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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 roots, discriminant, factorisation, quadratic equations, 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: 10
Difficulty Level: Moderate
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations introduces equations in which the highest power of the variable is 2. Students learn different methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
The chapter also covers the discriminant, which helps determine the nature of the roots. The relationship between roots and coefficients is useful for solving many conceptual and application-based problems.
This practice set contains 5 fresh MCQs based on important concepts from Class 10 Mathematics Chapter 4.
What You Will Learn
✔ Standard form of a quadratic equation
✔ Factorisation
✔ Quadratic formula
✔ Discriminant
✔ Nature of roots
✔ Relationship between roots and coefficients
✔ Formation of quadratic equations
✔ Application-based problems
Why This Topic Is Important
Quadratic Equations is an important chapter in Class 10 Mathematics. It develops algebraic reasoning and problem-solving skills.
The concepts of roots, discriminant, factorisation, and relationships between roots and coefficients are useful in board examinations and higher-level mathematics.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Mathematics Revision
Q1. Which of the following is the quadratic equation whose roots are $$4$$ and $$-3$$?
A) $$x^2-x-12=0$$
B) $$x^2+x-12=0$$
C) $$x^2-x+12=0$$
D) $$x^2+7x-12=0$$
Answer:
A) $$x^2-x-12=0$$
Useful Concept
If the roots of a quadratic equation are $$\alpha$$ and $$\beta$$, then the equation is:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Solution
Given:
$$\alpha=4,\quad \beta=-3$$
Sum of roots:
$$\alpha+\beta=4+(-3)=1$$
Product of roots:
$$\alpha\beta=4(-3)=-12$$
Therefore:
$$x^2-(1)x+(-12)=0$$
Hence:
$$x^2-x-12=0$$
Therefore, the correct answer is:
A) $$x^2-x-12=0$$
Q2. What is the nature of the roots of the quadratic 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 real root
Answer:
C) No real roots
Useful Formula
For the quadratic equation:
$$ax^2+bx+c=0$$
the discriminant is:
$$D=b^2-4ac$$
If:
$$D<0$$
the equation has no real roots.
Solution
Given:
$$5x^2-6x+2=0$$
Therefore:
$$a=5,\quad b=-6,\quad c=2$$
Now 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
Q3. If the roots of the quadratic equation $$x^2-9x+14=0$$ are $$\alpha$$ and $$\beta$$, then the value of $$\frac{1}{\alpha}+\frac{1}{\beta}$$ is:
A) $$\frac{9}{14}$$
B) $$\frac{14}{9}$$
C) $$\frac{7}{9}$$
D) $$\frac{9}{7}$$
Answer:
A) $$\frac{9}{14}$$
Useful Concept
We know:
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
The product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$x^2-9x+14=0$$
Therefore:
$$a=1,\quad b=-9,\quad c=14$$
Sum of roots:
$$\alpha+\beta=-\frac{-9}{1}$$
$$\alpha+\beta=9$$
Product of roots:
$$\alpha\beta=\frac{14}{1}$$
$$\alpha\beta=14$$
Now:
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$
Substituting the values:
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{9}{14}$$
Hence, the correct answer is:
A) $$\frac{9}{14}$$
Q4. The sum of two numbers is $$11$$ and their product is $$24$$. Which quadratic equation has these two numbers as its roots?
A) $$x^2+11x+24=0$$
B) $$x^2-11x+24=0$$
C) $$x^2+11x-24=0$$
D) $$x^2-24x+11=0$$
Answer:
B) $$x^2-11x+24=0$$
Useful Concept
If the roots of a quadratic equation are $$\alpha$$ and $$\beta$$, then:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Solution
Given:
$$\alpha+\beta=11$$
and:
$$\alpha\beta=24$$
Substitute these values into:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Therefore:
$$x^2-(11)x+24=0$$
Hence:
$$x^2-11x+24=0$$
Therefore, the correct answer is:
B) $$x^2-11x+24=0$$
Q5. A rectangular garden has an area of $$192,m^2$$. Its length is $$4,m$$ more than its breadth. What is the breadth of the garden?
A) $$12,m$$
B) $$14,m$$
C) $$16,m$$
D) $$18,m$$
Answer:
A) $$12,m$$
Useful Concept
The area of a rectangle is:
$$\text{Area}=\text{Length}\times\text{Breadth}$$
Solution
Let the breadth of the garden be:
$$x,m$$
Then the length is:
$$x+4,m$$
Given:
$$\text{Area}=192,m^2$$
Therefore:
$$x(x+4)=192$$
Expanding:
$$x^2+4x=192$$
Bring all terms to one side:
$$x^2+4x-192=0$$
Now factorise:
$$x^2+16x-12x-192=0$$
Grouping the terms:
$$x(x+16)-12(x+16)=0$$
Therefore:
$$(x+16)(x-12)=0$$
So:
$$x+16=0$$
or:
$$x-12=0$$
Therefore:
$$x=-16$$
or:
$$x=12$$
Since $$x$$ represents the breadth of a garden, its value cannot be negative.
Therefore:
$$x=12,m$$
Hence, the correct answer is:
A) $$12,m$$
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
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Reciprocal of Roots
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$
Nature of Roots
If:
$$D>0$$
the quadratic equation has two distinct real roots.
If:
$$D=0$$
the quadratic equation has two equal real roots.
If:
$$D<0$$
the quadratic equation has no real roots.
Related Practice Questions
Q1. Form a quadratic equation whose roots are $$5$$ and $$-2$$.
Q2. Determine the nature of roots of:
$$2x^2+3x+5=0$$
Q3. If the roots of:
$$3x^2-8x+4=0$$
are $$\alpha$$ and $$\beta$$, find:
$$\frac{1}{\alpha}+\frac{1}{\beta}$$
Q4. Form a quadratic equation whose sum and product of roots are $$7$$ and $$10$$ respectively.
Q5. A rectangle has an area of $$120,m^2$$ and its length is $$2,m$$ more than its breadth. Form the quadratic equation.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$2x^2-7x+3=0$$
Q2. What is the product of roots of:
$$4x^2-9x+5=0$$
Q3. If the roots are $$3$$ and $$-5$$, what is their sum?
Q4. What is the quadratic equation whose roots are $$2$$ and $$7$$?
Q5. If:
$$\alpha+\beta=8$$
and:
$$\alpha\beta=15$$
form the quadratic equation.
Exam Tips
✔ Identify $$a$$, $$b$$, and $$c$$ carefully before calculating the discriminant.
✔ Remember the negative sign in:
$$\alpha+\beta=-\frac{b}{a}$$
✔ When roots are given, use:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
✔ For reciprocal expressions involving roots, use the sum and product of roots.
✔ In word problems, define the variable clearly before forming the equation.
✔ Reject negative values when the variable represents a physical length or quantity.
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:
$$\alpha\beta=\frac{c}{a}$$
✔ Equation with roots $$\alpha,\beta$$:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
✔ 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 sign while forming a quadratic equation from given roots.
❌ Confusing the sum of roots with the product of roots.
❌ Using:
$$b^2+4ac$$
instead of:
$$b^2-4ac$$
❌ Making arithmetic mistakes while calculating the discriminant.
❌ Accepting a negative value when the variable represents a physical length.
Key Takeaways
✔ A quadratic equation can be formed when its roots are known.
✔ The sum and product of roots are directly related to the coefficients.
✔ The discriminant determines the nature of the roots.
✔ Reciprocal expressions involving roots can be solved using the sum and product of roots.
✔ Application-based problems can be converted into quadratic equations.
✔ Always check whether the final answer is meaningful in the context of the problem.
FAQs
Q. What is the standard form of a quadratic equation?
Answer:
The standard form is:
$$ax^2+bx+c=0,\quad a\ne0$$
Q. How can a quadratic equation be formed when its roots are known?
Answer:
If the roots are $$\alpha$$ and $$\beta$$, the equation is:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Q. What does the discriminant tell us?
Answer:
The discriminant determines the nature of the roots.
$$D>0$$
gives two distinct real roots.
$$D=0$$
gives two equal real roots.
$$D<0$$
gives no real roots.
Q. What is the product of roots?
Answer:
For:
$$ax^2+bx+c=0$$
the product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Q. How can reciprocal roots be added?
Answer:
Use:
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh, NCERT-based practice on formation of quadratic equations, roots and coefficients, discriminant, reciprocal roots, factorisation, and application-based problems.
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