Madhyamik Mathematics Suggestion Chapter 1 | Quadratic Equation in One Variable
Madhyamik Mathematics Suggestion – Chapter 1 focuses on Quadratic Equation in One Variable, an important area of Class 10 Mathematics. This chapter-based practice set includes objective questions, fill-in-the-blanks, True or False questions, short-answer problems and long-answer problems.
The questions are arranged according to their expected marks so that students can use the set for structured revision and written practice.
Quadratic Equation in One Variable: Important Questions
The following question set covers the major patterns presented in the source chapter. Students should practise the questions in writing rather than simply reading the answers.
Multiple Choice Questions: 1 Mark
Fill in the Blanks: 1 Mark
True or False: 1 Mark
Short Answer Questions: 2 Marks
The following questions require a little more reasoning than the objective section. Students should show the necessary mathematical steps instead of writing only the final result.
1. For what value of a will (a − 2)x² − 5x + 6 = 0 cease to be a quadratic equation?
For the equation to remain quadratic, the coefficient of x² must not be zero.
Therefore,
a − 2 = 0
So, a = 2.
2. If one root of ax² + bx + c = 0 is twice the other, prove that 2b² = 9ac.
Let the two roots be α and 2α.
Using the relationships between the roots and coefficients of a quadratic equation,
α + 2α = −b/a
and
2α² = c/a.
Eliminating α from these relationships gives the required result:
3. Solve x² − 3x + 2 = 0 using the quadratic formula.
For x² − 3x + 2 = 0, we have:
a = 1, b = −3, c = 2
Using the quadratic formula, the roots are obtained as:
Substitution gives the two roots:
x = 1 and x = 2.
4. If the roots of (b − c)x² + (c − a)x + (a − b) = 0 are equal, show that 2b = a + c.
For a quadratic equation to have equal roots, its discriminant must be zero.
Applying the equal-root condition to the given equation and simplifying the resulting relation gives:
5. Find k if 1 is a root of 5x² − 2x + k = 0.
Since x = 1 is a root, substitute x = 1 into the equation:
5(1)² − 2(1) + k = 0
5 − 2 + k = 0
Therefore,
6. The roots of x² − px + q = 0 are α and β. If α − β = 1, show that p² = 1 + 4q.
Using the root relationships:
α + β = p
and
αβ = q.
Now use the identity involving the difference of the roots:
Since α − β = 1, the required relationship follows:
7. Show that one root of x² − 6x + 8 = 0 is the square of the other.
Factorising the quadratic equation gives:
x² − 6x + 8 = 0
(x − 2)(x − 4) = 0
Hence the roots are 2 and 4. Since 4 = 2², one root is the square of the other.
8. Determine whether (2 − x)² = x² − 4x + 4 is an identity or an equation.
Expanding the left-hand side:
(2 − x)² = x² − 4x + 4
The two sides are identical for every real value of x.
Therefore, the given relation is an identity.
9. In terms of which variable can a higher-degree equation become a quadratic equation?
The source includes an equation involving powers of a variable where the expression can be rewritten using a suitable substitution. By treating the repeated higher power as a new variable, the equation can be represented in quadratic form with respect to that new variable.
Long Answer Questions: 5 Marks
These questions require complete mathematical working. During Madhyamik preparation, students should practise presenting the steps clearly, because the reasoning and intermediate calculations are important in a written Mathematics answer.
1. Problem Based on a Two-Digit Number
A two-digit number has a relationship between its tens digit and units digit. When the product of its two digits is subtracted from the number, the result is 15. Determine the units digit using the information given.
Let the required digits be represented algebraically according to the relationship given in the question. Form the corresponding quadratic equation and solve it using the appropriate quadratic-equation method.
The final answer should be obtained only after checking the resulting digit against the original condition.
2. Roots in the Ratio r : 1
If the ratio of the roots of ax² + bx + c = 0 is r : 1, prove the required relationship between a, b, c and r.
Let the two roots be rα and α. Apply the sum and product relationships of the roots with the coefficients of the quadratic equation. On eliminating α, the required relation is obtained.
3. Constructing a Quadratic Equation from the Sum of Roots
The sum of the two roots of a quadratic equation is 10, while the sum of the squares of the roots is 52. Find the quadratic equation.
Let the roots be α and β.
Given:
α + β = 10
and
α² + β² = 52
Using
we can determine αβ. The standard form of the required quadratic equation can then be constructed using the sum and product of its roots.
Chapter 1 Question Pattern at a Glance
| Question Type | Marks Mentioned in Source | Main Skill |
|---|---|---|
| Multiple Choice Questions | 1 Mark | Concept recognition and quick calculation |
| Fill in the Blanks | 1 Mark | Formula and property recall |
| True or False | 1 Mark | Concept checking |
| Short Answer Questions | 2 Marks | Calculation, proof and reasoning |
| Long Answer Questions | 5 Marks | Detailed mathematical solution |
Important Quadratic Equation Concepts for Revision
While practising this chapter, students should pay particular attention to the relationship between the roots and coefficients of a quadratic equation, the condition for an equation to be quadratic, equal roots, real and non-real roots, and the use of the quadratic formula.
Sum of roots = −b/a
Product of roots = c/a
These relationships are repeatedly useful when solving problems involving the roots of a quadratic equation. Students should practise applying them in different forms rather than memorising isolated answers.
How to Prepare Chapter 1 for Madhyamik Mathematics
For this chapter, written practice is especially important. Start with the objective questions, then move towards short proofs and numerical problems. Once the basic relationships between roots and coefficients become clear, practise problems where the roots are given through a condition such as a ratio, difference or square relationship.
Recommended revision sequence:
Concepts → Formula and root relationships → MCQ → Fill in the Blanks → True or False → Short Problems → Long Problems
Students preparing for the wider WBBSE Class 10 examination can also refer to Cademy's Madhyamik Suggestion 2027 All Subjects resource for subject-wise revision materials.
For broader Class 10 preparation, Cademy's WBBSE Class 10 Madhyamik 2027 Syllabus, Exam Pattern and Preparation Guide also provides general subject-wise preparation guidance.
Chapter 1: Final Revision Checklist
| Check | Revision Target |
|---|---|
| ✓ | Understand what makes an equation quadratic. |
| ✓ | Practise sum and product of roots. |
| ✓ | Revise the quadratic formula. |
| ✓ | Practise problems involving equal roots. |
| ✓ | Practise problems involving ratios and differences of roots. |
| ✓ | Practise forming a quadratic equation from given root information. |
| ✓ | Write complete solutions for 2-mark and 5-mark questions. |