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Mathematics · 2025 · 3 marks
CBSE 2025 · Region 5 · Set 1 · Q30
(a) If a + b + c = 0 such that | a |=3,| b |=5,| c |=7, then find the angle between a and b.
(a)
If $\displaystyle \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ such that $\displaystyle |\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=5,|\overrightarrow{\mathrm{c}}|=7$, then find the angle between $\displaystyle \vec{\mathrm{a}}$ and $\displaystyle \vec{\mathrm{b}}$.
OR
(b)
If $\displaystyle \vec{\mathrm{a}}$ and $\displaystyle \vec{\mathrm{b}}$ are unit vectors inclined with each other at an angle $\displaystyle \theta$, then prove that $\displaystyle \frac{1}{2}|\vec{\mathrm{a}}-\vec{\mathrm{b}}|=\sin \frac{\theta}{2}$.
Marking-scheme solution
\[\begin{aligned}
& \text { Given } \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}} \Rightarrow|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=|-\overrightarrow{\mathbf{c}}|^{2} \\
& \Rightarrow|\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}|^{2}=|\overrightarrow{\mathbf{c}}|^{2} \Rightarrow|\overrightarrow{\mathbf{a}}|^{2}+|\overrightarrow{\mathbf{b}}|^{2}+2 \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=|\overrightarrow{\mathbf{c}}|^{2} \\
& \Rightarrow 9+25+2 \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=49
\end{aligned}
\]
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CBSE Class 12 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.