Between the following two statements: Statement I : Let a = i +2 j -3 k and b =2 i + j - k . Then the vector r satisfying a × r = a × b and a · r =0 is of magnitude 10 . Statement II : In a triangle A B C, 2 A+ 2 B+ 2 C

Correct answer: A. Statement I is incorrect but Statement II is correct. aligned & a = i +2 j -3 k & a =2 i + j - k & a × r = a × b ; a · r =0 & a ×( r - b )= 0 & a =λ( r - b ) & a · a =λ( a · r - a · b ) & 14=-7 λ λ=-2 & - a 2 = r - b r = b - a 2 & = 2 b - a 2 = 3 i + k 2 aligned Statement (I) is incorrect aligned & 2 ~A+ 2 ~B+ 2 c ≥-(3)/(2) & 2 ~A+2 ~B+2 C=2 π & 2 ~A+ 2 ~B+ 2 C & =-1-4 A · B · C & ≥-1-4 × (1)/(2) × (1)/(2) × (1)/(2) & =-(3)/(2) aligned Statement (II) is correct.

JEE Main 20249 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 9 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2024 · 9 April · Evening Shift · Mathematics · Vector Algebra

Between the following two statements: Statement I : Let a = i +2 j...

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

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

Options

AStatement I is incorrect but Statement II is correct.
BBoth Statement I and Statement II are correct.
CStatement I is correct but Statement II is incorrect.
DBoth Statement I and Statement II are incorrect.

Correct answer

A. Statement I is incorrect but Statement II is correct.

Step-by-step solution

\begin{aligned} & \bar{a}=\hat{i}+2 \hat{j}-3 \hat{k} \\ & \bar{a}=2 \hat{i}+\hat{j}-\hat{k} \\ & \bar{a} \times \bar{r}=\bar{a} \times \bar{b} ; \bar{a} \cdot \bar{r}=0 \\ & \Rightarrow \bar{a} \times(\bar{r}-\bar{b})=\overline{0} \\ & \Rightarrow \bar{a}=\lambda(\bar{r}-\bar{b}) \\ & \bar{a} \cdot \bar{a}=\lambda(\bar{a} \cdot \bar{r}-\bar{a} \cdot \bar{b}) \\ & 14=-7 \lambda \Rightarrow \lambda=-2 \\ & \frac{-\bar{a}}{2}=\bar{r}-\bar{b} \Rightarrow \bar{r}=\bar{b}-\frac{\bar{a}}{2} \\ & =\frac{2 \bar{b}-\bar{a}}{2}=\frac{3 \hat{i}+\hat{k}}{2}\end{aligned}

Statement (I) is incorrect

\begin{aligned} & \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{c} \geq-\frac{3}{2} \\ & 2 \mathrm{~A}+2 \mathrm{~B}+2 \mathrm{C}=2 \pi \\ & \cos 2 \mathrm{~A}+\cos 2 \mathrm{~B}+\cos 2 \mathrm{C} \\ & =-1-4 \cos \mathrm{A} \cdot \cos \mathrm{B} \cdot \cos \mathrm{C} \\ & \geq-1-4 \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \\ & =-\frac{3}{2} \end{aligned}

Statement (II) is correct.

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