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Practice Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers based on the latest NCERT syllabus. Solve fresh, board exam-oriented MCQs on factorisation, roots, 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: 6
Difficulty Level: Moderate
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations helps students solve equations involving a variable raised to the power of 2. Different methods such as factorisation, completing the square, and the quadratic formula can be used to find the roots.
This practice set focuses on fresh questions involving the nature of roots, relationships between roots and coefficients, factorisation, and application-based equations.
What You Will Learn?
✔ Solving quadratic equations
✔ Factorisation method
✔ Quadratic formula
✔ Discriminant and nature of roots
✔ Sum and product of roots
✔ Forming quadratic equations
✔ Application-based problems
Why This Topic Is Important?
Quadratic Equations is an important chapter for Class 10 Mathematics examinations. Questions can involve direct calculations as well as situations based on numbers, dimensions, and other real-life relationships.
Strong knowledge of this chapter helps students solve both objective and subjective questions accurately.
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. Which of the following quadratic equations has two distinct real roots?
A) $$x^2+4x+4=0$$
B) $$x^2+2x+5=0$$
C) $$x^2-7x+10=0$$
D) $$x^2+6x+9=0$$
Answer:
C) $$x^2-7x+10=0$$
Useful Concept
A quadratic equation has two distinct real roots when:
$$D>0$$
where:
$$D=b^2-4ac$$
Solution
For option C:
$$x^2-7x+10=0$$
Here:
$$a=1,\quad b=-7,\quad c=10$$
Therefore:
$$D=(-7)^2-4(1)(10)$$
$$D=49-40$$
$$D=9$$
Since:
$$D>0$$
the equation has two distinct real roots.
Hence, the correct answer is:
C) $$x^2-7x+10=0$$
Q2. If the roots of the quadratic equation
$$x^2-10x+k=0$$
are $$2$$ and $$8$$, what is the value of $$k$$?
A) $$10$$
B) $$12$$
C) $$16$$
D) $$20$$
Answer:
C) $$16$$
Useful Formula
The product of roots is:
$$\alpha\beta=\frac{c}{a}$$
Solution
The roots are:
$$\alpha=2,\quad\beta=8$$
Their product is:
$$\alpha\beta=2\times8=16$$
For the equation:
$$x^2-10x+k=0$$
we have:
$$a=1,\quad c=k$$
Therefore:
$$\alpha\beta=\frac{k}{1}=k$$
Hence:
$$k=16$$
Therefore, the correct answer is:
C) $$16$$
Q3. The roots of the equation
$$4x^2-4x-3=0$$
are:
A) $$\frac{3}{2},-\frac{1}{2}$$
B) $$\frac{1}{2},-\frac{3}{2}$$
C) $$2,-1$$
D) $$-2,1$$
Answer:
A) $$\frac{3}{2},-\frac{1}{2}$$
Useful Concept
The equation can be solved using factorisation.
Solution
Given:
$$4x^2-4x-3=0$$
Split the middle term:
$$4x^2-6x+2x-3=0$$
Group the terms:
$$2x(2x-3)+1(2x-3)=0$$
Therefore:
$$(2x+1)(2x-3)=0$$
Thus:
$$2x+1=0$$
or
$$2x-3=0$$
Therefore:
$$x=-\frac{1}{2}$$
or
$$x=\frac{3}{2}$$
Hence, the roots are:
$$\frac{3}{2},-\frac{1}{2}$$
Therefore, the correct answer is:
A) $$\frac{3}{2},-\frac{1}{2}$$
Q4. If the sum of the roots of a quadratic equation is $$7$$ and their product is $$10$$, which is the equation?
A) $$x^2+7x+10=0$$
B) $$x^2-7x+10=0$$
C) $$x^2+7x-10=0$$
D) $$x^2-10x+7=0$$
Answer:
B) $$x^2-7x+10=0$$
Useful Formula
If the sum of roots is $$S$$ and product is $$P$$, then the quadratic equation is:
$$x^2-Sx+P=0$$
Solution
Given:
$$\alpha+\beta=7$$
and
$$\alpha\beta=10$$
Therefore:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Substituting:
$$x^2-7x+10=0$$
Hence, the correct answer is:
B) $$x^2-7x+10=0$$
Q5. The product of two consecutive positive integers is $$72$$. Which quadratic equation can be used to find the smaller integer?
A) $$x^2+x-72=0$$
B) $$x^2-x-72=0$$
C) $$x^2+x+72=0$$
D) $$x^2-x+72=0$$
Answer:
A) $$x^2+x-72=0$$
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 integer is:
$$x+1$$
According to the question:
$$x(x+1)=72$$
Expanding:
$$x^2+x=72$$
Bringing all terms to one side:
$$x^2+x-72=0$$
Therefore, the required quadratic equation is:
$$\boxed{x^2+x-72=0}$$
Hence, the correct answer is:
A) $$x^2+x-72=0$$
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
$$\alpha\beta=\frac{c}{a}$$
Equation from Sum and Product
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
Nature of Roots
$$D>0$$
Two distinct real roots.
$$D=0$$
Two equal real roots.
$$D<0$$
No real roots.
Related Practice Questions
Q1. Determine the nature of roots of:
$$2x^2-5x+2=0$$
Q2. Find the value of $$k$$ if the roots of:
$$x^2-6x+k=0$$
have product $$8$$.
Q3. Solve:
$$6x^2+x-2=0$$
Q4. Form a quadratic equation whose sum of roots is $$9$$ and product is $$14$$.
Q5. The product of two consecutive integers is $$110$$. Form the quadratic equation.
Mini Quiz Challenge
Try answering within 60 seconds.
Q1. What is the condition for two distinct real roots?
Q2. What is the product of roots of $$3x^2-8x+5=0$$?
Q3. If roots are $$4$$ and $$6$$, what is their sum?
Q4. If roots are $$4$$ and $$6$$, what is their product?
Q5. If one consecutive integer is $$x$$, what is the next integer?
Exam Tips
✔ Remember the signs in the relationship between roots and coefficients.
✔ For a known sum and product of roots, directly use:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
✔ In consecutive-number problems, carefully define the two numbers.
✔ Check the discriminant when asked about the nature of roots.
✔ Verify factorised answers by multiplication.
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:
$$\frac{c}{a}$$
✔ Equation from roots:
$$x^2-(\alpha+\beta)x+\alpha\beta=0$$
✔ Consecutive integers:
$$x,\ x+1$$
Common Mistakes Students Make
❌ Writing the wrong sign before the sum of roots.
❌ Confusing the coefficient of $$x$$ with the product of roots.
❌ Forgetting to expand brackets in consecutive-number problems.
❌ Ignoring the condition that integers may need to be positive.
❌ Making sign errors during factorisation.
Key Takeaways
✔ The discriminant determines the nature of roots.
✔ Sum and product of roots can be used to form a quadratic equation.
✔ Factorisation is useful for many quadratic equations.
✔ Quadratic equations can represent number-based real-life problems.
✔ Careful variable definition makes application questions easier.
FAQs
Q. When does a quadratic equation have two distinct real roots?
Answer:
When:
$$D>0$$
Q. How can a quadratic equation be formed from the sum and product of roots?
Answer:
If the sum of roots is $$S$$ and product is $$P$$, then:
$$x^2-Sx+P=0$$
Q. What are consecutive positive integers?
Answer:
If the smaller positive integer is $$x$$, the next consecutive integer is:
$$x+1$$
Q. What is the product of roots of $$ax^2+bx+c=0$$?
Answer:
$$\frac{c}{a}$$
Q. What is the condition for equal roots?
Answer:
$$D=0$$
Conclusion
These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh NCERT-based practice on the discriminant, factorisation, root relationships, equation formation, and consecutive-number applications.
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