THE GOAL • CBSE CLASS 10

QUADRATIC EQUATIONS

Chapter 4 • Basic Mathematics • 2025 PYQs

CBSE 2025 Basic Mathematics Previous Year Questions Chapter-wise Collection

Chapter 4: Quadratic Equations

Unique Previous Year Questions extracted from the CBSE Class 10 Basic Mathematics 2025 question paper sets, with major conceptual variations retained.

I

Multiple Choice Questions

1. Discriminant Calculation
1 Mark
The discriminant of the quadratic equation \[ x^2-3x-2=0 \] is:
(A) \(1\)
(B) \(17\)
(C) \(\sqrt{17}\)
(D) \(-\sqrt{17}\)
Set 430/1/1
2. Standard Form Conversion
1 Mark
The equation \[ x+\frac{1}{x}=3,\qquad x\neq0 \] is expressed as a quadratic equation in the form \[ ax^2+bx+c=0. \] The value of \(a-b+c\) is:
(A) \(5\)
(B) \(2\)
(C) \(1\)
(D) \(-1\)
Set 430/1/1
3. Nature of Roots – Coefficient \(a\)
1 Mark
The value of \(a\) for which \[ ax^2+3x+1=0 \] has real and equal roots is:
(A) \(\frac{4}{9}\)
(B) \(\frac{9}{4}\)
(C) \(\frac{3}{2}\)
(D) \(\frac{2}{3}\)
Set 430/2/1
4. Identification of Quadratic Equation
1 Mark
Which of the following equations is a quadratic equation?
(A) \(x^2=(x+1)^2\)
(B) \((x-1)(x+2)=2x+1\)
(C) \((x+2)^3=2x(x^2-1)\)
(D) \(\sqrt{x}=x^2\)
Set 430/2/1
5. Nature of Roots – Coefficient \(p\)
1 Mark
One of the values of \(p\) for which \[ px^2+4x+p=0 \] has real and equal roots is:
(A) \(4\)
(B) \(-4\)
(C) \(2\)
(D) \(0\)
Set 430/2/3
6. Degree / Algebraic Identification
1 Mark
If \[ (\sqrt{x}+1)^2=x^2+2\sqrt{x} \] is expressed as a quadratic equation in the form \[ ax^2+bx+c=0, \] then the value of \(a-b+c\) is:
(A) \(-1\)
(B) \(0\)
(C) \(1\)
(D) \(2\)
Set 430/3/2
7. Equal Roots with \(k\)
1 Mark
The value of \(k\) for which the roots of the quadratic equation \[ 6x^2+4kx+k=0 \] are real and equal, is:
(A) \(0\)
(B) \(\frac{3}{4}\)
(C) \(-\frac{3}{2}\)
(D) \(\frac{2}{3}\)
Set 430/5/1
8. Simplifying to a Quadratic Equation
1 Mark
If \[ (x+1)^3=x^3+1 \] is expressed as a quadratic equation in the form \[ px^2+qx+r=0, \] then the value of \(p-q+r\) is:
(A) \(0\)
(B) \(1\)
(C) \(3\)
(D) \(6\)
Set 430/3/3
9. Roots of a Basic Equation
1 Mark
The roots of the equation \[ x^2-8=0 \] are:
(A) Rational and distinct
(B) Irrational and distinct
(C) Real and equal
(D) Not real
Set 430/5/2
II

Assertion – Reason Question

10. Real Roots of a Quadratic Equation
1 Mark
Assertion (A): Every quadratic equation has two real roots.
Reason (R): A quadratic polynomial can have at most two zeroes.
Set 430/3/1
III

Very Short Answer Type Questions

11. Solving by Quadratic Formula
2 Marks
Solve the equation using quadratic formula:
\[ 4x^2-9x+3=0 \]
Set 430/6/1
12. Nature of Roots
2 Marks
Find the nature of roots of the equation:
\[ 3x^2-4\sqrt{3}\,x+4=0 \]
Set 430/6/1
13. Solving with Irrational Coefficients
2 Marks
Solve the quadratic equation using quadratic formula:
\[ \sqrt{3}\,x^2+10x+7\sqrt{3}=0 \]
Set 430/6/3
14. Nature of Roots – Literal Coefficients
2 Marks
Find the nature of roots of the equation \[ 4x^2-4a^2x+a^4-b^4=0, \qquad b\neq0. \]
Set 430/6/3
IV

Short Answer Type Questions

15. Word Problem – Consecutive Even Numbers
3 Marks
The sum of the squares of two consecutive even numbers is \(452\). Find the numbers.
Set 430/5/1
16. Word Problem – Consecutive Odd Numbers
3 Marks
The sum of the squares of two consecutive odd numbers is \(514\). Find the numbers.
Set 430/5/3
17. Figure-Data Based – Path Width
3 Marks
A rectangular field is \(16\) m long and \(10\) m wide. There is a path of equal width all around it, having an area of \(120\text{ m}^2\). Find the width of the path.
Set 430/5/2
18. Word Problem – Perimeter and Area
3 Marks
Find the length and breadth of a rectangular park whose perimeter is \(100\) m and area is \(600\text{ m}^2\).
Set 430/6/1
V

Long Answer Type Questions

19. Word Problem – Squares and Difference
5 Marks
The difference of the squares of two positive numbers is \(180\). The square of the smaller number is \(8\) times the greater number. Find the two numbers.
Set 430/1/1
20. Equal Roots and Finding the Roots
5 Marks
Find the value(s) of \(k\) for which the equation \[ 2x^2+kx+3=0 \] has real and equal roots. Hence, find the roots of the equations so obtained.
Set 430/1/1
21. Word Problem – Two Squares
5 Marks
The sum of areas of two squares is \[ 2650\text{ cm}^2. \] If the sum of their perimeters is \(280\) cm, find the sides of the two given squares.
Set 430/2/1
22. Algebraic Manipulation
5 Marks
Express the equation \[ \frac{1}{x}-\frac{1}{x-2}=3, \qquad x\neq0,2 \] as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.
Set 430/2/1
23. Word Problem – Consecutive Odd Integers
5 Marks
Find two consecutive odd integers, the sum of whose squares is \(290\).
Set 430/3/1
24. Word Problem – Rectangular Hall
5 Marks
A charity trust decides to build a rectangular hall having an area of \(300\text{ m}^2\). The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.
Set 430/3/1
25. Literal Coefficient – Discriminant and Roots
5 Marks
It is given that \[ p^2x^2+(p^2-q^2)x-q^2=0, \qquad p\neq0. \]
(i) Show that the discriminant \(D\) of the above equation is a perfect square.
(ii) Find the roots of the equation.
Set 430/4/1
26. Word Problem – Three Consecutive Positive Integers
5 Marks
Three consecutive positive integers are such that the sum of the square of the smallest and product of the other two is \(67\). Find the numbers, using quadratic equation.
Set 430/4/1