THE GOAL • CBSE CLASS 10

QUADRATIC EQUATIONS

Chapter 4 • Standard Mathematics • 2025 PYQs

CBSE 2025 Standard Mathematics Previous Year Questions Chapter-wise Collection

Chapter 4: Quadratic Equations

Unique Previous Year Questions extracted from the CBSE Class 10 Standard Mathematics 2025 question paper sets.

I

Multiple Choice Questions

1. Quadratic Equation from Roots
1 Mark
The quadratic equation whose roots are \(7\) and \(\frac{1}{7}\) is:
(A) \(7x^2-50x+7=0\)
(B) \(7x^2-50x+1=0\)
(C) \(7x^2+50x-7=0\)
(D) \(7x^2+50x-1=0\)
Series 30/2/1 (Q3) & 30/2/3 (Q18)
2. Sum and Product of Roots
1 Mark
The quadratic equation whose sum and product of roots are \(a\) and \(\frac{1}{a}\), respectively, is:
(A) \(ax^2-ax+1=0\)
(B) \(ax^2-a^2x+1=0\)
(C) \(ax^2+ax+1=0\)
(D) \(ax^2+a^2x-1=0\)
Series 30/2/2 (Q1)
3. Solving a Quadratic Equation
1 Mark
If \[ \frac{x}{12}-\frac{3}{x}=0, \] then the values of \(x\) are:
(A) \(\pm6\)
(B) \(\pm4\)
(C) \(\pm12\)
(D) \(\pm3\)
Series 30/3/1 (Q5)
4. Discriminant
1 Mark
The discriminant of the quadratic equation \(bx^2+ax+c=0,\ b\neq0\), is given by:
(A) \(b^2-4ac\)
(B) \(\sqrt{b^2-4ac}\)
(C) \(\sqrt{a^2-4bc}\)
(D) \(a^2-4bc\)
Series 30/3/2 (Q14)
5. Equal and Positive Roots
1 Mark
The value of \(a\) for which \[ ax^2+x+a=0 \] has equal and positive roots is:
(A) \(2\)
(B) \(-2\)
(C) \(\frac12\)
(D) \(-\frac12\)
Series 30/4/1 (Q3), 30/4/2 (Q5) & 30/4/3 (Q12)
6. Identifying a Quadratic Equation
1 Mark
Which of the following equations is a quadratic equation?
(A) \(x^2+1=(x-1)^2\)
(B) \((x+\sqrt{x})^2=2x\sqrt{x}\)
(C) \(x^3+3x^2=(x+1)^3\)
(D) \((x+1)(x-1)=(x+1)^2\)
Series 30/5/1 (Q4) & 30/5/2 (Q8)
7. Nature of Roots
1 Mark
If \[ x^2+bx+b=0 \] has two real and distinct roots, then the value of \(b\) can be:
(A) \(0\)
(B) \(4\)
(C) \(3\)
(D) \(-3\)
Series 30/5/1 (Q5) & 30/5/2 (Q7)
8. Real and Equal Roots
1 Mark
Which of the following quadratic equations has real and equal roots?
(A) \((x+1)^2=2x+1\)
(B) \(x^2+x=0\)
(C) \(x^2-4=0\)
(D) \(x^2+x+1=0\)
Series 30/6/1 (Q3)
9. Real and Distinct Roots
1 Mark
Which of the following quadratic equations has real and distinct roots?
(A) \(x^2+2x=0\)
(B) \(x^2+x+1=0\)
(C) \((x-1)^2=1-2x\)
(D) \(2x^2+x+1=0\)
Series 30/6/2 (Q5)
10. Equation without Real Roots
1 Mark
Which of the following equations does not have a real root?
(A) \(x^2=0\)
(B) \(2x-1=3\)
(C) \(x^2+1=0\)
(D) \(x^3+x^2=0\)
Series 30/6/3 (Q12)
II

Very Short Answer Type Questions

11. Equal Roots – Find \(k\)
2 Marks
Find the value(s) of \(k\) so that the quadratic equation \[ 4x^2+kx+1=0 \] has real and equal roots.
Series 30/3/1 (Q24a), 30/3/2 (Q22a) & 30/3/3 (Q22a)
III

Long Answer Type Questions

12. Right Triangle
5 Marks
The perimeter of a right triangle is \(60\) cm and its hypotenuse is \(25\) cm. Find the lengths of the other two sides of the triangle.
Series 30/1/1 (Q34a) & 30/1/3 (Q32a)
13. Train Speed
5 Marks
A train travels a distance of \(480\) km at a uniform speed. If the speed had been \(8\) km/h less, then it would have taken \(3\) hours more to cover the same distance. Find the speed of the train.
Series 30/1/1 (Q34b) & 30/1/3 (Q32b)
14. Areas and Perimeters of Squares
5 Marks
The sum of the areas of two squares is \(52\text{ cm}^2\) and the difference of their perimeters is \(8\) cm. Find the lengths of the sides of the two squares.
Series 30/2/1 (Q33a) & 30/2/2 (Q32a)
15. Upward and Downward Journey
5 Marks
The time taken by a person to travel an upward distance of \(150\) km was \(2\frac{1}{2}\) hours more than the time taken in the downward return journey. If he returned at a speed of \(10\) km/h more than the speed while going up, find the speeds in each direction.
Series 30/2/1 (Q33b) & 30/2/2 (Q32b)
16. Rational Equation to Quadratic Equation
5 Marks
Express the equation \[ \frac{x-2}{x-3}+\frac{x-4}{x-5}=\frac{10}{3}, \qquad x\neq3,5 \] as a quadratic equation in standard form. Hence, find the roots of the equation so formed.
Series 30/4/1 (Q32b), 30/4/2 (Q33b) & 30/4/3 (Q35b)
17. Equal Roots – Parameter \(p\)
5 Marks
Find the value(s) of \(p\) for which the quadratic equation \[ (p+4)x^2-(p+1)x+1=0 \] has real and equal roots. Also, find the roots of the equation(s) so obtained.
Series 30/5/1 (Q32b), 30/5/2 (Q33b) & 30/5/3 (Q35b)
18. Smallest Value of \(p\)
5 Marks
Find the smallest value of \(p\) for which the quadratic equation \[ x^2-2(p+1)x+p^2=0 \] has real roots. Hence, find the roots of the equation so obtained.
Series 30/6/1 (Q32b) & 30/6/2 (Q33b)
IV

Case Study Based Question

19. Rectangular Lawn and Walkway
4 Marks
The figure shows a rectangular lawn surrounded by a walkway of uniform width \(x\). A garden designer is planning a rectangular lawn with dimensions \(12\) metres by \(10\) metres. The total area of the lawn and the uniform walkway is \(360\) square metres.
(i) Formulate the quadratic equation representing the total area of the lawn and the walkway, taking the width of walkway as \(x\) m. (1 mark)
(ii) (a) Solve the quadratic equation to find the width of the walkway \(x\). (2 marks)
OR
(b) If the cost of paving the walkway at the rate of ₹50 per square metre is ₹12,000, calculate the area of the walkway. (2 marks)
(iii) Find the perimeter of the lawn. (1 mark)
Series 30/3/1 (Q36) & 30/3/2 (Q37)