MHTCET 14 May 2023 Shift 2 Solved Paper

To find the value of $|\overline{a} \times (\overline{b} \times \overline{c})|$, we need to utilize given information step by step. First, let's understand the vectors and their relationships stated in the problem.Given: $\;|\vec{a}|=\sqrt{3}$$\;|\vec{b}|=5$$\;\vec{b} \cdot \vec{c}=10$The angle between $\vec{b}$ and $\vec{c}$ is $\;\frac{\pi}{3}$ $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$To find $|\vec{b} \cdot \vec{c}|$ given the angle between them and the magnitude of $\vec{b}$, we apply the formula for dot product in terms of the angle between the vectors:$\vec{b} \cdot \vec{c}=|\vec{b}||\vec{c}| \cos (\theta)$Substituting the known values, we get:$10=5|\vec{c}| \cos \left(\;\frac{\pi}{3}\right)$ Since $\cos \left(\;\frac{\pi}{3}\right) =\;\frac{1}{2}$, we have:$\;10=5|\vec{c}| \;\frac{1}{2}$$\;|\vec{c}|=4$So the magnitude of $\vec{c}$ is 4 .Given that $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$, we can find the magnitude of the cross product $|\vec{b} \times \vec{c}|$ using the formula for the cross product in terms of magnitudes and angles:$|\vec{b} \times \vec{c}|=|\vec{b}||\vec{c}| \sin (\theta)$ Substituting the known values, we get:$|\vec{b} \times \vec{c}|=5 \cdot 4 \cdot \sin \left(\;\frac{\pi}{3}\right)$Since $\sin \left(\;\frac{\pi}{3}\right) =\;\frac{\sqrt{3}}{2}$, we have:$|\vec{b} \times \vec{c}|=20 \cdot \;\frac{\sqrt{3}}{2}=10 \sqrt{3}$ Now, to find $|\vec{a} \times (\vec{b} \times \vec{c})|$, we use the fact that the magnitude of the cross product of two vectors is given by the product of their magnitudes and the sine of the angle between them. Since $\vec{a}$ is perpendicular to $\vec{b} \times \vec{c}$, the angle between them is $\;\frac{\pi}{2}$, and $\sin \left(\;\frac{\pi}{2}\right) =1$. Therefore, the magnitude of their cross product is:$\;|\vec{a} \times (\vec{b} \times \vec{c})|=|\vec{a}||\vec{b} \times \vec{c}| \sin \left(\;\frac{\pi}{2}\right)$$\;=\sqrt{3} \cdot 10 \sqrt{3} \cdot 1=3 \cdot 10=30$