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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 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: 11
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
A quadratic equation is an equation in which the highest power of the variable is 2. Class 10 Mathematics Chapter 4 covers important methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
The chapter also introduces the discriminant and explains how the value of the discriminant determines the nature of the roots. These concepts are important for solving both direct and application-based questions.
This practice set contains 5 fresh MCQs designed for Class 10 examination preparation.
What You Will Learn
✔ Quadratic equations in standard form
✔ Factorisation method
✔ Quadratic formula
✔ Discriminant
✔ Nature of roots
✔ Sum and product of roots
✔ Finding unknown coefficients
✔ Application-based quadratic equations
Why This Topic Is Important
Quadratic Equations is an important chapter for Class 10 Mathematics. It develops algebraic thinking and helps students solve problems involving numbers, dimensions, areas, and other real-life situations.
Understanding the relationship between roots and coefficients also makes many questions easier to solve without finding the roots directly.
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
$$x^2-7x+12=0$$
are:
A) $$2,5$$
B) $$3,4$$
C) $$-3,-4$$
D) $$1,12$$
Answer:
B) $$3,4$$
Useful Concept
A quadratic equation can be solved by factorisation.
Solution
Given:
$$x^2-7x+12=0$$
We need two numbers whose product is $$12$$ and whose sum is $$-7$$.
The numbers are:
$$-3\quad\text{and}\quad-4$$
Therefore:
$$x^2-3x-4x+12=0$$
Grouping the terms:
$$x(x-3)-4(x-3)=0$$
Therefore:
$$(x-3)(x-4)=0$$
So:
$$x-3=0$$
or:
$$x-4=0$$
Hence:
$$x=3$$
or:
$$x=4$$
Therefore, the roots are:
$$3,\quad4$$
Hence, the correct answer is:
B) $$3,4$$
Q2. For the quadratic equation
$$2x^2+7x+3=0$$
what is the sum of its roots?
A) $$\frac{7}{2}$$
B) $$-\frac{7}{2}$$
C) $$\frac{3}{2}$$
D) $$-\frac{3}{2}$$
Answer:
B) $$-\frac{7}{2}$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the sum of the roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Solution
Given:
$$2x^2+7x+3=0$$
Therefore:
$$a=2,\quad b=7,\quad c=3$$
Using:
$$\alpha+\beta=-\frac{b}{a}$$
we get:
$$\alpha+\beta=-\frac{7}{2}$$
Hence, the correct answer is:
B) $$-\frac{7}{2}$$
Q3. For what value of $$k$$ will the quadratic equation
$$x^2+6x+k=0$$
have equal roots?
A) $$6$$
B) $$9$$
C) $$12$$
D) $$18$$
Answer:
B) $$9$$
Useful Formula
A quadratic equation has equal roots when:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
Given:
$$x^2+6x+k=0$$
Therefore:
$$a=1,\quad b=6,\quad c=k$$
For equal roots:
$$D=0$$
Therefore:
$$b^2-4ac=0$$
Substituting the values:
$$6^2-4(1)(k)=0$$
$$36-4k=0$$
Therefore:
$$4k=36$$
Hence:
$$k=9$$
Therefore, the correct answer is:
B) $$9$$
Q4. If one root of the quadratic equation
$$3x^2-10x+k=0$$
is $$2$$, then the value of $$k$$ is:
A) $$4$$
B) $$6$$
C) $$8$$
D) $$10$$
Answer:
C) $$8$$
Useful Concept
If a given number is a root of a quadratic equation, substituting that value for $$x$$ makes the equation equal to zero.
Solution
Given:
$$3x^2-10x+k=0$$
One root is:
$$x=2$$
Substitute $$x=2$$:
$$3(2)^2-10(2)+k=0$$
$$3(4)-20+k=0$$
$$12-20+k=0$$
$$k-8=0$$
Therefore:
$$k=8$$
Hence, the correct answer is:
C) $$8$$
Q5. The product of two consecutive positive integers is $$90$$. What is the smaller integer?
A) $$8$$
B) $$9$$
C) $$10$$
D) $$11$$
Answer:
B) $$9$$
Useful Concept
If the smaller positive integer is $$x$$, then the next consecutive integer is:
$$x+1$$
Solution
Let the smaller integer be:
$$x$$
Then the next integer is:
$$x+1$$
According to the question:
$$x(x+1)=90$$
Expanding:
$$x^2+x=90$$
Bring all terms to one side:
$$x^2+x-90=0$$
Factorising:
$$x^2+10x-9x-90=0$$
Grouping the terms:
$$x(x+10)-9(x+10)=0$$
Therefore:
$$(x+10)(x-9)=0$$
So:
$$x+10=0$$
or:
$$x-9=0$$
Thus:
$$x=-10$$
or:
$$x=9$$
Since the question asks for a positive integer, we take:
$$x=9$$
Therefore, the smaller integer is:
$$9$$
Hence, the correct answer is:
B) $$9$$
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
$$\alpha\beta=\frac{c}{a}$$
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.
Consecutive Positive Integers
If the smaller integer is:
$$x$$
then the next integer is:
$$x+1$$
Related Practice Questions
Q1. Find the roots of:
$$x^2-11x+30=0$$
Q2. Find the sum of roots of:
$$5x^2-8x+3=0$$
Q3. Find the value of $$k$$ if:
$$x^2+4x+k=0$$
has equal roots.
Q4. If one root of:
$$2x^2-7x+k=0$$
is $$2$$, find $$k$$.
Q5. The product of two consecutive positive integers is $$132$$. Find the smaller integer.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$x^2-10x+25=0$$
Q2. What is the product of roots of:
$$3x^2-5x+7=0$$
Q3. What is the sum of roots of:
$$4x^2+9x-2=0$$
Q4. For what value of $$k$$ will:
$$x^2-8x+k=0$$
have equal roots?
Q5. If one root of:
$$x^2-6x+k=0$$
is $$2$$, find $$k$$.
Exam Tips
✔ Always identify $$a$$, $$b$$, and $$c$$ correctly.
✔ Remember that the coefficient of $$x$$ includes its sign.
✔ For equal roots, use:
$$D=0$$
✔ If one root is given, substitute it directly into the equation.
✔ In factorisation, carefully check the signs of the factors.
✔ In word problems, reject values that do not satisfy the conditions of 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:
$$\alpha+\beta=-\frac{b}{a}$$
✔ 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 negative sign in:
$$-\frac{b}{a}$$
❌ Writing the wrong value of $$b$$ when the coefficient is negative.
❌ Using the wrong condition for equal roots.
❌ Making sign errors during factorisation.
❌ Forgetting to substitute the given root correctly.
❌ Accepting a negative answer when the question asks for a positive integer.
Key Takeaways
✔ Factorisation can be used to find the roots of many quadratic equations.
✔ The sum of roots is:
$$-\frac{b}{a}$$
✔ The product of roots is:
$$\frac{c}{a}$$
✔ Equal roots occur when:
$$D=0$$
✔ A known root can be substituted directly into the equation to find an unknown coefficient.
✔ Consecutive integer problems can be converted into quadratic equations.
FAQs
Q. What is the condition for equal roots?
Answer:
For a quadratic equation:
$$ax^2+bx+c=0$$
the roots are equal when:
$$D=0$$
That is:
$$b^2-4ac=0$$
Q. What is the sum of roots of a quadratic equation?
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 can we find an unknown coefficient when one root is given?
Answer:
Substitute the given root into the quadratic equation and set the equation equal to zero.
Q. How are two consecutive positive integers represented?
Answer:
If the smaller integer is $$x$$, the next consecutive integer is:
$$x+1$$
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 quadratic equations.
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