Variations

Maths, SSC-KPK Class 10. This chapter is split into 34 topics and has 883 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Dividing a quantity in a given ratio 29 questions
2. Inverse variation concepts 20 questions
3. Alternendo and invertendo 28 questions
4. Proportion with algebraic terms 25 questions
5. Summary and Review Exercise 3 22 questions
6. Joint variation numerical problems 27 questions
7. Componendo-dividendo property 30 questions
8. Identifying direct or inverse variation 29 questions
9. Applying proportion properties to a/b = c/d 25 questions
10. Direct variation with squares and roots 28 questions
11. Third proportional 27 questions
12. Ratio and simplest form 29 questions
13. Mean proportional 29 questions
14. Real life problems on inverse variation 26 questions
15. Finding numbers from ratio conditions 29 questions
16. Proportion and cross product 27 questions
17. Solving equations by componendo-dividendo 21 questions
18. Joint variation concepts 21 questions
19. Constants and variables 21 questions
20. Real life problems on direct variation 26 questions
21. Ratio and proportion word problems 30 questions
22. Inverse variation with squares and cubes 26 questions
23. Direct variation tables 25 questions
24. Number patterns and fun facts 21 questions
25. K-method 30 questions
26. Direct variation concepts 23 questions
27. Fourth proportional 26 questions
28. Componendo and dividendo 23 questions
29. Inverse variation numerical problems 28 questions
30. Joint variation with powers and inverse parts 22 questions
31. Continued proportion 30 questions
32. K-method proofs and identities 24 questions
33. Direct variation numerical problems 29 questions
34. Variation in science and geometry 27 questions

Sample questions: Variations

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Which statement about the proportion a : b :: c : d is FALSE?

• A. ad = bc
• B. a + b = c + d(correct answer)
• C. a/b = c/d
• D. b and c are the means

Answer: B.Only the product of means equals the product of extremes; the sums a + b and c + d are not necessarily equal (for 1 : 2 :: 2 : 4, 3 ≠ 6).

Q2. Which of the following is a constant according to the book?

• A. 22/7(correct answer)
• B. Speed of a train
• C. Population of a town
• D. Demand of a commodity

Answer: A.The book lists 4, 2.718 and 22/7 as constants, and speed of a train, demand of a commodity and population of a town as variables.

Q3. How many kinds of variation are discussed in this unit (besides joint variation)?

• A. Four: direct, inverse, joint and partial
• B. Three: direct, inverse and constant
• C. Two: direct and inverse(correct answer)
• D. One: only direct

Answer: C.The book names two kinds of variation, direct variation and inverse variation; joint variation is a combination discussed later.

Q4. Variation is necessary for us because

• A. it removes the need for units in all kinds of measurement
• B. it applies only to numbers that are greater than 100
• C. it shows how one quantity changes with another in daily life(correct answer)
• D. it keeps every quantity in a given problem fixed at one value

Answer: C.The 'Why it's important' section says we often use one type of variation or another in daily life, so knowing variation is necessary.

Q5. If y varies directly as x and y = 0 when x = 0, which value must k have for the relation to hold at x = 7?

• A. Only k = 1, the value for every direct variation
• B. Only k = 7, the value of x given here
• C. Any constant except zero(correct answer)
• D. Only k = 0, because y = 0 at x = 0

Answer: C.k is determined by one pair of values; 0 = k × 0 is true for every k, and k = 0 is excluded because then y would always be zero.

Q6. Solve ((3x + 3)² + (3x − 4)²)/((3x + 3)² − (3x − 4)²) = 41/9.

• A. x = −8/27 or x = −32/3
• B. x = −31/3 or x = 1/27
• C. x = 32/3 only
• D. x = 8/27 or x = 32/3(correct answer)

Answer: D.Componendo-dividendo gives (3x + 3)²/(3x − 4)² = (41 + 9)/(41 − 9) = 25/16; hence (3x + 3)/(3x − 4) = ±5/4 and x = 8/27 or 32/3.

Q7. Solve ((2x + 1)² + (3x + 4)²)/((2x + 1)² − (3x + 4)²) = 25/7.

• A. x = 13/6 or x = 19/18
• B. x = −13/6 or x = −19/18(correct answer)
• C. x = −8 or x = −16/17
• D. x = −19/18 only

Answer: B.Componendo-dividendo gives (2x + 1)²/(3x + 4)² = (25 + 7)/(25 − 7) = 16/9; hence (2x + 1)/(3x + 4) = ±4/3 and x = −13/6 or −19/18.

Q8. Solve ((3x − 5)² + (x − 1)²)/((3x − 5)² − (x − 1)²) = −13/5.

• A. x = 17/11 or x = 13/7(correct answer)
• B. x = −17/11 or x = −13/7
• C. x = 13/9 or x = 7/3
• D. x = 13/7 only

Answer: A.Componendo-dividendo gives (3x − 5)²/(x − 1)² = (13 + (−5))/(13 − (−5)) = 4/9; hence (3x − 5)/(x − 1) = ±2/3 and x = 17/11 or 13/7.

Q9. Solve ((2x + 5)² + (x + 4)²)/((2x + 5)² − (x + 4)²) = −13/5.

• A. x = 23/8 or x = 7/4
• B. x = −7/4 only
• C. x = −23/8 or x = −7/4(correct answer)
• D. x = −22/7 or x = 2

Answer: C.Componendo-dividendo gives (2x + 5)²/(x + 4)² = (13 + (−5))/(13 − (−5)) = 4/9; hence (2x + 5)/(x + 4) = ±2/3 and x = −23/8 or −7/4.

Q10. If a/b = c/d, which of the following is true? (Example 11)

• A. (2a + 3b) : b = (2c − 3d) : d
• B. (2a + 3b) : b = (2c + 3d) : d(correct answer)
• C. (2a + 3b) : a = (2c + 3d) : d
• D. (2a + 3b) : 2b = (2c + 3d) : 3d

Answer: B.Multiply a/b = c/d by 2/3: 2a/(3b) = 2c/(3d); componendo gives (2a + 3b)/(3b) = (2c + 3d)/(3d), hence (2a + 3b) : b = (2c + 3d) : d.