figure 

Q19.Which of the following statements are true (T) and which are false (F) ?  

(a) The ray has one starting point.

(b) The successor of a three digit number is always a three digit number 

Q20.write the definition of line segment and ray also draw a diagram.

Q21. Number of line segments in figure is

(a) 5                      (b) 10                     (c) 15                (d) 20

Q22 How many lines can be drawn that pass through the point?
(a) 1       (b) 0        (c) 2       (d) infinite

Q23. The measure of a Right angle is-

60°    (b) 120°       (c) 270°     (d) 90°

Q24. An angle is formed with two _ and vertex

(a)   sides (b) lines ( c ) rays (d) line segment 

Q25.The obtuse angle is

(a) 90◦         (b) less than 180 but greater than 90◦  (c) greater than 180◦            (d) 180◦

Q26 How many rays are drawn from .P ? 

Q27. Draw a model  for a ray from real life examples.

Q28..Draw the clock and find a)   the angle at  4 o’clock                   b) the angle at  6 o’clock

Q29 Name a) four points   b)   Two rays    c) A line segment   d)  the angle     

Q30. Which angle is greater: ∠XOY or ∠AOB? Give reasons.
Q31.The number of angles in the given figure is

(A) 3          (B)4      (C) 5       (D)6

Q32. Which of the following statements are true (T) and which are false (F) ?  

Q33. Find the angle measure between the hands of the clock in each figure :

Q34. Draw angles with the following measure:   a)140°           ( or)70°

Q35. Name the following questions:a) Starting point of each ray        b)   Two rays      c) Write a two example of ray from our nature
Q36.Name the angles marked in the given figure 

1a. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?
1b.Sheetal has marked two points on a piece of paper. How many different lines can she draw that pass through both of the points ?.
(a)   2   (b) 3                  (c) many                     (d) 1

 Q37. Name the rays shown in Fig., Is T the starting point of each of these rays 

Q38. In Fig. name: a. Five points     b. A line     c. Four rays     d. Five line segments 

Q39.. Find the angles in the given pictures? Draw the rays forming any one of the angles and name the vertex of the angle. 

Q40. Which angle is greater: ∠XOY or ∠AOB? Give reasons.

41. Find out the number of acute angles in each of the figures below. What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers?

Q42.Write the degree measures down near the indicated angles