JEE Main 2022 · 24 Jun · Shift 2 · Mathematics Q16

← JEE Main 2022 · 24 Jun · Shift 2

MathematicsSingle correct+4 / -1Question 16 of 90

Let $\hat a$ and $\hat b$ be two unit vectors such that $\left|\left(\hat a+\hat b\right)+2\left(\hat a\times\hat b\right)\right|=2$ . If $\theta\in(0,\pi)$ is the angle between $\hat a$ and $\hat b$ , then among the statements:

(S1) : $2\left|\hat a\times\hat b\right|=\left|\hat a-\hat b\right|$

(S2) : The projection of $\hat a$ on $\left(\hat a+\hat b\right)$ is $\frac12$

Only (S1) is true.

Only (S2) is true.

Both (S1) and (S2) are true.

Both (S1) and (S2) are false.

Answer and worked solution

Official answer: C

Square the condition. $\hat a+\hat b$ is perpendicular to $\hat a\times\hat b$ , so $|\hat a+\hat b|^2+4|\hat a\times\hat b|^2=4$ : $2+2\cos\theta+4\sin^2\theta=4$ , i.e. $2\cos^2\theta-\cos\theta-1=0$ .

Angle. $\cos\theta=-\frac12$ (since $\theta\ne0$ ), $\theta=\frac{2\pi}3$ .

(S1). $2|\hat a\times\hat b|=2\sin\theta=\sqrt3$ and $|\hat a-\hat b|=\sqrt{2-2\cos\theta}=\sqrt3$ : true.

(S2). $\dfrac{\hat a\cdot(\hat a+\hat b)}{|\hat a+\hat b|}=\dfrac{1-\frac12}{1}=\dfrac12$ : true.

(C)

Topics:Cross product Dot product

← Previous question Next question →

JEE Main 2022 · 24 Jun · Shift 2, Mathematics question 16