A - -aij-3- 3 is a diagonal matrix such that -A- - 27000 and x1- x2- x3 - N are elements along principal diagonal- then

|A|=x_1x_2x_3=27000 Now, x_1+x_2+x_33≥ (x_1x_2x_3)^1/3 x_1+x_2+x_3 ≥ 90 (x_1+x_2+x_3)_min=90 Also x_1x_2x_3=2^33^35^3 No. of solution ^3+3 1C_3 1× ^3+3 1C_3 1× ...

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Standard XII

Mathematics

Properties of Cube Root of a Complex Number

A = [aij]3× 3...

Question

$A = [a_{i j}]_{3 \times 3}$ is a diagonal matrix such that $| A | = 27000$ and $x_{1}, x_{2}, x_{3} \in N$ are elements along principal diagonal, then

Number of such matrix is

^{'} N^{'}

then

^{'} N^{'}

is perfect square

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Minimum value of

x_{1} + x_{2} + x_{3}

60

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Number of such matrix is

^{'} N^{'}

then

^{'} N^{'}

is perfect cube

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Minimum value of

x_{1} + x_{2} + x_{3}

90

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Solution

The correct option is D Minimum value of $x_{1} + x_{2} + x_{3}$ is $90$
$| A | = x_{1} x_{2} x_{3} = 27000$
Now, $\frac{x_{1} + x_{2} + x_{3}}{3} \geq (x_{1} x_{2} x_{3})^{1 / 3}$
$x_{1} + x_{2} + x_{3} \geq 90$
$(x_{1} + x_{2} + x_{3})_{min} = 90$
Also $x_{1} x_{2} x_{3} = 2^{3} 3^{3} 5^{3}$
No. of solution $^{3 + 3 - 1} C_{3 - 1} \times^{3 + 3 - 1} C_{3 - 1} \times^{3 + 3 - 1} C_{3 - 1}$
$=^{5} C_{2} \times^{5} C_{2} \times^{5} C_{2}$
$= 10^{3}$ perfect cube

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A=[aij]3×3 is a diagonal matrix such that |A|=27000 and x1,x2,x3∈N are elements along principal diagonal, then

The correct option is D Minimum value of x1+x2+x3 is 90|A|=x1x2x3=27000 Now, x1+x2+x33≥(x1x2x3)1/3 x1+x2+x3≥90 (x1+x2+x3)min=90 Also x1x2x3=233353 No. of solution 3+3−1C3−1× 3+3−1C3−1× 3+3−1C3−1 = 5C2× 5C2× 5C2 =103 perfect cube

$A = [a_{i j}]_{3 \times 3}$ is a diagonal matrix such that $| A | = 27000$ and $x_{1}, x_{2}, x_{3} \in N$ are elements along principal diagonal, then