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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions based on the latest NCERT syllabus. Solve fresh and exam-oriented MCQs on roots, discriminant, factorisation, quadratic formula, 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: 9
Difficulty Level: Moderate to Difficult
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations deals with equations in which the highest power of the variable is 2. In Class 10 Mathematics, students learn different methods for solving quadratic equations, including factorisation and the quadratic formula.
The chapter also introduces the discriminant, which helps determine the nature of the roots. The relationship between roots and coefficients is also useful for solving many conceptual and application-based questions.
This practice set contains 5 fresh MCQs designed to strengthen students’ understanding of important concepts from Chapter 4.
What Will You Learn?
✔ Quadratic equations in standard form
✔ Factorisation method
✔ Quadratic formula
✔ Discriminant
✔ Nature of roots
✔ Relationship between roots and coefficients
✔ Formation of quadratic equations
✔ Application-based problems
Why Is This Topic Important?
Quadratic Equations is an important chapter for Class 10 Mathematics examinations. Questions can be asked directly from formulas as well as through application-based situations involving numbers, dimensions, and other mathematical relationships.
Understanding the relationship between roots and coefficients helps students solve many questions quickly and accurately.
Exam Relevance
These questions are useful for:
✔ CBSE Board Exams
✔ State Board Exams
✔ School Unit Tests
✔ Half-Yearly Exams
✔ Annual Exams
✔ Mathematics Revision
Q1. If the roots of the quadratic equation
$$2x^2-7x+3=0$$
are $$\alpha$$ and $$\beta$$, then the value of
$$\alpha^2+\beta^2$$
is:
A) $$\frac{37}{4}$$
B) $$\frac{25}{4}$$
C) $$\frac{13}{4}$$
D) $$\frac{49}{4}$$
Answer:
A) $$\frac{37}{4}$$
Useful Concept
For a quadratic equation:
$$ax^2+bx+c=0$$
the sum and product of roots are:
$$\alpha+\beta=-\frac{b}{a}$$
and
$$\alpha\beta=\frac{c}{a}$$
Also:
$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$
Solution
Given:
$$2x^2-7x+3=0$$
Therefore:
$$a=2,\quad b=-7,\quad c=3$$
Sum of roots:
$$\alpha+\beta=-\frac{-7}{2}$$
$$\alpha+\beta=\frac{7}{2}$$
Product of roots:
$$\alpha\beta=\frac{3}{2}$$
Now:
$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$
Substituting the values:
$$\alpha^2+\beta^2=\left(\frac{7}{2}\right)^2-2\left(\frac{3}{2}\right)$$
$$=\frac{49}{4}-3$$
$$=\frac{49}{4}-\frac{12}{4}$$
$$=\frac{37}{4}$$
Hence, the correct answer is:
A) $$\frac{37}{4}$$
Q2. For which values of $$k$$ will the quadratic equation
$$x^2-(k+2)x+9=0$$
have equal roots?
A) $$k=4$$ only
B) $$k=-8$$ only
C) $$k=4\text{ or }k=-8$$
D) $$k=6$$
Answer:
C) $$k=4\text{ or }k=-8$$
Useful Formula
For equal roots:
$$D=0$$
where:
$$D=b^2-4ac$$
Solution
Given:
$$x^2-(k+2)x+9=0$$
Here:
$$a=1$$
$$b=-(k+2)$$
$$c=9$$
For equal roots:
$$D=0$$
Therefore:
$$[-(k+2)]^2-4(1)(9)=0$$
$$
(k+2)^2-36=0
$$
Therefore:
$$
(k+2)^2=36
$$
Taking square roots:
$$
k+2=\pm6
$$
Therefore:
$$
k+2=6
$$
or
$$
k+2=-6
$$
Hence:
$$
k=4
$$
or
$$
k=-8
$$
Therefore, both values make the roots equal.
Hence, the correct answer is:
C) $$k=4\text{ or }k=-8$$
Q3. If one root of the quadratic equation
$$2x^2-5x+k=0$$
is $$2$$, then the value of $$k$$ is:
A) $$1$$
B) $$2$$
C) $$3$$
D) $$4$$
Answer:
B) $$2$$
Useful Concept
If a number is a root of a quadratic equation, substituting that value for the variable makes the equation equal to zero.
Solution
Given:
$$2x^2-5x+k=0$$
One root is:
$$x=2$$
Substitute:
$$2(2)^2-5(2)+k=0$$
$$8-10+k=0$$
$$k-2=0$$
Therefore:
$$k=2$$
Hence, the correct answer is:
B) $$2$$
Q4. The quadratic equation
$$3x^2-4x+5=0$$
has:
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
The discriminant of:
$$ax^2+bx+c=0$$
is:
$$D=b^2-4ac$$
If:
$$D<0$$
the equation has no real roots.
Solution
Given:
$$3x^2-4x+5=0$$
Therefore:
$$a=3,\quad b=-4,\quad c=5$$
Calculate the discriminant:
$$D=(-4)^2-4(3)(5)$$
$$D=16-60$$
$$D=-44$$
Since:
$$D<0$$
the equation has no real roots.
Hence, the correct answer is:
C) No real roots
Q5. The product of two consecutive positive integers is $$156$$. Which quadratic equation represents the smaller integer?
A) $$x^2+x-156=0$$
B) $$x^2-x-156=0$$
C) $$x^2+x+156=0$$
D) $$x^2-x+156=0$$
Answer:
A) $$x^2+x-156=0$$
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:
$$x(x+1)=156$$
Expanding:
$$x^2+x=156$$
Bringing all terms to one side:
$$x^2+x-156=0$$
Therefore, the required quadratic equation is:
$$x^2+x-156=0$$
Hence, the correct answer is:
A) $$x^2+x-156=0$$
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
$$\alpha\beta=\frac{c}{a}$$
Square of Roots
$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$
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 integer is:
$$x+1$$
Related Practice Questions
Q1. Find the roots of:
$$x^2-9x+20=0$$
Q2. Find the value of $$k$$ if:
$$x^2-(k+1)x+16=0$$
has equal roots.
Q3. If the roots of:
$$3x^2-7x+2=0$$
are $$\alpha$$ and $$\beta$$, find:
$$\alpha^2+\beta^2$$
Q4. Determine the nature of roots of:
$$4x^2+4x+7=0$$
Q5. The product of two consecutive positive integers is $$210$$. Form the quadratic equation.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant of:
$$x^2-8x+16=0$$
Q2. What is the condition for equal roots?
Q3. What is the product of roots of:
$$5x^2-9x+4=0$$
Q4. If the smaller of two consecutive integers is $$x$$, what is the larger integer?
Q5. What is the sum of roots of:
$$2x^2-11x+7=0$$
Exam Tips
✔ First identify the values of $$a$$, $$b$$, and $$c$$ before using the discriminant.
✔ Remember that the coefficient of $$x$$ is $$b$$, including its sign.
✔ For equal roots, always use:
$$D=0$$
✔ When one root is given, substitute it directly into the equation.
✔ In application-based questions, define the variable carefully.
✔ If the variable represents a length, age, or quantity, reject values that are not 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:
$$-\frac{b}{a}$$
✔ Product of roots:
$$\frac{c}{a}$$
✔ For equal roots:
$$D=0$$
✔ For two distinct real roots:
$$D>0$$
✔ For no real roots:
$$D<0$$
✔ Consecutive integers:
$$x,\ x+1$$
Common Mistakes Students Make
❌ Forgetting the negative sign in:
$$-\frac{b}{a}$$
❌ Using:
$$b^2+4ac$$
instead of:
$$b^2-4ac$$
❌ Forgetting to square the complete value of $$b$$.
❌ Assuming that $$D<0$$ means the roots are negative.
❌ Forming the wrong equation for consecutive integers.
❌ Accepting a value that is not meaningful for the situation described in the question.
Key Takeaways
✔ The discriminant determines the nature of the roots.
✔ The sum and product of roots can be obtained directly from the coefficients.
✔ Algebraic identities can simplify expressions involving roots.
✔ A given root can be substituted directly into the equation to find an unknown coefficient.
✔ Consecutive-number problems can be converted into quadratic equations.
✔ Parameter-based questions may have more than one valid value, so all solutions should be checked.
FAQs
Q. What is the discriminant of a quadratic equation?
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 real roots when:
$$D=0$$
Q. What is the sum of roots of a quadratic equation?
Answer:
For:
$$ax^2+bx+c=0$$
the sum of roots is:
$$-\frac{b}{a}$$
Q. What is the product of roots?
Answer:
The product of roots is:
$$\frac{c}{a}$$
Q. How are consecutive integers represented?
Answer:
If one 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 and NCERT-based practice on discriminant, nature of roots, relationships between roots and coefficients, unknown coefficients, and application-based quadratic equations.
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