Between the following two statements: Statement-I: Let and . Then the vector satisfying and is of magnitude . Statement-II: In a triangle ABC, cos 2A + cos 2B + cos 2C . [2024]

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Solution

Q.
Between the following two statements:

Statement-I: Let $\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - \hat{k}$ . Then the vector $\vec{r}$ satisfying $\vec{a} \times \vec{r} = \vec{a} \times \vec{b}$ and $\vec{a} \cdot \vec{r} = 0$ is of magnitude $\sqrt{10}$ .

Statement-II: In a triangle ABC, cos 2A + cos 2B + cos 2C $\geq - \frac{3}{2}$ . [2024]

1 Both Statement-I and Statement-II are incorrect.

2 Statement-I is incorrect but Statement-II is correct.

3 Statement-I is correct but Statement-II is incorrect.

4 Both Statement-I and Statement-II are correct.

Ans.
(2)

We have, $\vec{a} \times \vec{r} = \vec{a} \times \vec{b}$

$\Rightarrow \vec{a} \times (\vec{r} - \vec{b}) = 0 \Rightarrow \vec{a} = λ (\vec{r} - \vec{b})$

$\Rightarrow \vec{a} \cdot \vec{a} = λ (\vec{a} \cdot \vec{r} - \vec{a} \cdot \vec{b})$

$\Rightarrow | \vec{a} |^{2} = λ (0 - 7) \Rightarrow 14 = - 7 λ$

$\Rightarrow λ = - 2 \Rightarrow \vec{r} = \vec{b} - \frac{\vec{a}}{2} = \frac{2 \vec{b} - \vec{a}}{2} = \frac{3 \hat{i} + \hat{k}}{2}$

So, $| \vec{r} | = \frac{\sqrt{10}}{2}$

So, Statement-I is incorrect.

Let $△$ ABC be given triangle and O be its circumcentre

Now, OA = $\vec{a}$ , OB = $\vec{b}$ and OC = $\vec{c}$

Now, |OA| = |OB| = |OC|

(distance from circumcentre will be same for all vertices.)

Consider (OA + OB + OC)

$= {(\vec{a} + \vec{b} + \vec{c})}^{2} \geq 0$

$\Rightarrow | \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 \vec{a} \cdot \vec{b} + 2 \vec{b} \cdot \vec{c} + 2 \vec{c} \cdot \vec{a} \geq 0$

$\Rightarrow 3 | \vec{a} |^{2} + 2 | \vec{a} | | \vec{b} | \cos 2 C + 2 | \vec{b} | | \vec{c} | \cos 2 A + 2 | \vec{c} | | \vec{a} | \cos 2 B \geq 0$

$\Rightarrow 3 | \vec{a} |^{2} + 2 | \vec{a} |^{2} [\cos 2 A + \cos 2 B + \cos 2 C] \geq 0$

$\Rightarrow 2 [\cos 2 A + \cos 2 B + \cos 2 C] \geq - 3$ $[∵ | \vec{a} | > 0]$

$\Rightarrow$ cos 2A + cos 2B + cos 2C $\geq - \frac{3}{2}$

Hence, Statement-II is correct.

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