CBSE Class 12 Mathematics 2024: question 24

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. $(\vec{a}+\vec{b})\perp \vec{a}\Rightarrow (\vec{a}+\vec{b})\cdot \vec{a}=0\Rightarrow |\vec{a}{|}^2+\vec{a}\cdot \vec{b}=0$ ...(1)

   ½ mark

2. $(2\vec{a}+\vec{b})\perp \vec{b}\Rightarrow (2\vec{a}+\vec{b})\cdot \vec{b}=0\Rightarrow 2\vec{a}\cdot \vec{b}+|\vec{b}{|}^2=0$ ...(2)

   ½ mark

3. From (1), $\vec{a}\cdot \vec{b}=-|\vec{a}{|}^2$; substituting in (2): $-2|\vec{a}{|}^2+|\vec{b}{|}^2=0$

   ½ mark

4. $|\vec{b}{|}^2=2|\vec{a}{|}^2\Rightarrow |\vec{b}|=\sqrt{2}|\vec{a}|$ (magnitudes are positive). Hence proved.

   ½ mark

Answer: $|\vec{b}|=\sqrt{2}|\vec{a}|$ (proved)

Revise the chapter

• Chapter: Vector Algebra (notes, formulas, where marks are lost and every past question)
• Topic: Scalar (Dot) Product of Vectors