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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers based on the latest NCERT syllabus. Solve fresh and board exam-oriented MCQs on roots, factorisation, discriminant, quadratic formula, and application-based problems with detailed solutions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 8
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations deals with equations in which the highest power of the variable is 2. Students learn to solve these equations using factorisation and the quadratic formula and also learn how the discriminant determines the nature of their roots.
This practice set contains fresh questions covering roots, coefficients, discriminant, factorisation, and real-life applications.
What You Will Learn?
✔ Solving quadratic equations by factorisation
✔ Applying the quadratic formula
✔ Finding the nature of roots
✔ Using relationships between roots and coefficients
✔ Finding unknown coefficients
✔ Forming quadratic equations
✔ Solving application-based problems
Why This Topic Is Important?
Quadratic equations are an important part of Class 10 Mathematics. They test algebraic reasoning and calculation skills and are useful in both direct questions and real-life application problems.
A strong understanding of this chapter helps students solve board-examination questions with greater accuracy and confidence.
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. The roots of the quadratic equation
$$3x^2-10x+3=0$$
are:
A) $$3,\frac{1}{3}$$
B) $$2,\frac{1}{2}$$
C) $$-3,-\frac{1}{3}$$
D) $$3,-\frac{1}{3}$$
Answer:
A) $$3,\frac{1}{3}$$
Useful Concept
For factorisation, split the middle term so that the resulting factors can be obtained.
Solution
Given:
$$3x^2-10x+3=0$$
Split the middle term:
$$3x^2-9x-x+3=0$$
Group the terms:
$$3x(x-3)-1(x-3)=0$$
Therefore:
$$(3x-1)(x-3)=0$$
Thus:
$$3x-1=0$$
or
$$x-3=0$$
Therefore:
$$x=\frac{1}{3}$$
or
$$x=3$$
Hence, the roots are:
$$3,\frac{1}{3}$$
Therefore, the correct answer is:
A) $$3,\frac{1}{3}$$
Q2. If the roots of
$$2x^2-7x+k=0$$
are reciprocal to each other, what is the value of $$k$$?
A) $$1$$
B) $$2$$
C) $$\frac{1}{2}$$
D) $$7$$
Answer:
B) $$2$$
Useful Concept
If two non-zero roots are reciprocal to each other, their product is:
$$1$$
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\frac{c}{a}$$
Solution
Given:
$$2x^2-7x+k=0$$
Here:
$$a=2,\quad c=k$$
The roots are reciprocal to each other.
Therefore:
$$\alpha\beta=1$$
Using:
$$\alpha\beta=\frac{c}{a}$$
we get:
$$\frac{k}{2}=1$$
Therefore:
$$k=2$$
Hence, the correct answer is:
B) $$2$$
Q3. What is the nature of the roots of the equation
$$5x^2-6x+2=0$$?
A) Two distinct real roots
B) Two equal real roots
C) No real roots
D) One positive and one negative root
Answer:
C) No real roots
Useful Formula
The discriminant is:
$$D=b^2-4ac$$
If:
$$D<0$$
the quadratic equation has no real roots.
Solution
Given:
$$5x^2-6x+2=0$$
Therefore:
$$a=5,\quad b=-6,\quad c=2$$
Calculate the discriminant:
$$D=(-6)^2-4(5)(2)$$
$$D=36-40$$
$$D=-4$$
Since:
$$D<0$$
the equation has no real roots.
Hence, the correct answer is:
C) No real roots
Q4. If the roots of a quadratic equation are $$2+\sqrt{3}$$ and $$2-\sqrt{3}$$, which is the equation?
A) $$x^2-4x+1=0$$
B) $$x^2+4x+1=0$$
C) $$x^2-4x-1=0$$
D) $$x^2+4x-1=0$$
Answer:
A) $$x^2-4x+1=0$$
Useful Concept
If the roots are $$\alpha$$ and $$\beta$$, the quadratic equation is:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Solution
Given:
$$\alpha=2+\sqrt{3}$$
and
$$\beta=2-\sqrt{3}$$
First find the sum:
$$\alpha+\beta=(2+\sqrt{3})+(2-\sqrt{3})$$
$$=4$$
Now find the product:
$$\alpha\beta=(2+\sqrt{3})(2-\sqrt{3})$$
Using:
$$(a+b)(a-b)=a^2-b^2$$
we get:
$$=2^2-(\sqrt{3})^2$$
$$=4-3$$
$$=1$$
Therefore, the required equation is:
$$x^2-4x+1=0$$
Hence, the correct answer is:
A) $$x^2-4x+1=0$$
Q5. The difference between the squares of two consecutive positive integers is $$31$$. What is the smaller integer?
A) $$14$$
B) $$15$$
C) $$16$$
D) $$17$$
Answer:
B) $$15$$
Useful Concept
If the smaller integer is $$x$$, the next consecutive integer is:
$$x+1$$
The difference between their squares is:
$$(x+1)^2-x^2$$
Solution
Let the smaller integer be:
$$x$$
Then the next integer is:
$$x+1$$
According to the question:
$$(x+1)^2-x^2=31$$
Expand:
$$x^2+2x+1-x^2=31$$
Therefore:
$$2x+1=31$$
$$2x=30$$
$$x=15$$
Hence, the smaller integer is:
$$15$$
Therefore, the correct answer is:
B) $$15$$
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 from Given Roots
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Reciprocal Roots
If roots are reciprocal:
$$\alpha\beta=1$$
Therefore:
$$\frac{c}{a}=1$$
Nature of Roots
$$D>0$$
Two distinct real roots.
$$D=0$$
Two equal real roots.
$$D<0$$
No real roots.
Related Practice Questions
Q1. Solve:
$$2x^2-9x+4=0$$
Q2. Find the value of $$k$$ if the roots of:
$$3x^2-5x+k=0$$
are reciprocal to each other.
Q3. Determine the nature of roots of:
$$4x^2+4x+5=0$$
Q4. Form a quadratic equation whose roots are:
$$3+\sqrt{2}$$
and
$$3-\sqrt{2}$$
Q5. The difference between the squares of two consecutive integers is $$45$$. Find the smaller integer.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$x^2-6x+9=0$$
Q2. If roots are reciprocal, what is their product?
Q3. What is the sum of roots of:
$$4x^2-12x+7=0$$
Q4. What is the product of roots of:
$$4x^2-12x+7=0$$
Q5. If the smaller of two consecutive integers is $$x$$, what is the larger integer?
Exam Tips
✔ Remember the three discriminant conditions.
✔ For reciprocal roots, use the product of roots directly.
✔ When irrational roots are given, first calculate their sum and product.
✔ Use identities such as:
$$(a+b)(a-b)=a^2-b^2$$
to simplify calculations.
✔ In application questions, define the variable clearly before forming an equation.
✔ Always check whether the answer satisfies the conditions given in the question.
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:
$$-\frac{b}{a}$$
✔ Product of roots:
$$\frac{c}{a}$$
✔ Reciprocal roots:
$$\alpha\beta=1$$
✔ Two distinct real roots:
$$D>0$$
✔ Equal real roots:
$$D=0$$
✔ No real roots:
$$D<0$$
Common Mistakes Students Make
❌ Forgetting the negative sign in the sum of roots formula.
❌ Confusing reciprocal roots with equal roots.
❌ Making errors while multiplying surds.
❌ Using the wrong condition for the discriminant.
❌ Not defining consecutive integers correctly.
❌ Accepting an answer without checking the original condition.
Key Takeaways
✔ The discriminant determines the nature of the roots.
✔ Reciprocal roots have product equal to 1.
✔ The sum and product of roots can be used to form a quadratic equation.
✔ Surd-based roots can often be simplified using algebraic identities.
✔ Real-life number problems can also be represented using quadratic equations.
FAQs
Q. What happens when the discriminant is negative?
Answer:
If:
$$D<0$$
the quadratic equation has no real roots.
Q. What is the product of reciprocal roots?
Answer:
If the roots are reciprocal to each other, their product is:
$$1$$
Q. How can an equation be formed when the roots are given?
Answer:
Use:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
where $$\alpha$$ and $$\beta$$ are the given roots.
Q. What are consecutive integers?
Answer:
Two integers that differ by 1 are called consecutive integers. They can be represented as:
$$x,\ x+1$$
Q. What is the discriminant formula?
Answer:
For:
$$ax^2+bx+c=0$$
the discriminant is:
$$D=b^2-4ac$$
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh NCERT-based practice covering factorisation, reciprocal roots, discriminant, irrational roots, and application-based questions.
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