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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 quadratic formula, discriminant, roots, factorisation, 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: 12
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 an equation in which the highest power of the variable is 2.
Students learn different methods to solve quadratic equations, including factorisation, completing the square, and the quadratic formula. The discriminant is also used to determine the nature of the roots.
This practice set contains 5 fresh MCQs covering important concepts and exam-oriented applications 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 quadratic equations
Why This Topic Is Important
Quadratic Equations develops algebraic reasoning and problem-solving skills. Questions from this chapter can test both calculation and conceptual understanding.
The relationship between roots and coefficients is particularly useful because it allows students to solve many questions without calculating 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. Which of the following quadratic equations has roots $$2$$ and $$5$$?
A) $$x^2+7x+10=0$$
B) $$x^2-7x+10=0$$
C) $$x^2-7x-10=0$$
D) $$x^2+7x-10=0$$
Answer:
B) $$x^2-7x+10=0$$
Useful Concept
If the roots are $$\alpha$$ and $$\beta$$, then the quadratic equation is:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Solution
Given:
$$\alpha=2,\quad \beta=5$$
Sum of roots:
$$\alpha+\beta=2+5=7$$
Product of roots:
$$\alpha\beta=2\times5=10$$
Therefore:
$$x^2-7x+10=0$$
Hence, the correct answer is:
B) $$x^2-7x+10=0$$
Q2. The discriminant of the quadratic equation
$$2x^2-5x+3=0$$
is:
A) $$1$$
B) $$7$$
C) $$13$$
D) $$25$$
Answer:
A) $$1$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the discriminant is:
$$D=b^2-4ac$$
Solution
Given:
$$2x^2-5x+3=0$$
Therefore:
$$a=2,\quad b=-5,\quad c=3$$
Using:
$$D=b^2-4ac$$
we get:
$$D=(-5)^2-4(2)(3)$$
$$D=25-24$$
Therefore:
$$D=1$$
Hence, the correct answer is:
A) $$1$$
Q3. If the roots of the quadratic equation
$$3x^2-11x+6=0$$
are $$\alpha$$ and $$\beta$$, then the value of
$$\alpha\beta$$
is:
A) $$\frac{6}{3}$$
B) $$\frac{11}{3}$$
C) $$\frac{6}{11}$$
D) $$\frac{11}{6}$$
Answer:
A) $$2$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the product of the roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$3x^2-11x+6=0$$
Therefore:
$$a=3,\quad b=-11,\quad c=6$$
Using:
$$\alpha\beta=\frac{c}{a}$$
we get:
$$\alpha\beta=\frac{6}{3}$$
Therefore:
$$\alpha\beta=2$$
Hence, the correct answer is:
A) $$2$$
Q4. For what value of $$k$$ will the quadratic equation
$$x^2-10x+k=0$$
have equal roots?
A) $$20$$
B) $$25$$
C) $$50$$
D) $$100$$
Answer:
B) $$25$$
Useful Formula
A quadratic equation has equal roots when:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
Given:
$$x^2-10x+k=0$$
Therefore:
$$a=1,\quad b=-10,\quad c=k$$
For equal roots:
$$D=0$$
Therefore:
$$(-10)^2-4(1)(k)=0$$
$$100-4k=0$$
Therefore:
$$4k=100$$
Hence:
$$k=25$$
Therefore, the correct answer is:
B) $$25$$
Q5. The length of a rectangular field is $$5,m$$ more than its breadth. If its area is $$84,m^2$$, what is its breadth?
A) $$5,m$$
B) $$6,m$$
C) $$7,m$$
D) $$8,m$$
Answer:
B) $$7,m$$
Useful Concept
Area of a rectangle is:
$$\text{Area}=\text{Length}\times\text{Breadth}$$
Solution
Let the breadth be:
$$x,m$$
Then the length is:
$$x+5,m$$
Given:
$$x(x+5)=84$$
Expanding:
$$x^2+5x=84$$
Bring all terms to one side:
$$x^2+5x-84=0$$
Factorising:
$$x^2+12x-7x-84=0$$
Grouping:
$$x(x+12)-7(x+12)=0$$
Therefore:
$$(x+12)(x-7)=0$$
So:
$$x+12=0$$
or:
$$x-7=0$$
Therefore:
$$x=-12$$
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:
$$x^2-13x+40=0$$
Q2. Find the discriminant of:
$$3x^2+2x-1=0$$
Q3. If the roots of:
$$4x^2-9x+2=0$$
are $$\alpha$$ and $$\beta$$, find:
$$\alpha+\beta$$
Q4. Find the value of $$k$$ if:
$$x^2+8x+k=0$$
has equal roots.
Q5. A rectangle has an area of $$96,m^2$$. Its length is $$4,m$$ more than its breadth. Form the quadratic equation.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$x^2-6x+9=0$$
Q2. What is the product of roots of:
$$2x^2+7x+5=0$$
Q3. What is the sum of roots of:
$$5x^2-12x+7=0$$
Q4. For what value of $$k$$ will:
$$x^2-12x+k=0$$
have equal roots?
Q5. If the roots are $$4$$ and $$-2$$, form the quadratic equation.
Exam Tips
✔ Identify $$a$$, $$b$$, and $$c$$ carefully.
✔ Remember that the sign of the coefficient must be included.
✔ Use:
$$D=b^2-4ac$$
to determine the nature of roots.
✔ For equal roots, always put:
$$D=0$$
✔ For a word problem, define the unknown quantity before forming the equation.
✔ Check whether the obtained answer is meaningful in the given situation.
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 $$b$$ while calculating the discriminant.
❌ Using $$b^2+4ac$$ instead of $$b^2-4ac$$.
❌ Confusing the sum of roots with the product of roots.
❌ Making sign errors while factorising.
❌ Forgetting that a length cannot have a negative value.
❌ Not checking the final answer in the original condition.
Key Takeaways
✔ The discriminant determines the nature of roots.
✔ The product of roots is:
$$\frac{c}{a}$$
✔ The sum of roots is:
$$-\frac{b}{a}$$
✔ Equal roots occur when:
$$D=0$$
✔ Quadratic equations can be formed from real-life situations.
✔ Factorisation is an effective method when the quadratic expression can be easily factorised.
FAQs
Q. What is the discriminant?
Answer:
For:
$$ax^2+bx+c=0$$
the discriminant is:
$$D=b^2-4ac$$
Q. When does a quadratic equation have equal roots?
Answer:
A quadratic equation has equal roots when:
$$D=0$$
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. What is the sum of roots?
Answer:
The sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Q. Which method can be used to solve a quadratic equation?
Answer:
A quadratic equation can be solved using factorisation, completing the square, or the quadratic formula, depending on the equation.
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh, NCERT-based practice on factorisation, discriminant, roots and coefficients, equal roots, and application-based quadratic equations.
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