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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, exam-oriented MCQs on factorisation, quadratic formula, discriminant, nature of roots, and application-based questions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 15
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
Quadratic Equations is an important chapter in Class 10 Mathematics. A quadratic equation is generally written in the form:
$$ax^2+bx+c=0,\quad a\ne0$$
Students learn different methods to solve quadratic equations, including factorisation, completing the square, and the quadratic formula. The chapter also introduces the discriminant, which helps determine the nature of the roots.
This practice set contains 5 fresh MCQs covering important concepts, calculations, and application-based problems.
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 quadratic equations
Why This Topic Is Important
Quadratic Equations develops algebraic reasoning and problem-solving skills. It is useful for solving mathematical problems involving numbers, dimensions, areas, and other real-life situations.
Understanding the discriminant and the relationship between roots and coefficients is especially important for examination preparation.
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) $$2,6$$
B) $$3,4$$
C) $$-3,-4$$
D) $$1,12$$
Answer:
B) $$3,4$$
Useful Concept
For factorisation, find two numbers whose product is the constant term and whose sum is the coefficient of $$x$$.
Solution
Given:
$$x^2-7x+12=0$$
We need two numbers whose product is:
$$12$$
and whose sum is:
$$-7$$
The required numbers are:
$$-3\quad\text{and}\quad-4$$
Therefore:
$$x^2-3x-4x+12=0$$
Grouping:
$$x(x-3)-4(x-3)=0$$
Therefore:
$$(x-3)(x-4)=0$$
Thus:
$$x-3=0$$
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. What is the value of the discriminant of the quadratic equation
$$3x^2-5x+2=0$$
?
A) $$1$$
B) $$5$$
C) $$7$$
D) $$9$$
Answer:
A) $$1$$
Useful Formula
The discriminant of:
$$ax^2+bx+c=0$$
is:
$$D=b^2-4ac$$
Solution
Given:
$$3x^2-5x+2=0$$
Therefore:
$$a=3,\quad b=-5,\quad c=2$$
Using:
$$D=b^2-4ac$$
we get:
$$D=(-5)^2-4(3)(2)$$
$$D=25-24$$
Therefore:
$$D=1$$
Hence, the correct answer is:
A) $$1$$
Since:
$$D>0$$
the equation has two distinct real roots.
Q3. If the roots of the quadratic equation
$$2x^2-7x+3=0$$
are $$\alpha$$ and $$\beta$$, then what is the value of
$$\alpha\beta$$
?
A) $$\frac{3}{2}$$
B) $$\frac{7}{2}$$
C) $$\frac{2}{3}$$
D) $$-\frac{3}{2}$$
Answer:
A) $$\frac{3}{2}$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$2x^2-7x+3=0$$
Therefore:
$$a=2,\quad b=-7,\quad c=3$$
Using:
$$\alpha\beta=\frac{c}{a}$$
we get:
$$\alpha\beta=\frac{3}{2}$$
Hence, the correct answer is:
A) $$\frac{3}{2}$$
Q4. 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
For equal roots:
$$D=0$$
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:
$$(-6)^2-4(1)(k)=0$$
$$36-4k=0$$
Therefore:
$$4k=36$$
Hence:
$$k=9$$
Therefore, the correct answer is:
B) $$9$$
Q5. The product of two consecutive positive integers is $$182$$. What is the smaller integer?
A) $$12$$
B) $$13$$
C) $$14$$
D) $$15$$
Answer:
B) $$13$$
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 consecutive integer is:
$$x+1$$
According to the question:
$$x(x+1)=182$$
Expanding:
$$x^2+x=182$$
Bring all terms to one side:
$$x^2+x-182=0$$
Factorising:
$$x^2+14x-13x-182=0$$
Grouping:
$$x(x+14)-13(x+14)=0$$
Therefore:
$$(x+14)(x-13)=0$$
Thus:
$$x+14=0$$
or:
$$x-13=0$$
Therefore:
$$x=-14$$
or:
$$x=13$$
Since the question asks for a positive integer, we reject:
$$x=-14$$
Therefore:
$$x=13$$
Hence, the correct answer is:
B) $$13$$
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.
Consecutive Positive Integers
If the smaller integer is:
$$x$$
then the next consecutive integer is:
$$x+1$$
Related Practice Questions
Q1. Find the roots of:
$$x^2-11x+30=0$$
Q2. Find the discriminant of:
$$2x^2+3x-2=0$$
Q3. Find the product of roots of:
$$5x^2-8x+3=0$$
Q4. Find the value of $$k$$ if:
$$x^2+12x+k=0$$
has equal roots.
Q5. The product of two consecutive positive integers is $$240$$. Find the smaller integer.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. Find the discriminant of:
$$x^2-10x+25=0$$
Q2. Find the sum of roots of:
$$2x^2-9x+4=0$$
Q3. Find the product of roots of:
$$7x^2-5x+3=0$$
Q4. For what value of $$k$$ will:
$$x^2-8x+k=0$$
have equal roots?
Q5. If one root of:
$$x^2-9x+k=0$$
is $$4$$, find the value of $$k$$.
Exam Tips
✔ First identify $$a$$, $$b$$, and $$c$$ correctly.
✔ Pay special attention to the sign of $$b$$.
✔ For equal roots, remember:
$$D=0$$
✔ Use the relationship between roots and coefficients whenever it saves calculation.
✔ In factorisation, verify the factors before selecting the answer.
✔ In word problems, define the variable clearly.
✔ Reject mathematically possible 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:
$$D=0$$
✔ Two distinct real roots:
$$D>0$$
✔ No real roots:
$$D<0$$
Common Mistakes Students Make
❌ Forgetting the negative sign in:
$$-\frac{b}{a}$$
❌ Using the wrong value of $$a$$ while calculating the product of roots.
❌ Writing:
$$D=b^2+4ac$$
instead of:
$$D=b^2-4ac$$
❌ Making sign errors during factorisation.
❌ Forgetting to reject a negative value when the question asks for a positive quantity.
❌ Not checking whether the selected MCQ option exactly matches the calculated answer.
Key Takeaways
✔ A quadratic equation has the general form:
$$ax^2+bx+c=0$$
✔ Factorisation can be used to find roots directly.
✔ The discriminant determines the nature of roots.
✔ The sum of roots is:
$$-\frac{b}{a}$$
✔ The product of roots is:
$$\frac{c}{a}$$
✔ Equal roots occur when:
$$D=0$$
✔ Real-life number problems can be represented using quadratic equations.
FAQs
Q. What is a quadratic equation?
Answer:
A quadratic equation is an equation of the form:
$$ax^2+bx+c=0,\quad a\ne0$$
Q. What is the discriminant?
Answer:
The discriminant is:
$$D=b^2-4ac$$
It helps determine the nature of the roots.
Q. When does a quadratic equation have equal roots?
Answer:
A quadratic equation has equal real roots when:
$$D=0$$
Q. What is the product of the roots?
Answer:
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Q. What is the sum of the roots?
Answer:
The sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh, NCERT-based practice on factorisation, discriminant, nature of roots, relationships between roots and coefficients, equal roots, and application-based quadratic equations.
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