CBSE Class 10 Mathematics Standard 2025: question 32(b)

Hints

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

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.

1. For real roots, discriminant $D=b^2-4ac\ge 0$.

   ½ mark

2. Here $a=1,b=-2(p+1),c=p^2$: $[-2(p+1){]}^2-4p^2\ge 0$

   ½ mark

3. $4(p^2+2p+1)-4p^2\ge 0\Rightarrow 8p+4\ge 0\Rightarrow p\ge -\frac{1}{2}$

   1 mark

4. Smallest value of $p=-\frac{1}{2}$.

   ½ mark

5. At $p=-\frac{1}{2}$: $x^2-2\left(-\frac{1}{2}+1\right)x+{\left(-\frac{1}{2}\right)}^2=0$

   ½ mark

6. $x^2-x+\frac{1}{4}=0\Rightarrow 4x^2-4x+1=0$

   1 mark

7. $(2x-1)(2x-1)=0$

   ½ mark

8. Roots are $\frac{1}{2},\frac{1}{2}$.

   ½ mark

Answer: $p=-\frac{1}{2}$; roots $\frac{1}{2},\frac{1}{2}$

Revise the chapter

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