Worksheet on Direct and Inverse Variation ICSE Class 8

 Certainly! Here's a worksheet on the topic of Direct and Inverse Variation for 8th-grade students following the ICSE board curriculum. The problems are categorized into easy, medium, and difficult levels.

**Direct and Inverse Variation Worksheet**

**Instructions:**

1. Solve all the problems.

2. Show your work and calculations where necessary.

3. Circle or underline your final answers.

**Easy Problems:**

1. If y varies directly with x, and y = 12 when x = 4, find the constant of variation (k).

2. If y varies inversely with x, and y = 10 when x = 5, find the constant of variation (k).

3. If y varies directly with x, and y = 25 when x = 5, find y when x = 8.

4. If y varies inversely with x, and y = 6 when x = 9, find y when x = 12.

5. If y varies directly with x, and y = 15 when x = 3, find x when y = 30.

**Medium Problems:**

6. The cost (C) of printing flyers is directly proportional to the number of flyers (n). If it costs $40 to print 200 flyers, find the cost to print 600 flyers.

7. The time (t) it takes to complete a task is inversely proportional to the number of workers (w). If it takes 8 hours for 6 workers to complete the task, how long will it take for 12 workers to finish the same task?

8. A car travels at a constant speed. If it covers 60 miles in 2 hours, how long will it take to cover 150 miles at the same speed?

9. The force (F) of attraction between two objects is directly proportional to the product of their masses (m1 and m2) and inversely proportional to the square of the distance (d) between them. If F = 12 when m1 = 4, m2 = 6, and d = 3, find F when m1 = 8, m2 = 9, and d = 5.

10. The pressure (P) in a closed container is inversely proportional to its volume (V). If P = 48 kPa when V = 4 liters, find the pressure when V = 10 liters.

**Difficult Problems:**

11. A car's fuel efficiency (miles per gallon) varies inversely with its speed (in miles per hour). If the car gets 30 miles per gallon at 60 mph, find the fuel efficiency at 70 mph.

12. The force (F) of gravity between two objects varies directly with the product of their masses (m1 and m2) and inversely with the square of the distance (d) between them. If F = 9.8 N when m1 = 5 kg, m2 = 10 kg, and d = 1 m, find F when m1 = 3 kg, m2 = 8 kg, and d = 2 m.

13. The time (t) it takes for a pendulum to complete one full swing varies directly with the square root of its length (L). If a pendulum takes 2 seconds to complete one swing when L = 9 meters, find the time it takes when L = 16 meters.

14. The resistance (R) in an electrical circuit is inversely proportional to the square of the current (I). If R = 25 ohms when I = 5 amperes, find R when I = 10 amperes.

15. The force (F) required to lift an object with a pulley system varies directly with the weight (W) of the object and inversely with the number (n) of supporting ropes. If F = 120 N when W = 600 N and n = 4, find F when W = 800 N and n = 6.

**Answers:**

**Easy Problems:**

1. k = 3

2. k = 50

3. y = 40

4. y = 4

5. x = 6

**Medium Problems:**

6. $150

7. 4 hours

8. 5 hours

9. F = 5.76

10. P = 19.2 kPa

**Difficult Problems:**

11. Fuel efficiency at 70 mph = 25 mpg

12. F = 4.35 N

13. Time = 3 seconds

14. R = 6.25 ohms

15. F = 160 N

Worksheet on Sets ICSE Class 8

 

**Sets Worksheet – ICSE – Class 8**

 

**Instructions:**

1. Answer all the questions.

2. Circle or underline your final answer.

3. Show your work or reasoning if required.

4. Answers to all the problems are given at the end. You should look at the solutions only after attempting all the problems.

 

 

**Questions:**

1. Define a "set" in your own words. Provide an example.

 

2. Classify the following into sets:

   a) The days of the week

   b) Even numbers less than 20

   c) Vowels in the English alphabet

 

3. List the elements of the set A = {2, 4, 6, 8, 10}. Also, find the cardinality of set A.

 

4. Create a set B with the first five prime numbers. Write it in the set-builder notation.

 

5. Determine whether the following statements are true or false:

   a) {1, 2, 3} ⊆ {1, 2, 3, 4, 5}

   b) {a, b, c} ⊇ {b, c, d}

   c) {2, 4, 6} ⊂ {1, 2, 3, 4, 5, 6}

 

6. Find the union of sets P = {1, 2, 3, 4, 5} and Q = {4, 5, 6, 7}. Write the result in roster form.

 

7. Calculate the intersection of sets X = {a, b, c} and Y = {b, c, d}. Write the result in set-builder notation.

 

8. Consider two sets: M = {1, 2, 3, 4} and N = {3, 4, 5, 6}. Find the difference M - N.

 

9. Solve the following set equation for set Z: Z ∩ {2, 3, 4} = {3, 4}. Write the result in roster form.

 

10. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, find the complement of the set V = {2, 4, 6, 8}.

 

**Additional Challenging Questions:**

 

11. Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8, 9}. Find A ∪ B, A ∩ B, and A - B.

 

12. Consider a universal set U = {x | x is a positive integer less than 10}. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∩ B and A ∪ B.

 

13. Let U = {a, b, c, d, e, f, g} be the universal set. If A = {a, b, c, d} and B = {b, c, e, f}, find A ∪ B and A' (complement of A).

 

14. Define three sets A, B, and C as follows:

    A = {x | x is a multiple of 2 and 3}

    B = {x | x is a multiple of 2 and 5}

    C = {x | x is a multiple of 3 and 5}

   Find A ∩ B, A ∪ C, and B ∩ C.

 

15. Let U be the set of all students in a school, A be the set of students who play chess, and B be the set of students who play cricket. If there are 120 students in total, 60 play chess, and 80 play cricket, how many students play both chess and cricket?

 

16. Consider the set P = {x | x is a prime number less than 20} and the set Q = {x | x is an odd number less than 20}. Find P ∩ Q.

 

17. Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. Find the symmetric difference of sets A and B.

 

18. Determine whether the following statement is true or false: For any two sets A and B, A ∪ B = B ∪ A.

**Answers:**

 

1. A set is a collection of distinct objects or elements. Example: Set of even numbers less than 10 = {2, 4, 6, 8}.

 

2.

   a) Set of days of the week = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

   b) Set of even numbers less than 20 = {2, 4, 6, 8, 10, 12, 14, 16, 18}

   c) Set of vowels in the English alphabet = {a, e, i, o, u}

 

3. Set A = {2, 4, 6, 8, 10}, Cardinality of A = 5.

 

4. Set B = {2, 3, 5, 7}. In set-builder notation: B = {x | x is a prime number and 1 < x < 10}.

 

5.

   a) True

   b) False

   c) True

 

6. P ∪ Q = {1, 2, 3, 4, 5, 6, 7}.

 

7. X ∩ Y = {b, c}. In set-builder notation: {x | x is an element of X and x is an element of Y}.

 

8. M - N = {1, 2}. (Elements in M but not in N)

 

9. Z = {3, 4}.

 

10. U - V = {1, 3, 5, 7, 9}.

 

**Answers to Additional Questions:**

 

11. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A ∩ B = {4, 5, 6}, A - B = {1, 2, 3}

 

12. A ∩ B = {3, 4}, A ∪ B = {1, 2, 3, 4, 5, 6}

 

13. A ∪ B = {a, b, c, d, e, f}, A' = {e, f, g}

 

14. A ∩ B = {x | x is a multiple of 2, 3, and 5}, A ∪ C = {x | x is a multiple of 2 or 3 or 5}, B ∩ C = {x | x is a multiple of 3 and 5}

 

15. Students who play both chess and cricket = 20 students.

 

16. P ∩ Q = {3, 5, 7, 11, 13, 17, 19}

 

17. Symmetric difference of sets A and B = {1, 2, 6, 7}

 

18. True. The union of two sets is commutative, so A ∪ B = B ∪ A.