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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers based on the NCERT syllabus. Solve fresh MCQs on quadratic formula, discriminant, roots, factorisation, and application-based concepts with detailed solutions.
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Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 3
Difficulty Level: Moderate
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations focuses on equations of degree 2 and different methods used to find their roots. After learning basic factorisation and relationships between roots and coefficients, students should also understand how to apply the quadratic formula and discriminant.
This practice set focuses on slightly more challenging calculations involving the quadratic formula, discriminant, and conditions related to the roots.
What You Will Learn?
✔ Quadratic formula
✔ Discriminant-based questions
✔ Nature of roots
✔ Finding unknown coefficients
✔ Relationship between roots and coefficients
✔ Application of quadratic equations
✔ Board examination concepts
Why This Topic Is Important?
Quadratic Equations is an important Class 10 Mathematics chapter because questions can test both calculation and reasoning. Understanding the discriminant and quadratic formula allows students to solve equations that cannot be easily factorised.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Mathematics Practice Tests
Q1. The roots of the quadratic equation
$$2x^2-7x+3=0$$
are:
A) $$3,\frac{1}{2}$$
B) $$2,\frac{3}{2}$$
C) $$1,\frac{3}{2}$$
D) $$3,2$$
Answer:
A) $$3,\frac{1}{2}$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the quadratic formula is:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Solution
Given:
$$2x^2-7x+3=0$$
Therefore:
$$a=2,\quad b=-7,\quad c=3$$
Using the quadratic formula:
$$x=\frac{-(-7)\pm\sqrt{(-7)^2-4(2)(3)}}{2(2)}$$
$$x=\frac{7\pm\sqrt{49-24}}{4}$$
$$x=\frac{7\pm\sqrt{25}}{4}$$
$$x=\frac{7\pm5}{4}$$
Therefore:
$$x=\frac{7+5}{4}=3$$
and
$$x=\frac{7-5}{4}=\frac{1}{2}$$
Hence, the roots are:
$$3,\frac{1}{2}$$
Therefore, the correct answer is:
A) $$3,\frac{1}{2}$$
Q2. 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:
$$(-6)^2-4(1)(k)=0$$
$$36-4k=0$$
$$4k=36$$
$$k=9$$
Hence, the correct answer is:
B) $$9$$
Q3. If one root of the equation
$$x^2-8x+15=0$$
is $$3$$, what is the other root?
A) $$4$$
B) $$5$$
C) $$6$$
D) $$7$$
Answer:
B) $$5$$
Useful Formula
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Solution
Given:
$$x^2-8x+15=0$$
Here:
$$a=1,\quad b=-8$$
Therefore:
$$\alpha+\beta=-\frac{-8}{1}=8$$
One root is:
$$\alpha=3$$
Let the other root be $$\beta$$.
Then:
$$3+\beta=8$$
Therefore:
$$\beta=5$$
Hence, the other root is:
$$\boxed{5}$$
Therefore, the correct answer is:
B) $$5$$
Q4. The discriminant of the quadratic equation
$$3x^2+2x+5=0$$
is:
A) $$-56$$
B) $$56$$
C) $$64$$
D) $$-64$$
Answer:
A) $$-56$$
Useful Formula
$$D=b^2-4ac$$
Solution
Given:
$$3x^2+2x+5=0$$
Therefore:
$$a=3,\quad b=2,\quad c=5$$
Using:
$$D=b^2-4ac$$
we get:
$$D=(2)^2-4(3)(5)$$
$$D=4-60$$
$$D=-56$$
Hence, the correct answer is:
A) $$-56$$
Since:
$$D<0$$
the equation has no real roots.
Q5. If the roots of the quadratic equation
$$x^2-(m+3)x+12=0$$
are $$3$$ and $$4$$, what is the value of $$m$$?
A) $$2$$
B) $$3$$
C) $$4$$
D) $$5$$
Answer:
B) $$4$$
Useful Formula
The sum of roots is:
$$\alpha+\beta=-\frac{b}{a}$$
Solution
The given roots are:
$$3,\quad4$$
Therefore, their sum is:
$$3+4=7$$
The equation is:
$$x^2-(m+3)x+12=0$$
Here:
$$a=1$$
and
$$b=-(m+3)$$
Therefore:
$$\alpha+\beta=-\frac{b}{a}$$
$$7=-\frac{-(m+3)}{1}$$
$$7=m+3$$
Therefore:
$$m=4$$
Hence, the correct answer is:
C) $$4$$
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:
$$D>0$$
Two distinct real roots.
If:
$$D=0$$
Two equal real roots.
If:
$$D<0$$
No real roots.
Related Practice Questions
Q1. Solve using the quadratic formula:
$$x^2-4x-5=0$$
Q2. Find the value of $$k$$ if:
$$x^2-10x+k=0$$
has equal roots.
Q3. Find the other root if one root of:
$$2x^2-9x+10=0$$
is $$2$$.
Q4. Find the discriminant of:
$$4x^2-3x+2=0$$
Q5. If the roots are $$5$$ and $$-2$$, form the quadratic equation.
Mini Quiz Challenge
Try these questions without using a calculator.
Q1. What is the quadratic formula?
Q2. What is the discriminant of $$x^2-4x+4=0$$?
Q3. What happens when $$D<0$$?
Q4. If the sum of roots is 9 and one root is 4, what is the other root?
Q5. If roots are 2 and 6, what is their product?
Challenge: Complete all five within 60 seconds.
Exam Tips
✔ Write the values of $$a$$, $$b$$ and $$c$$ separately before using the quadratic formula.
✔ Be careful with negative values of $$b$$.
✔ Calculate the discriminant before proceeding with the quadratic formula.
✔ If one root is already known, use the sum or product of roots to find the other root quickly.
✔ Always verify the answer when the calculation is complicated.
Quick Revision Notes
✔ Quadratic formula:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
✔ Discriminant:
$$D=b^2-4ac$$
✔ Equal roots:
$$D=0$$
✔ Distinct real roots:
$$D>0$$
✔ No real roots:
$$D<0$$
✔ Sum of roots:
$$-\frac{b}{a}$$
✔ Product of roots:
$$\frac{c}{a}$$
Common Mistakes Students Make
❌ Forgetting the denominator $$2a$$ in the quadratic formula.
❌ Using $$b$$ instead of $$-b$$ in the numerator.
❌ Making calculation errors while finding the discriminant.
❌ Forgetting that a negative discriminant means no real roots.
❌ Using the wrong coefficient for $$a$$.
Key Takeaways
✔ The quadratic formula can solve any quadratic equation.
✔ The discriminant helps determine the nature of roots.
✔ The sum and product of roots can make many problems easier.
✔ Unknown coefficients can be found using the relationships between roots and coefficients.
✔ Careful sign handling is essential in quadratic equation problems.
FAQs
Q. What is the quadratic formula?
Answer:
For:
$$ax^2+bx+c=0$$
the roots are given by:
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Q. What does a negative discriminant indicate?
Answer:
If:
$$D<0$$
the quadratic equation has no real roots.
Q. What is the condition for equal roots?
Answer:
The condition is:
$$D=0$$
Q. How can one root be used to find the other root?
Answer:
Use the sum of roots:
$$\alpha+\beta=-\frac{b}{a}$$
or the product:
$$\alpha\beta=\frac{c}{a}$$
Q. What is the importance of the discriminant?
Answer:
The discriminant determines whether the quadratic equation has two distinct real roots, equal real roots, or no real roots.
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh NCERT-based practice on the quadratic formula, discriminant, roots, and relationships between roots and coefficients.
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