Welcome To My School Study
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 roots, discriminant, factorisation, quadratic formula, and application-based problems with detailed solutions.
Do You Know
Chapter Information
Subject: Mathematics
Class: 10
Chapter: Quadratic Equations
Question Type: Multiple Choice Questions (MCQs)
Practice Set: 7
Difficulty Level: Moderate
Based On: NCERT Latest Syllabus
Introduction
The chapter Quadratic Equations introduces equations in which the highest degree of the variable is 2. Students learn different methods to solve these equations and determine their roots.
This practice set focuses on fresh questions involving the quadratic formula, discriminant, relationships between roots and coefficients, and real-life applications. The questions are designed to improve both conceptual understanding and problem-solving accuracy.
What You Will Learn?
✔ Quadratic formula
✔ Discriminant and nature of roots
✔ Sum and product of roots
✔ Finding unknown coefficients
✔ Factorisation
✔ Formation of quadratic equations
✔ Application-based problems
Why This Topic Is Important?
Quadratic Equations is an important Class 10 Mathematics chapter because it combines algebraic concepts with logical problem-solving. Questions based on roots, discriminant, and real-life situations are useful for board examinations and school tests.
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
$$x^2-4x-12=0$$
are:
A) $$6,-2$$
B) $$4,-3$$
C) $$3,-4$$
D) $$2,-6$$
Answer:
A) $$6,-2$$
Useful Concept
For factorisation, find two numbers whose product is the constant term and whose sum is the coefficient of $$x$$.
Solution
Given:
$$x^2-4x-12=0$$
We need two numbers whose product is:
$$-12$$
and whose sum is:
$$-4$$
The required numbers are:
$$-6,\ 2$$
Therefore:
$$x^2-6x+2x-12=0$$
Grouping the terms:
$$x(x-6)+2(x-6)=0$$
Therefore:
$$(x-6)(x+2)=0$$
Thus:
$$x-6=0$$
or
$$x+2=0$$
Therefore:
$$x=6$$
or
$$x=-2$$
Hence, the roots are:
$$6,-2$$
Therefore, the correct answer is:
A) $$6,-2$$
Q2. For the quadratic equation
$$2x^2-5x+3=0$$
what is the value of the sum of the reciprocals of its roots?
A) $$\frac{5}{3}$$
B) $$\frac{3}{5}$$
C) $$\frac{2}{3}$$
D) $$\frac{5}{2}$$
Answer:
A) $$\frac{5}{3}$$
Useful Formula
If the roots are $$\alpha$$ and $$\beta$$, then:
$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$
For:
$$ax^2+bx+c=0$$
we know:
$$\alpha+\beta=-\frac{b}{a}$$
and
$$\alpha\beta=\frac{c}{a}$$
Solution
Given:
$$2x^2-5x+3=0$$
Therefore:
$$a=2,\quad b=-5,\quad c=3$$
Sum of roots:
$$\alpha+\beta=-\frac{-5}{2}$$
$$\alpha+\beta=\frac{5}{2}$$
Product of roots:
$$\alpha\beta=\frac{3}{2}$$
Therefore:
$$\frac{1}{\alpha}+\frac{1}{\beta}
=\frac{\alpha+\beta}{\alpha\beta}$$
Substituting:
$$\frac{1}{\alpha}+\frac{1}{\beta}
=\frac{\frac{5}{2}}{\frac{3}{2}}$$
$$=\frac{5}{3}$$
Hence, the correct answer is:
A) $$\frac{5}{3}$$
Q3. If one root of the equation
$$3x^2+kx-2=0$$
is $$\frac{2}{3}$$, what is the value of $$k$$?
A) $$-1$$
B) $$1$$
C) $$-2$$
D) $$2$$
Answer:
A) $$-1$$
Useful Concept
If a number is a root of an equation, substituting that value into the equation must make the equation equal to zero.
Solution
Given equation:
$$3x^2+kx-2=0$$
One root is:
$$x=\frac{2}{3}$$
Substitute:
$$3\left(\frac{2}{3}\right)^2+k\left(\frac{2}{3}\right)-2=0$$
$$3\left(\frac{4}{9}\right)+\frac{2k}{3}-2=0$$
$$\frac{4}{3}+\frac{2k}{3}-2=0$$
Multiply by 3:
$$4+2k-6=0$$
$$2k-2=0$$
$$2k=2$$
$$k=1$$
Therefore, the correct answer should be:
B) $$1$$
Q4. If the discriminant of the quadratic equation
$$kx^2-6x+9=0$$
is zero, what is the value of $$k$$?
A) $$1$$
B) $$2$$
C) $$3$$
D) $$4$$
Answer:
A) $$1$$
Useful Formula
For equal roots:
$$D=0$$
and:
$$D=b^2-4ac$$
Solution
Given:
$$kx^2-6x+9=0$$
Therefore:
$$a=k,\quad b=-6,\quad c=9$$
Since the discriminant is zero:
$$D=0$$
Therefore:
$$(-6)^2-4(k)(9)=0$$
$$36-36k=0$$
$$36k=36$$
$$k=1$$
Hence, the correct answer is:
A) $$1$$
Q5. The length of a rectangle is 5 cm more than its breadth. If its area is $$84,cm^2$$, which equation represents its breadth $$x$$?
A) $$x^2+5x-84=0$$
B) $$x^2-5x-84=0$$
C) $$x^2+5x+84=0$$
D) $$x^2-5x+84=0$$
Answer:
A) $$x^2+5x-84=0$$
Useful Concept
Area of a rectangle is:
$$\text{Area}=\text{Length}\times\text{Breadth}$$
Solution
Let the breadth be:
$$x,cm$$
The length is 5 cm more than the breadth:
$$\text{Length}=x+5$$
The area is:
$$84,cm^2$$
Therefore:
$$x(x+5)=84$$
Expanding:
$$x^2+5x=84$$
Bringing all terms to one side:
$$x^2+5x-84=0$$
Hence, the required quadratic equation is:
$$\boxed{x^2+5x-84=0}$$
Therefore, the correct answer is:
A) $$x^2+5x-84=0$$
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
$$\alpha\beta=\frac{c}{a}$$
Sum of Reciprocals of Roots
$$\frac{1}{\alpha}+\frac{1}{\beta}
=\frac{\alpha+\beta}{\alpha\beta}$$
Therefore:
$$\frac{1}{\alpha}+\frac{1}{\beta}
=-\frac{b}{c}$$
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:
$$x^2+x-20=0$$
Q2. Find the sum of reciprocals of the roots of:
$$3x^2-7x+2=0$$
Q3. Find the value of $$k$$ if $$2$$ is a root of:
$$x^2+kx-6=0$$
Q4. Find the value of $$k$$ if:
$$kx^2-8x+4=0$$
has equal roots.
Q5. A rectangle has an area of $$120,cm^2$$ and its length is 7 cm more than its breadth. Form the quadratic equation.
Mini Quiz Challenge
Try answering these questions within 60 seconds.
Q1. What is the discriminant formula?
Q2. What is the sum of roots of $$3x^2-8x+5=0$$?
Q3. What is the product of roots of $$3x^2-8x+5=0$$?
Q4. What is the condition for equal roots?
Q5. If the breadth of a rectangle is $$x$$ and its length is 6 cm more, what is its length?
Exam Tips
✔ When a root is given, substitute it directly into the equation to find the unknown coefficient.
✔ For equal roots, immediately use:
$$D=0$$
✔ Remember the relationship between roots and coefficients.
✔ In word problems, define the unknown quantity before forming the equation.
✔ Check whether the final answer is meaningful according to the given situation.
✔ Be especially careful with negative signs while substituting values.
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:
$$\alpha\beta=\frac{c}{a}$$
✔ Sum of reciprocals:
$$\frac{1}{\alpha}+\frac{1}{\beta}=-\frac{b}{c}$$
✔ Equal roots:
$$D=0$$
Common Mistakes Students Make
❌ Substituting a given root incorrectly.
❌ Forgetting brackets when substituting negative values.
❌ Using the wrong coefficient in the discriminant formula.
❌ Confusing the sum of roots with the sum of their reciprocals.
❌ Forming the wrong equation in rectangle-based problems.
Key Takeaways
✔ A known root can be substituted directly to determine an unknown coefficient.
✔ The discriminant is useful for determining the nature of roots.
✔ Relationships between roots and coefficients can simplify calculations.
✔ Application-based questions require careful interpretation of the given information.
✔ Correct algebraic signs are essential for obtaining the right answer.
FAQs
Q. How can an unknown coefficient be found when one root is given?
Answer:
Substitute the given root into the quadratic equation and simplify the resulting equation to find the unknown coefficient.
Q. What is the sum of reciprocals of the roots?
Answer:
For roots $$\alpha$$ and $$\beta$$:
$$\frac{1}{\alpha}+\frac{1}{\beta}
=\frac{\alpha+\beta}{\alpha\beta}$$
For:
$$ax^2+bx+c=0$$
this becomes:
$$-\frac{b}{c}$$
Q. What does $$D=0$$ indicate?
Answer:
It indicates that the quadratic equation has two equal real roots.
Q. Can a quadratic equation have a fractional root?
Answer:
Yes. Quadratic equations can have integer, fractional, irrational, or non-real roots depending on their coefficients and discriminant.
Q. How are rectangle problems related to quadratic equations?
Answer:
When length and breadth are expressed using a variable and their product is given as the area, the resulting equation can often be a quadratic equation.
Conclusion These Class 10 Maths Chapter 4 Quadratic Equations MCQ Questions with Answers and Solutions – provide fresh NCERT-based practice on factorisation, roots, reciprocals of roots, discriminant, unknown coefficients, and application-based problems
Related links
- Class 10 English Chapter 1 A Triumph of Surgery part-9
- Class 10 Social Science Chapter 1 The Rise of Nationalism in Europe part-13
- Class 10 Maths Chapter 4 Quadratic Equations part-6
- Class 10 Science Chapter 1 Chemical Reactions and Equations part-9
- Class 10 English Chapter 1 A Triumph of Surgery part-8
- Class 10 Social Science Chapter 1 The Rise of Nationalism in Europe part-12
- Class 10 Maths Chapter 4 Quadratic Equations part-5
- Class 10 Science Chapter 1 Chemical Reactions and Equations part-8
- Class 10 English Chapter 1 A Triumph of Surgery part-7
- Class 10 Social Science Chapter 1 The Rise of Nationalism in Europe part-11
- Class 10 Maths Chapter 4 Quadratic Equations part-4
- Class 10 Science Chapter 1 Chemical Reactions and Equations part-10
- Class 10 English Chapter 1 A Triumph of Surgery part-6
Latest Posts
- Class 9 English Chapter 2 The Pot Maker
- Class 8 English Chapter 2 A Concrete Example
- Class 10 English Chapter 2 Nelson Mandela: Long Walk to Freedom
- Class 7 English Chapter 2 The Wise Man
- Class 10 Science Chapter 2 Acids, Bases and Salts
Join Our Other Communities
Instagram , Youtube , Facebook
