Between the following two statements: Statement-I : Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$. Then the vector $\overrightarrow{\mathrm{r}}$ satisfying $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{r}}=0$ is of magnitude $\sqrt{10}$. Statement-II : In a triangle $\mathrm{ABC}, \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B} +\cos 2 \mathrm{C} \geq-\frac{3}{2}$.

(A) Both Statement-I and Statement-II are incorrect

(C) Both Statement-I and Statement-II are correct

(D) Statement-I is correct but Statement-II is incorrect

Vector Algebra — JEE Main · 2024 · 09 Apr 2024 · Shift 2 Question with Solution

Between the following two statements:

Statement-I : Let

a = \hat{i} + 2 \hat{j} - 3 \hat{k}

and

b = 2 \hat{i} + \hat{j} - \hat{k}

. Then the vector

r

satisfying

a \times r = a \times b

and

a \cdot r = 0

is of magnitude

10

.

Statement-II : In a triangle

ABC, cos 2 A + cos 2 B + cos 2 C \geq - \frac{3}{2}

Answer: (B)

Statement I:

a = \hat{i} + 2 \hat{j} - 3 \hat{k}

b = 2 \hat{i} + \hat{j} - \hat{k}

. Since

a \times (r - b) = 0

r - b

is parallel to

a

r = b + λ a

.

Using

a \cdot r = 0

a \cdot b + λ (a \cdot a) = 0

a \cdot b = (1) (2) + (2) (1) + (- 3) (- 1) = 7

;

a \cdot a = 1 + 4 + 9 = 14

; so

λ = - \frac{7}{14} = - \frac{1}{2}

r = b - \frac{1}{2} a = (2 - \frac{1}{2}) \hat{i} + (1 - 1) \hat{j} + (- 1 + \frac{3}{2}) \hat{k} = \frac{3}{2} \hat{i} + \frac{1}{2} \hat{k}

∣ r ∣ = (\frac{3}{2})^{2} + (\frac{1}{2})^{2} = \frac{10}{4} = \frac{10}{2}

This is

\frac{10}{2}

, not

10

, so Statement I is incorrect.

Statement II: For a triangle,

A + B + C = π

, and

cos 2 A + cos 2 B + cos 2 C = - 1 - 4 cos A cos B cos C

Since

cos A cos B cos C \leq \frac{1}{8}

(maximum at

A = B = C = π /3

cos 2 A + cos 2 B + cos 2 C \geq - 1 - 4 (\frac{1}{8}) = - \frac{3}{2}

Statement II is correct.

Statement I is incorrect but Statement II is correct.

JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Single correct option