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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions based on the NCERT syllabus. Solve fresh and exam-oriented MCQs on factorisation, roots, discriminant, quadratic formula, and application-based quadratic equations with detailed solutions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 14
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
Quadratic Equations is an important chapter of Class 10 Mathematics. A quadratic equation is generally written in the form:
$$ax^2+bx+c=0,\quad a\ne0$$
Students learn different methods of solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
The discriminant helps determine the nature of the roots, while the relationships between roots and coefficients help solve many questions without directly calculating the roots.
This practice set contains 5 fresh MCQs covering important concepts from the chapter.
What You Will Learn
✔ Standard form of quadratic equations
✔ Factorisation method
✔ Quadratic formula
✔ Discriminant
✔ Nature of roots
✔ Sum and product of roots
✔ Equal roots
✔ Application-based problems
Why This Topic Is Important
Quadratic Equations strengthens students’ algebraic reasoning and problem-solving skills. It is useful for solving mathematical situations involving numbers, dimensions, areas, and other real-life applications.
The concepts of discriminant and relationships between roots and coefficients are particularly useful in examination questions.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Mathematics Revision
Q1. The roots of the quadratic equation
$$x^2-10x+21=0$$
are:
A) $$3,7$$
B) $$2,8$$
C) $$-3,-7$$
D) $$1,21$$
Answer:
A) $$3,7$$
Useful Concept
For factorisation, find two numbers whose product is the constant term and whose sum is the coefficient of $$x$$.
Solution
Given:
$$x^2-10x+21=0$$
We need two numbers whose product is:
$$21$$
and whose sum is:
$$-10$$
The required numbers are:
$$-3\quad\text{and}\quad-7$$
Therefore:
$$x^2-3x-7x+21=0$$
Grouping:
$$x(x-3)-7(x-3)=0$$
Therefore:
$$(x-3)(x-7)=0$$
Thus:
$$x=3$$
or:
$$x=7$$
Hence, the roots are:
$$3,\quad7$$
Therefore, the correct answer is:
A) $$3,7$$
Q2. What is the product of the roots of the quadratic equation
$$4x^2-11x+6=0$$
?
A) $$\frac{11}{4}$$
B) $$\frac{3}{2}$$
C) $$\frac{4}{6}$$
D) $$-\frac{6}{4}$$
Answer:
B) $$\frac{3}{2}$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$4x^2-11x+6=0$$
Therefore:
$$a=4,\quad b=-11,\quad c=6$$
Using:
$$\alpha\beta=\frac{c}{a}$$
we get:
$$\alpha\beta=\frac{6}{4}$$
Simplifying:
$$\alpha\beta=\frac{3}{2}$$
Hence, the correct answer is:
B) $$\frac{3}{2}$$
Q3. For what value of $$k$$ will the quadratic equation
$$x^2+8x+k=0$$
have equal roots?
A) $$8$$
B) $$12$$
C) $$16$$
D) $$32$$
Answer:
C) $$16$$
Useful Formula
A quadratic equation has equal roots when:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
Given:
$$x^2+8x+k=0$$
Therefore:
$$a=1,\quad b=8,\quad c=k$$
For equal roots:
$$D=0$$
Therefore:
$$8^2-4(1)(k)=0$$
$$64-4k=0$$
Therefore:
$$4k=64$$
Hence:
$$k=16$$
Therefore, the correct answer is:
C) $$16$$
Q4. If one root of the quadratic equation
$$2x^2-7x+k=0$$
is $$3$$, then the value of $$k$$ is:
A) $$2$$
B) $$3$$
C) $$6$$
D) $$8$$
Answer:
C) $$3$$
Useful Concept
If a number is a root of a quadratic equation, substituting that value for $$x$$ makes the equation equal to zero.
Solution
Given:
$$2x^2-7x+k=0$$
One root is:
$$x=3$$
Substitute:
$$2(3)^2-7(3)+k=0$$
$$2(9)-21+k=0$$
$$18-21+k=0$$
$$k-3=0$$
Therefore:
$$k=3$$
Hence, the correct answer is:
B) $$3$$
Q5. The length of a rectangular garden is $$3,m$$ more than its breadth. If the area of the garden is $$70,m^2$$, what is the breadth?
A) $$5,m$$
B) $$7,m$$
C) $$10,m$$
D) $$14,m$$
Answer:
B) $$7,m$$
Useful Concept
The area of a rectangle is:
$$\text{Area}=\text{Length}\times\text{Breadth}$$
Solution
Let the breadth be:
$$x,m$$
Then the length is:
$$x+3,m$$
Given:
$$x(x+3)=70$$
Expanding:
$$x^2+3x=70$$
Bring all terms to one side:
$$x^2+3x-70=0$$
Factorising:
$$x^2+10x-7x-70=0$$
Grouping:
$$x(x+10)-7(x+10)=0$$
Therefore:
$$(x+10)(x-7)=0$$
So:
$$x+10=0$$
or:
$$x-7=0$$
Therefore:
$$x=-10$$
or:
$$x=7$$
Since breadth represents a length, the negative value is not possible.
Therefore:
$$x=7,m$$
Hence, the correct answer is:
B) $$7,m$$
Important Formulas & Concepts
Standard Form
$$ax^2+bx+c=0,\quad a\ne0$$
Quadratic Formula
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Discriminant
$$D=b^2-4ac$$
Sum of Roots
$$\alpha+\beta=-\frac{b}{a}$$
Product of Roots
$$\alpha\beta=\frac{c}{a}$$
Equation When Roots Are Known
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Nature of Roots
If:
$$D>0$$
the equation has two distinct real roots.
If:
$$D=0$$
the equation has two equal real roots.
If:
$$D<0$$
the equation has no real roots.
Related Practice Questions
Q1. Find the roots of:
$$x^2-12x+35=0$$
Q2. Find the sum of roots of:
$$5x^2-13x+6=0$$
Q3. Find the product of roots of:
$$3x^2+8x-5=0$$
Q4. Find the value of $$k$$ if:
$$x^2-10x+k=0$$
has equal roots.
Q5. A rectangle has an area of $$108,m^2$$ and its length is $$3,m$$ more than its breadth. Find its breadth.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. Find the discriminant of:
$$2x^2-8x+8=0$$
Q2. Find the sum of roots of:
$$3x^2+10x+7=0$$
Q3. Find the product of roots of:
$$6x^2-5x+2=0$$
Q4. For what value of $$k$$ will:
$$x^2+10x+k=0$$
have equal roots?
Q5. If one root of:
$$x^2-8x+k=0$$
is $$3$$, find $$k$$.
Exam Tips
✔ Identify $$a$$, $$b$$, and $$c$$ before applying any formula.
✔ Keep the sign of each coefficient correct.
✔ For equal roots, use:
$$D=0$$
✔ For finding an unknown coefficient when a root is given, substitute the root directly.
✔ In factorisation, verify the factors by multiplying them back.
✔ In application-based questions, reject values that are not physically meaningful.
Quick Revision Notes
✔ Standard form:
$$ax^2+bx+c=0$$
✔ Quadratic formula:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
✔ Discriminant:
$$D=b^2-4ac$$
✔ Sum of roots:
$$\alpha+\beta=-\frac{b}{a}$$
✔ Product of roots:
$$\alpha\beta=\frac{c}{a}$$
✔ Equal roots:
$$D=0$$
✔ Distinct real roots:
$$D>0$$
✔ No real roots:
$$D<0$$
Common Mistakes Students Make
❌ Forgetting the sign of $$b$$.
❌ Using the wrong formula for the product of roots.
❌ Writing $$D=b^2+4ac$$ instead of:
$$D=b^2-4ac$$
❌ Making sign errors during factorisation.
❌ Forgetting to reject negative lengths.
❌ Not verifying the answer after solving.
Key Takeaways
✔ Factorisation is useful for solving suitable quadratic equations.
✔ The sum of roots is:
$$-\frac{b}{a}$$
✔ The product of roots is:
$$\frac{c}{a}$$
✔ The discriminant determines the nature of roots.
✔ Equal roots occur when:
$$D=0$$
✔ Real-life situations involving dimensions can be represented using quadratic equations.
FAQs
Q. What is the formula for the discriminant?
Answer:
For:
$$ax^2+bx+c=0$$
the discriminant is:
$$D=b^2-4ac$$
Q. When are the roots equal?
Answer:
The roots are equal when:
$$D=0$$
Q. What is the sum of roots?
Answer:
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Q. What is the product of roots?
Answer:
The product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Q. How do we solve a quadratic equation when one root is given?
Answer:
Substitute the given root into the equation and solve for the unknown coefficient.
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh, NCERT-based practice on factorisation, roots and coefficients, discriminant, equal roots, unknown coefficients, and application-based problems.
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