It is given that $a, b$ are real and $p, q$ are purely imaginary. Which pair of inequalities must always be true? A. $a^2 p^2 + b^2 q^2 \leq 2abpq$, $a^2 b^2 + p^2 q^2 \geq 2abpq$ B. $a^2 p^2 + b^2 q^2 \leq 2abpq$, $a^2 b^2 + p^2 q^2 \leq 2abpq$ C. $a^2 p^2 + b^2 q^2 \geq 2abpq$, $a^2 b^2 + p^2 q^2 \leq 2abpq$ D. $a^2 p^2 + b^2 q^2 \geq 2abpq$, $a^2 b^2 + p^2 q^2 \geq 2abpq$

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It is given that $a, b$ are real and $p, q$ are purely imaginary - HSC - SSCE Mathematics Extension 2 - Question 9 - 2018 - Paper 1

Question 9

It is given that $a, b$ are real and $p, q$ are purely imaginary. Which pair of inequalities must always be true? A. $a^2 p^2 + b^2 q^2 \leq 2abpq$, $a^2 b^2 + p^2 ... show full transcript

Worked Solution & Example Answer:It is given that $a, b$ are real and $p, q$ are purely imaginary - HSC - SSCE Mathematics Extension 2 - Question 9 - 2018 - Paper 1

Step 1

A. $a^2 p^2 + b^2 q^2 \leq 2abpq$, $a^2 b^2 + p^2 q^2 \geq 2abpq$

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Answer

This pair cannot always be true as it contradicts the relationship established by the inequalities.

Step 2

B. $a^2 p^2 + b^2 q^2 \leq 2abpq$, $a^2 b^2 + p^2 q^2 \leq 2abpq$

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Answer

This pair must also hold true since both terms involve the positive relationship of the variables involved.

Step 3

C. $a^2 p^2 + b^2 q^2 \geq 2abpq$, $a^2 b^2 + p^2 q^2 \leq 2abpq$

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Answer

This combination is inconsistent and cannot be true simultaneously.

Step 4

D. $a^2 p^2 + b^2 q^2 \geq 2abpq$, $a^2 b^2 + p^2 q^2 \geq 2abpq$

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Answer

This pair cannot always be true due to contradictions arising from the definitions of real and imaginary components.

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