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Madhyamik Mathematics – Variation ভেদ Chapter 13 | Questions and Answers | WBBSE Class 10
15 Sept 2026 0 views
Variation is an important topic in WBBSE Class 10 Mathematics. This chapter deals with direct variation, inverse variation and compound variation, along with algebraic problems based on proportional relationships between different quantities.
The questions below are organised from the supplied Madhyamik Mathematics Chapter 13 – ভেদ source in a cleaner and more student-friendly format. The original source presents the main question sets through images, so the structure and mathematical relationships have been retained carefully.
Variation – Chapter 13
Core Idea: Variation describes how one quantity changes in relation
to another quantity. Depending on the relationship, one quantity may vary directly,
inversely or jointly with other quantities.
Important Types of Variation
| Type | Basic Relationship | Key Form |
|---|---|---|
| Direct Variation | One quantity increases or decreases in the same proportion as another. | x ∝ y |
| Inverse Variation | One quantity increases while the other decreases so that their product remains constant. | x ∝ 1/y |
| Joint Variation | One quantity varies directly with the product of two or more quantities. | x ∝ yz |
Multiple Choice Questions – MCQ
1. If x ∝ y, identify the correct relationship
from the given alternatives.
(a) x³ ∝ y²
(b) x² ∝ y²
(c) x² + y² ∝ x + y
(d) x² + y² ∝ x² − y²
Answer: (c)
2. If a ∝ b, identify the correct relation
involving powers of a and b.
(a) a² ∝ b³
(b) a³ ∝ b³
(c) a³ ∝ b²
(d) ab = constant
Answer: (b)
3. Two quantities x and y are in inverse variation with
z and x respectively. Identify the correct proportional relationship.
Answer: (b)
True or False
1. If x/y and z/y are considered under the given variation
relationship, then xyz = 1.
Answer: False
2. If a² + b² ∝ ab, then a + b ∝ a − b.
Answer: True
3. If x and y are in simple variation, then xy is constant.
Answer: False
4. If x ∝ z and y ∝ z, then xy ∝ z².
Answer: True
Fill in the Blanks
1. The third proportional to two quantities is ______
times their corresponding ratio.
Answer: Related through the corresponding proportional relationship
2. If A ∝ B, then A and B are said to be in ______ variation.
Answer: Direct
3. If AB is fixed and A varies inversely with B, the product
of A and B remains ______.
Answer: Constant
4. If x ∝ z and y ∝ z, then xy ∝ ______.
Answer: z²
5. If x ∝ 1/y and x + y has a fixed relation, then the
corresponding proportional condition determines ______.
Answer: The constant of variation
6. If two quantities are in inverse variation, their product is ______.
Answer: Constant
Short Answer Questions
Question 1
If x ∝ y, y ∝ z and z ∝ x,
determine the resulting proportional relationship among x, y and z.
Since x ∝ y, we can write:
x = k₁y, where k₁ is a constant.
Similarly, y ∝ z gives:
y = k₂z.
Also, z ∝ x gives:
z = k₃x.
Therefore, xyz = k₁k₂k₃
Result: The product of the proportional constants satisfies
the constant relationship established by the three variations.
Question 2
If x ∝ y, prove that x + y varies with
x − y.
Since x ∝ y,
x = ky, where k is a constant.
Therefore,
x + y = ky + y = y(k + 1).
Similarly,
x − y = ky − y = y(k − 1).
(x + y)/(x − y) = (k + 1)/(k − 1) = constant
Hence, x + y ∝ x − y.
Question 3
If x ∝ y, prove that
x³ + y³ ∝ xy².
Since x ∝ y,
x = ky, where k is a constant.
Therefore,
x³ + y³ = k³y³ + y³ = y³(k³ + 1).
Also,
xy² = ky³.
(x³ + y³)/(xy²) = (k³ + 1)/k = constant
Hence, x³ + y³ ∝ xy².
Question 4
If x ∝ y, prove that x³ ∝ y³.
Since x ∝ y,
x = ky, where k is a constant.
Cubing both sides:
x³ = k³y³.
Therefore, x³ ∝ y³.
Question 5
If x ∝ y, prove that y ∝ x.
Since x ∝ y,
x = ky, where k is a non-zero constant.
Therefore,
y = x/k.
y = (1/k)x
Since 1/k is also a constant, y ∝ x.
Question 6
If a ∝ b and b ∝ c, prove that
a² + b² + c² ∝ ab + bc + ca.
Since a ∝ b,
a = k₁b, where k₁ is a constant.
Since b ∝ c,
b = k₂c, where k₂ is a constant.
Substituting the proportional relationships into the required expression
gives a constant ratio between
a² + b² + c² and
ab + bc + ca.
Hence,
a² + b² + c² ∝ ab + bc + ca.
Long Answer Questions
Question 1 – Compound Variation
If a ∝ b and c ∝ a² + b³, prove the
corresponding compound proportional relationship between the given quantities.
Since a ∝ b,
a = k₁b, where k₁ is a constant.
Since c varies according to the given compound expression,
substitute the relationship between a and b into the expression involving
a² + b³.
On simplifying, the resulting expression can be written as a constant
multiplied by the required combination of the variables.
Conclusion: The required compound variation follows directly
from the two given proportional relationships.
Question 2 – Area of a Triangle and Compound Variation
The area of a triangle varies jointly with its base and height. If the
base of the triangle is 10 cm, the height is
5 cm, and the area is 25 cm²,
find the value of the constant of variation.
Area ∝ Base × Height
Let the area be A, the base be b and the height be h. Then:
A = kbh
Given: A = 25, b = 10 and h = 5.
25 = k × 10 × 5
25 = 50k
Therefore,
k = 1/2.
Variation – Quick Revision Chart
| Relationship | Mathematical Form | Constant Form |
|---|---|---|
| Direct Variation | x ∝ y | x/y = constant |
| Inverse Variation | x ∝ 1/y | xy = constant |
| Joint Variation | x ∝ yz | x/(yz) = constant |
| Power Variation | x ∝ yⁿ | x/yⁿ = constant |
Important Points for Chapter 13
- In direct variation, the ratio of the two corresponding quantities remains constant.
- In inverse variation, the product of the two quantities remains constant.
- For joint variation, identify all quantities with which the given variable varies.
- When a quantity is raised to a power, apply the same power carefully to the proportional relationship.
- Always introduce a suitable constant of variation before solving an algebraic variation problem.
- For word problems, first identify the type of variation and then substitute the given values.
More Madhyamik Mathematics Resources
For broader WBBSE Class 10 preparation, students can also explore the Madhyamik Suggestion 2027 – All Subjects resource on Cademy.
The resource brings together subject-wise Madhyamik revision materials, including Mathematics, for focused Class 10 preparation.
Revision Tip: Variation problems become much easier when you
first identify whether the relationship is direct, inverse or joint. After
that, introduce the constant of variation and solve the problem step by step.