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Question

For k = $blank fraction numerator 1 over denominator square root of 50 end fraction$ , the value of a, b, c such that PP' = I, where P = $open square brackets table row cell 2 divided by 3 end cell cell 3 k end cell a row cell negative 1 divided by 3 end cell cell negative 4 k end cell b row cell 2 divided by 3 end cell cell negative 5 k end cell c end table close square brackets$ is-

± $\frac{16}{5 \sqrt{2}}$ , ± $\frac{13}{5 \sqrt{2}}$ , $\mp$ $\frac{1}{3 \sqrt{2}}$
$\mp$ $\frac{1}{3 \sqrt{2}}$ , ± $\frac{13}{5 \sqrt{2}}$ , ± $\frac{16}{5 \sqrt{2}}$
± $\frac{13}{5 \sqrt{2}}$ , ± $\frac{16}{5 \sqrt{2}}$ , $\mp$ $\frac{1}{3 \sqrt{2}}$
None of these

The correct answer is: ± $fraction numerator 13 over denominator 5 square root of 2 end fraction$ , ± $fraction numerator 16 over denominator 5 square root of 2 end fraction$ , $negative-or-plus$ $fraction numerator 1 over denominator 3 square root of 2 end fraction$

For PP' = 1,
$open square brackets table row cell 2 divided by 3 end cell cell 3 k end cell a row cell negative 1 divided by 3 end cell cell negative 4 k end cell b row cell 2 divided by 3 end cell cell negative 5 k end cell c end table close square brackets blank open square brackets table row cell 2 divided by 3 end cell cell negative 1 divided by 3 end cell cell 2 divided by 3 end cell row cell 3 k end cell cell negative 4 k end cell cell negative 5 k end cell row a b c end table close square brackets$
= $open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets$ .
Performing matrix multiplication, we have
$table row cell fraction numerator 4 over denominator 9 end fraction plus 9 k to the power of 2 end exponent plus a to the power of 2 end exponent equals 1 comma fraction numerator 1 over denominator 9 end fraction plus 16 k to the power of 2 end exponent plus b to the power of 2 end exponent equals 1 comma end cell row cell fraction numerator 4 over denominator 9 end fraction plus 25 k to the power of 2 end exponent plus c to the power of 2 end exponent equals 0 end cell row cell rightwards double arrow a to the power of 2 end exponent equals fraction numerator 169 over denominator 450 to the power of ´ end exponent end fraction comma b to the power of 2 end exponent equals fraction numerator 256 over denominator 450 to the power of ´ end exponent end fraction comma c to the power of 2 end exponent equals fraction numerator 25 over denominator 450 end fraction. end cell row cell text end text text A end text text l end text text s end text text o end text text end text fraction numerator 4 over denominator 9 end fraction minus 15 k to the power of 2 end exponent plus a c equals 0 comma negative fraction numerator 2 over denominator 9 end fraction plus 20 k to the power of 2 end exponent plus b c equals stack stack 0 comma negative with _ below minus fraction numerator 2 over denominator 9 end fraction minus 12 k to the power of 2 end exponent plus a b equals 0 with _ below end cell row cell rightwards double arrow a b equals fraction numerator 208 over denominator 450 to the power of ´ end exponent end fraction comma b c equals negative fraction numerator 80 over denominator 450 to the power of ´ end exponent end fraction comma a c equals fraction numerator negative 65 over denominator 450 end fraction. end cell row cell text end text text H end text text e end text text n end text text c end text text e end text text end text a equals plus-or-minus fraction numerator 13 over denominator 5 square root of 2 end fraction comma b equals plus-or-minus fraction numerator 16 over denominator 5 square root of 2 end fraction comma c minus-or-plus fraction numerator 1 over denominator 3 square root of 2 end fraction. end cell end table$
Hence (C) is correct answer.

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