MathsCBSE Class 10 · Basic · Board exam 2025, set 430/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2025: question 32(a)

Set in the CBSE Class 10 board exam of 2025, paper 430/4/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerQuadratic Equations

It is given that $p^{2} x^{2} + (p^{2} - q^{2}) x - q^{2} = 0$ ; $(p \neq = 0)$ . (i) Show that the discriminant (D) of above equation is a perfect square. (ii) Find the roots of the equation.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Identify $a$ , $b$ , $c$ carefully; note that $c = - q^{2}$ .

Hint 2

Expand $(p^{2} - q^{2})^{2} + 4 p^{2} q^{2}$ and look for the identity $(A + B)^{2}$ .

Hint 3

Use $x = \frac{- b \pm D}{2 a}$ and simplify each sign separately.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) Here $a = p^{2}$ , $b = p^{2} - q^{2}$ , $c = - q^{2}$ . $D = b^{2} - 4 a c = (p^{2} - q^{2})^{2} - 4 (p^{2}) (- q^{2}) = (p^{2} - q^{2})^{2} + 4 p^{2} q^{2}$
1 mark
$= p^{4} - 2 p^{2} q^{2} + q^{4} + 4 p^{2} q^{2} = p^{4} + 2 p^{2} q^{2} + q^{4} = (p^{2} + q^{2})^{2}$ , which is a perfect square.
1 mark
(ii) $x = \frac{- b \pm D}{2 a} = \frac{- ( p ^{2} - q ^{2} ) \pm ( p ^{2} + q ^{2} ) ^{2}}{2 p ^{2}} = \frac{- ( p ^{2} - q ^{2} ) \pm ( p ^{2} + q ^{2} )}{2 p ^{2}}$
1 mark
Taking +: $x = \frac{- p ^{2} + q ^{2} + p ^{2} + q ^{2}}{2 p ^{2}} = \frac{2 q ^{2}}{2 p ^{2}} = \frac{q ^{2}}{p ^{2}}$
1 mark
Taking −: $x = \frac{- p ^{2} + q ^{2} - p ^{2} - q ^{2}}{2 p ^{2}} = \frac{- 2 p ^{2}}{2 p ^{2}} = - 1$
1 mark

Answer: (i) $D = (p^{2} + q^{2})^{2}$ , a perfect square; (ii) roots $\frac{q ^{2}}{p ^{2}}$ and $- 1$

Where the marks are

Writes $D = (p^{2} - q^{2})^{2} + 4 p^{2} q^{2}$ (1 mark)
Simplifies to $(p^{2} + q^{2})^{2}$ (1 mark)
Applies the quadratic formula correctly (1 mark)
Root $\frac{q ^{2}}{p ^{2}}$ (1 mark)
Root $- 1$ (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Nature of Roots and Discriminant

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/4/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2025: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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