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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 roots, factorisation, discriminant, quadratic formula, and real-life applications with detailed solutions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 13
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
A quadratic equation is an equation of the form:
$$ax^2+bx+c=0,\quad a\ne0$$
In Class 10 Mathematics, students learn how to solve quadratic equations using factorisation, completing the square, and the quadratic formula. The chapter also explains how the discriminant can be used to determine whether the roots are real and distinct, equal, or not real.
This practice set contains five fresh MCQs covering important concepts and application-based questions from Quadratic Equations.
What You Will Learn
✔ Standard form of a quadratic equation
✔ Factorisation
✔ Quadratic formula
✔ Discriminant
✔ Nature of roots
✔ Sum and product of roots
✔ Equal roots
✔ Application-based problems
Why This Topic Is Important
Quadratic Equations is an important algebraic topic in Class 10 Mathematics. It helps students develop logical thinking and systematic problem-solving skills.
The concepts in this chapter are also useful for solving problems involving numbers, areas, dimensions, and other real-life situations.
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 a factorised form of the quadratic equation
$$x^2-9x+20=0$$
A) $$(x-4)(x-5)=0$$
B) $$(x+4)(x+5)=0$$
C) $$(x-2)(x-10)=0$$
D) $$(x+2)(x-10)=0$$
Answer:
A) $$(x-4)(x-5)=0$$
Useful Concept
For factorisation, we look for two numbers whose product is the constant term and whose sum is the coefficient of $$x$$.
Solution
Given:
$$x^2-9x+20=0$$
We need two numbers whose:
Product is:
$$20$$
and sum is:
$$-9$$
The required numbers are:
$$-4\quad\text{and}\quad-5$$
Therefore:
$$x^2-4x-5x+20=0$$
Grouping:
$$x(x-4)-5(x-4)=0$$
Therefore:
$$(x-4)(x-5)=0$$
Hence, the correct answer is:
A) $$(x-4)(x-5)=0$$
Q2. If the roots of the quadratic equation
$$2x^2-9x+7=0$$
are $$\alpha$$ and $$\beta$$, what is the value of
$$\alpha+\beta$$
?
A) $$\frac{7}{2}$$
B) $$\frac{9}{2}$$
C) $$-\frac{9}{2}$$
D) $$\frac{2}{9}$$
Answer:
B) $$\frac{9}{2}$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the sum of the roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Solution
Given:
$$2x^2-9x+7=0$$
Therefore:
$$a=2,\quad b=-9,\quad c=7$$
Using:
$$\alpha+\beta=-\frac{b}{a}$$
we get:
$$\alpha+\beta=-\frac{-9}{2}$$
Therefore:
$$\alpha+\beta=\frac{9}{2}$$
Hence, the correct answer is:
B) $$\frac{9}{2}$$
Q3. For what value of $$k$$ will the quadratic equation
$$x^2-4x+k=0$$
have equal roots?
A) $$2$$
B) $$4$$
C) $$8$$
D) $$16$$
Answer:
B) $$4$$
Useful Formula
For equal roots:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
Given:
$$x^2-4x+k=0$$
Therefore:
$$a=1,\quad b=-4,\quad c=k$$
For equal roots:
$$D=0$$
Therefore:
$$(-4)^2-4(1)(k)=0$$
$$16-4k=0$$
Therefore:
$$4k=16$$
Hence:
$$k=4$$
Therefore, the correct answer is:
B) $$4$$
Q4. The quadratic equation
$$4x^2+4x+1=0$$
has:
A) Two distinct real roots
B) Two equal real roots
C) No real roots
D) One positive and one negative root
Answer:
B) Two equal real roots
Useful Formula
The discriminant is:
$$D=b^2-4ac$$
If:
$$D=0$$
the quadratic equation has two equal real roots.
Solution
Given:
$$4x^2+4x+1=0$$
Therefore:
$$a=4,\quad b=4,\quad c=1$$
Calculate the discriminant:
$$D=4^2-4(4)(1)$$
$$D=16-16$$
Therefore:
$$D=0$$
Since the discriminant is zero, the equation has two equal real roots.
Hence, the correct answer is:
B) Two equal real roots
Q5. The product of two consecutive positive integers is $$132$$. What is the smaller integer?
A) $$10$$
B) $$11$$
C) $$12$$
D) $$13$$
Answer:
B) $$11$$
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:
$$x(x+1)=132$$
Expanding:
$$x^2+x=132$$
Bring all terms to one side:
$$x^2+x-132=0$$
Factorising:
$$x^2+12x-11x-132=0$$
Grouping:
$$x(x+12)-11(x+12)=0$$
Therefore:
$$(x+12)(x-11)=0$$
So:
$$x+12=0$$
or:
$$x-11=0$$
Therefore:
$$x=-12$$
or:
$$x=11$$
Since the question asks for a positive integer:
$$x=11$$
Hence, the correct answer is:
B) $$11$$
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:
$$x^2-8x+15=0$$
Q2. Find the sum of roots of:
$$3x^2-10x+4=0$$
Q3. Find the product of roots of:
$$5x^2+7x-6=0$$
Q4. Find the value of $$k$$ if:
$$x^2+6x+k=0$$
has equal roots.
Q5. The product of two consecutive positive integers is $$156$$. Find the smaller integer.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$x^2-12x+36=0$$
Q2. What is the sum of roots of:
$$4x^2-5x+2=0$$
Q3. What is the product of roots of:
$$3x^2-7x+5=0$$
Q4. For what value of $$k$$ will:
$$x^2-6x+k=0$$
have equal roots?
Q5. If the roots are $$6$$ and $$-2$$, form the quadratic equation.
Exam Tips
✔ Identify $$a$$, $$b$$, and $$c$$ carefully.
✔ Remember the sign of $$b$$ while calculating the discriminant.
✔ For equal roots, use:
$$D=0$$
✔ In factorisation, check both the sum and product of the numbers.
✔ When solving word problems, define the unknown quantity first.
✔ Reject negative values when the variable represents a length or positive 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:
$$\alpha\beta=\frac{c}{a}$$
✔ Equal roots:
$$D=0$$
✔ Distinct real roots:
$$D>0$$
✔ No real roots:
$$D<0$$
Common Mistakes Students Make
❌ Forgetting the negative sign in:
$$-\frac{b}{a}$$
❌ Using the wrong pair of numbers during factorisation.
❌ Writing $$b^2+4ac$$ instead of:
$$b^2-4ac$$
❌ Confusing equal roots with distinct roots.
❌ Forgetting to reject a negative value in an application-based question.
❌ Not checking the final answer against the original equation.
Key Takeaways
✔ Factorisation can quickly solve suitable quadratic equations.
✔ The sum of roots is:
$$-\frac{b}{a}$$
✔ The product of roots is:
$$\frac{c}{a}$$
✔ The discriminant determines the nature of roots.
✔ Equal roots occur when:
$$D=0$$
✔ Consecutive integer problems can be converted into quadratic equations.
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. What is the condition for equal roots?
Answer:
The condition for equal roots is:
$$D=0$$
or:
$$b^2-4ac=0$$
Q. How can the sum of roots be found?
Answer:
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Q. How can the product of roots be found?
Answer:
The product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Q. How are 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, and application-based quadratic equations.
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