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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers based on the NCERT syllabus. Solve fresh MCQs on roots, factorisation, quadratic formula, discriminant, and relationships between roots and coefficients with detailed solutions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 2
Difficulty Level: Easy to Moderate
Based On: NCERT Latest Syllabus
Introduction
Quadratic Equations is an important chapter of Class 10 Mathematics. It deals with equations of degree 2 and teaches students how to find their roots using different algebraic methods.
In this practice set, students will work with factorisation, relationships between roots and coefficients, discriminant, and basic application of the quadratic formula.
What You Will Learn?
✔ Finding roots by factorisation
✔ Relationship between roots and coefficients
✔ Sum and product of roots
✔ Discriminant
✔ Nature of roots
✔ Quadratic formula
✔ Board examination concepts
Why This Topic Is Important?
Quadratic Equations questions test both conceptual understanding and calculation skills. Students should be comfortable identifying coefficients, finding roots, and using the relationship between roots and coefficients.
These concepts are frequently useful in board-level Mathematics questions.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Mathematics Practice Tests
Q1. If the roots of the quadratic equation
$$x^2-7x+12=0$$
are $$\alpha$$ and $$\beta$$, what is the value of
$$\alpha+\beta$$?
A) $$12$$
B) $$7$$
C) $$-7$$
D) $$-12$$
Answer:
B) $$7$$
Useful Formula
For the quadratic equation:
$$ax^2+bx+c=0$$
the sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Solution
Given:
$$x^2-7x+12=0$$
Comparing with:
$$ax^2+bx+c=0$$
we get:
$$a=1,\quad b=-7,\quad c=12$$
Therefore:
$$\alpha+\beta=-\frac{-7}{1}$$
$$\alpha+\beta=7$$
Hence, the correct answer is:
B) $$7$$
Q2. If the roots of
$$2x^2-9x+4=0$$
are $$\alpha$$ and $$\beta$$, then what is the value of
$$\alpha\beta$$?
A) $$2$$
B) $$\frac{9}{2}$$
C) $$4$$
D) $$\frac{9}{4}$$
Answer:
D) $$2$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$2x^2-9x+4=0$$
Therefore:
$$a=2,\quad b=-9,\quad c=4$$
Using:
$$\alpha\beta=\frac{c}{a}$$
we get:
$$\alpha\beta=\frac{4}{2}$$
$$\alpha\beta=2$$
Hence, the correct answer is:
A) $$2$$
Q3. Which of the following quadratic equations has equal roots?
A) $$x^2+4x+4=0$$
B) $$x^2+3x+2=0$$
C) $$x^2-5x+6=0$$
D) $$x^2-2x-3=0$$
Answer:
A) $$x^2+4x+4=0$$
Useful Formula
A quadratic equation has equal roots when:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
For option A:
$$x^2+4x+4=0$$
Here:
$$a=1,\quad b=4,\quad c=4$$
Therefore:
$$D=b^2-4ac$$
$$D=(4)^2-4(1)(4)$$
$$D=16-16$$
$$D=0$$
Since the discriminant is zero, the equation has equal roots.
Hence, the correct answer is:
A) $$x^2+4x+4=0$$
Q4. The roots of a quadratic equation are $$3$$ and $$-2$$. Which of the following is the corresponding quadratic equation?
A) $$x^2-x-6=0$$
B) $$x^2+x-6=0$$
C) $$x^2-x+6=0$$
D) $$x^2+5x+6=0$$
Answer:
A) $$x^2-x-6=0$$
Useful Concept
If the roots are $$\alpha$$ and $$\beta$$, then the quadratic equation can be written as:
$$(x-\alpha)(x-\beta)=0$$
Solution
Given roots:
$$\alpha=3$$
and
$$\beta=-2$$
Therefore:
$$(x-3)(x-(-2))=0$$
$$ (x-3)(x+2)=0$$
Multiplying:
$$x^2+2x-3x-6=0$$
$$x^2-x-6=0$$
Hence, the correct answer is:
A) $$x^2-x-6=0$$
Q5. The roots of the equation
$$3x^2-5x+2=0$$
are:
A) $$1,\frac{2}{3}$$
B) $$2,\frac{1}{3}$$
C) $$-1,-\frac{2}{3}$$
D) $$3,\frac{2}{5}$$
Answer:
A) $$1,\frac{2}{3}$$
Useful Concept
For factorisation, split the middle term so that the product of the first and last coefficients is obtained.
Solution
Given:
$$3x^2-5x+2=0$$
Split the middle term:
$$3x^2-3x-2x+2=0$$
Group the terms:
$$3x(x-1)-2(x-1)=0$$
Therefore:
$$(3x-2)(x-1)=0$$
Thus:
$$3x-2=0$$
or
$$x-1=0$$
Therefore:
$$x=\frac{2}{3}$$
or
$$x=1$$
Hence, the roots are:
$$1,\frac{2}{3}$$
Therefore, the correct answer is:
A) $$1,\frac{2}{3}$$
Important Formulas & Concepts
Standard Form
$$ax^2+bx+c=0,\quad a\ne0$$
Sum of Roots
$$\alpha+\beta=-\frac{b}{a}$$
Product of Roots
$$\alpha\beta=\frac{c}{a}$$
Quadratic Formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Discriminant
$$D=b^2-4ac$$
Equal Roots
$$D=0$$
Two Distinct Real Roots
$$D>0$$
No Real Roots
$$D<0$$
Equation When Roots Are Known
If roots are $$\alpha$$ and $$\beta$$:
$$(x-\alpha)(x-\beta)=0$$
or
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Related Practice Questions
Try solving these questions yourself:
Q1. Find the sum of roots of:
$$3x^2-8x+5=0$$
Q2. Find the product of roots of:
$$5x^2-7x+2=0$$
Q3. Determine whether the roots of the following equation are equal:
$$x^2-10x+25=0$$
Q4. Form a quadratic equation whose roots are $$4$$ and $$-3$$.
Q5. Find the roots of:
$$2x^2-7x+3=0$$
Mini Quiz Challenge
Answer these questions as quickly as possible.
Q1. What is the sum of roots of $$ax^2+bx+c=0$$?
Q2. What is the product of roots?
Q3. What is the condition for equal roots?
Q4. If roots are $$2$$ and $$5$$, what is their sum?
Q5. If roots are $$2$$ and $$5$$, what is their product?
Challenge: Try answering all five questions within 60 seconds.
Exam Tips
✔ Memorise the relationships between roots and coefficients.
✔ Always identify $$a$$, $$b$$ and $$c$$ carefully.
✔ Check the signs of the roots.
✔ When roots are given, use:
$$(x-\alpha)(x-\beta)=0$$
✔ In factorisation, multiply the factors back to verify the equation.
✔ Do not confuse the sum of roots with the product of roots.
Quick Revision Notes
✔ For $$ax^2+bx+c=0$$:
$$\alpha+\beta=-\frac{b}{a}$$
✔ Product of roots:
$$\alpha\beta=\frac{c}{a}$$
✔ Discriminant:
$$D=b^2-4ac$$
✔ Equal roots:
$$D=0$$
✔ Two distinct real roots:
$$D>0$$
✔ If roots are $$\alpha$$ and $$\beta$$, the equation is:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Common Mistakes Students Make
❌ Forgetting the negative sign in:
$$-\frac{b}{a}$$
❌ Writing the product of roots as $$-\frac{c}{a}$$.
❌ Making sign mistakes while forming an equation from given roots.
❌ Forgetting to multiply all terms during factorisation.
❌ Confusing equal roots with no real roots.
Key Takeaways
✔ The sum of roots depends on $$-\frac{b}{a}$$.
✔ The product of roots is $$\frac{c}{a}$$.
✔ Given roots can be used to construct a quadratic equation.
✔ The discriminant helps determine the nature of roots.
✔ Factorisation is often the quickest method when factors are easily identifiable.
FAQs
Q. What is the sum of roots of a quadratic equation?
Answer:
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$-\frac{b}{a}$$
Q. What is the product of roots?
Answer:
The product of roots is:
$$\frac{c}{a}$$
Q. How can a quadratic equation be formed when its roots are known?
Answer:
If the roots are $$\alpha$$ and $$\beta$$, then:
$$(x-\alpha)(x-\beta)=0$$
Q. When does a quadratic equation have equal roots?
Answer:
When:
$$D=0$$
Q. What is the discriminant formula?
Answer:
$$D=b^2-4ac$$
Conclusion
This Class 10 Maths Chapter 4 Quadratic Equations MCQ provides fresh NCERT-based questions covering roots, factorisation, relationships between roots and coefficients, discriminant, and formation of quadratic equations.
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