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If $straight x open parentheses 3 minus 2 over straight x close parentheses space equals space 3 over straight x comma space$ then the value of $straight x squared plus 1 over straight x squared$ will be

$3 1 over 9$
$3 2 over 9$
$2 1 over 9$
$2 1 over 9$

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42.

If $fraction numerator straight a over denominator straight q minus straight r end fraction equals space fraction numerator straight b over denominator straight r minus straight p end fraction equals fraction numerator straight c over denominator straight p minus straight q end fraction$ , then the value of pa + qb + rc is

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43.

The simplest value of $fraction numerator 3 square root of 8 minus 2 square root of 12 plus square root of 20 over denominator 3 square root of 18 minus 2 square root of 27 plus square root of 45 end fraction space$ is

$3 over 2$
$2 over 3$
$1 third$
$1 third$

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44.

If $open parentheses straight a plus 1 over straight a close parentheses squared space equals space 3$ , then the value of $straight a cubed plus 1 over straight a cubed$ is

0
$3 open parentheses straight a plus 1 over straight a close parentheses$
$3 open parentheses straight a squared plus 1 over straight a squared close parentheses$
$3 open parentheses straight a squared plus 1 over straight a squared close parentheses$

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45.

If $fraction numerator straight a squared plus straight b squared over denominator straight c squared end fraction space equals space fraction numerator straight b squared plus straight c squared over denominator straight a squared end fraction space equals fraction numerator straight c squared plus straight a squared over denominator straight b squared end fraction space equals space 1 over straight k comma space left parenthesis straight k not equal to 0 right parenthesis space then space straight k space equals space ?$

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46.

If $2 straight x plus fraction numerator 2 over denominator 9 straight x end fraction space equals 4 comma$ then the value of $27 straight x cubed plus fraction numerator 1 over denominator 27 straight x cubed end fraction$

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47.

If $straight x squared plus 1 over straight x squared equals space space 2 comma space$ then the value of $straight x minus 1 over straight x$ is

-2
0
1
1

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48.

If $pq left parenthesis straight p plus straight q right parenthesis space equals space 1 comma space$ then the value of $fraction numerator 1 over denominator straight p cubed straight q cubed end fraction minus straight p cubed minus straight q cubed$ is equal to

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49.
If $straight x equals space square root of 2 plus 1 comma$ then the value of $straight x to the power of 4 minus 1 over straight x to the power of 4$ is

$8 square root of 2$
$18 square root of 2$
$6 square root of 2$
$6 square root of 2$

6 square root of 2

straight x space equals space square root of 2 plus 1

1 over straight x space equals space fraction numerator 1 over denominator square root of 2 begin display style plus end style begin display style 1 end style end fraction cross times fraction numerator square root of 2 begin display style minus end style begin display style 1 end style over denominator square root of 2 begin display style minus end style begin display style 1 end style end fraction 1 over straight x space equals space square root of 2 minus 1

straight x squared space equals space left parenthesis square root of 2 plus 1 right parenthesis squared space equals space 2 plus 1 plus 2 square root of 2

straight x squared space equals space 3 plus 2 square root of 2 straight x to the power of 4 space equals space 9 plus 8 plus 12 square root of 2 space equals space 17 plus 12 square root of 2 space space space space 1 over straight x squared space equals space left parenthesis square root of 2 minus 1 right parenthesis squared space equals space 2 plus 1 minus 2 square root of 2 space space space space space 1 over straight x squared space equals space 3 minus 2 square root of 2 space space space space space space 1 over straight x to the power of 4 space equals space 9 plus 8 minus 12 square root of 2 space space space space space space 1 over straight x to the power of 4 space equals space 17 minus 12 square root of 2

straight x to the power of 4 minus 1 over straight x to the power of 4 space equals space 17 plus 12 square root of 2 minus 17 plus 12 square root of 2

therefore space straight x to the power of 4 minus 1 over straight x to the power of 4 space equals space 24 square root of 2

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50.

If $9 straight x squared plus 16 straight y squared space equals space 60$ and $3 straight x plus 4 straight y space equals space 6 comma$ then the value of xy is