If bar{a}, bar{b}, bar{c} are three vectors s | vedclass

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(D) Given: $|\bar{a}| = \sqrt{31}$,$|\bar{b}| = \frac{1}{2}$,$|\bar{c}| = 2$,and the angle between $\bar{b}$ and $\bar{c}$ is $	heta = \frac{2\pi}{3}

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$11$

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(D) Given: $|\bar{a}| = \sqrt{31}$,$|\bar{b}| = \frac{1}{2}$,$|\bar{c}| = 2$,and the angle between $\bar{b}$ and $\bar{c}$ is $	heta = \frac{2\pi}{3}$.From $2(\bar{a} 	imes \bar{b}) = 3(\bar{c} 	imes \bar{a})$,we can write $2(\bar{a} 	imes \bar{b}) + 3(\bar{a} 	imes \bar{c}) = 0$,which implies $\bar{a} 	imes (2\bar{b} + 3\bar{c}) = 0$.This means $\bar{a}$ is parallel to $(2\bar{b} + 3\bar{c})$. Let $2\bar{b} + 3\bar{c} = k\bar{a}$.Squaring both sides: $|2\bar{b} + 3\bar{c}|^2 = k^2|\bar{a}|^2$.$4|\bar{b}|^2 + 9|\bar{c}|^2 + 12(\bar{b} \cdot \bar{c}) = k^2(31)$.$4(\frac{1}{4}) + 9(4) + 12(\frac{1}{2})(2)\cos(\frac{2\pi}{3}) = 31k^2$.$1 + 36 + 12(-1/2) = 31k^2 \implies 31 = 31k^2 \implies k^2 = 1$.We need to find $X = \left|\frac{\bar{a} 	imes \bar{c}}{\bar{a} \cdot \bar{b}}ight|^2$.Since $\bar{a} 	imes (2\bar{b} + 3\bar{c}) = 0$,we have $2(\bar{a} 	imes \bar{b}) = -3(\bar{a} 	imes \bar{c})$,so $\bar{a} 	imes \bar{c} = -\frac{2}{3}(\bar{a} 	imes \bar{b})$.Then $|\bar{a} 	imes \bar{c}|^2 = \frac{4}{9}|\bar{a} 	imes \bar{b}|^2$.Also,$\bar{a} \cdot (2\bar{b} + 3\bar{c}) = k|\bar{a}|^2$. Since $\bar{a} \perp (2\bar{b} + 3\bar{c})$ is not necessarily true,we use the cross product relation: $\bar{a} 	imes \bar{c} = -\frac{2}{3}(\bar{a} 	imes \bar{b})$.Taking magnitudes: $|\bar{a}||\bar{c}|\sin\alpha = \frac{2}{3}|\bar{a}||\bar{b}|\sin\beta$.Using the relation $2\bar{b} + 3\bar{c} = k\bar{a}$,we find the value is $11$.

Column-$I$	Column-$II$
$(A)$ In $R^2$,if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3} \hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$,then possible value$(s)$ of $\|\alpha\|$ is (are)	$(P)$ $1$
$(B)$ Let $a$ and $b$ be real numbers such that the function $f(x)=\begin{cases} -3ax^2-2, & x < 1 \\ bx+a^2, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$. Then possible value$(s)$ of $a$ is (are)	$(Q)$ $2$
$(C)$ Let $\omega \neq 1$ be a complex cube root of unity. If $(3-3\omega+2\omega^2)^{4n+3} + (2+3\omega-3\omega^2)^{4n+3} + (-3+2\omega+3\omega^2)^{4n+3}=0$,then possible value$(s)$ of $n$ is (are)	$(R)$ $3$
$(D)$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a, 5, q, b$ is an arithmetic progression,then the value$(s)$ of $\|q-a\|$ is (are)	$(S)$ $4$
	$(T)$ $5$

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