Assertion -Let g be the volume of the parallelopiped formed by the vector a - a 1 i- a 2 j- a 3 k- b - b 1 i- b 2 j- b 3 k- c - c 1 i- c 2 j- c 3 k-If a r - b r - c r where r-1-2-3 are non-negative real numbers and - r-1 3 a r - b r - c r -3L- then V- L 3 Reason- x 3 - y 3 - z 3 - x-y-z

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Standard XII

Mathematics

Skew Symmetric Matrix

Assertion :Le...

Question

Assertion :Let $g$ be the volume of the parallelopiped formed by the vector $\to a = a_{1} \to i + a_{2} \to j + a_{3} \to k, \to b = b_{1} \to i + b_{2} \to j + b_{3} \to k, \to c = c_{1} \to i + c_{2} \to j + c_{3} \to k$ .
If $a_{r}, b_{r}, c_{r}$ where $r = 1, 2, 3$ are non-negative real numbers and $3 \sum r = 1 (a_{r} + b_{r} + c_{r}) = 3 L;$ then $V \leq L^{3}$
Reason: $x^{3} + y^{3} + z^{3} \leq (x + y + z)$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$V = ∥ ∥ ∥ ∥ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∥ ∥ ∥ ∥$
$= | a_{1} b_{2} c_{3} - a_{1} c_{2} b_{3} - a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2} - a_{3} b_{2} c_{1} |$
$= | (a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2}) - (a_{1} c_{2} b_{3} + a_{2} b_{1} c_{3} + a_{3} b_{2} c_{1}) |$
Without loss of generality let $a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2} \geq a_{1} b_{2} c_{3} + a_{2} b_{1} c_{3} + a_{3} b_{2} c_{1}$
$∴ V \leq (a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2})$ ...(1)
By $A . M . \leq G . M ., (a_{1} + b_{2} + c_{1}) \geq {(a_{1} b_{2} c_{3})}_{1}^{1 / 3}$
Similarly $(a_{2} + b_{3} + c_{1}) \geq 3 {(a_{2} b_{2} c_{1})}^{1 / 3}$ and $(a_{3} + b_{1} + c_{2}) \geq 3 {(a_{3} b_{1} c_{2})}^{1 / 3}$
$\Rightarrow {(a_{1} + b_{2} + c_{3})}^{3} \geq 27 (a_{1} b_{2} c_{3}); {(a_{2} + b_{3} + c_{1})}^{3} \geq 27 (a_{1} b_{2} c_{3})$ and ${(a_{2} + b_{3} + c_{1})}^{3} \geq 27 (a_{3} b_{1} c_{2})$
Adding we get,
${(a_{1} + b_{2} + c_{3})}^{3} + {(a_{2} + b_{3} + c_{1})}^{3} + {(a_{3} + b_{1} + c_{2})}^{3} \geq 27 (a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2})$ ....(2)
$∴$ from (1) and (2)
$V \leq \frac{1}{27} [{(a_{1} + b_{2} + c_{3})}^{3} + {(a_{2} + b_{3} + c_{1})}^{3} + {(a_{3} + b_{1} + c_{2})}^{3}]$
$\Rightarrow V \leq \frac{1}{27} {[(a_{1} + b_{1} + c_{1}) + (a_{2} + b_{2} + c_{2}) + (a_{3} + b_{3} + c_{3})]}^{3}$
$[∵ x^{3} + y^{3} + z^{3} \leq {(x + y + z)}^{3}$ for $x, y, z \geq 0]$
$\Rightarrow V \leq \frac{1}{27} {[3 L]}^{3} = L^{3} \Rightarrow V \leq L^{3}$
$\Rightarrow$ Assertion as well as reason both are correct and reason is the correct explanation of assertion
$\Rightarrow$ option (a) is correct.

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Similar questions

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Assertion :Let g be the volume of the parallelopiped formed by the vector →a=a1→i+a2→j+a3→k,→b=b1→i+b2→j+b3→k,→c=c1→i+c2→j+c3→k.If ar,br,cr where r=1,2,3 are non-negative real numbers and 3∑r=1(ar+br+cr)=3L; then V≤L3 Reason: x3+y3+z3≤(x+y+z)