Question : If $8 a^{3} + 27 b^{3} = 16$ and $2 a + 3 b = 4$ , then find the value of $16 a^{4} + 81 b^{4}$ .
Option 1: 26
Option 2: 30
Option 3: 28
Option 4: 32

Question

Question : The mode of the following data is __________.
13, 15, 31, 12, 27, 13, 27, 30, 27, 28 and 16

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Answer 1

Correct Answer: 32

Solution : Given, $8 a^{3} + 27 b^{3} = 16$ and $2 a + 3 b = 4$
Consider $8 a^{3} + 27 b^{3} = 16$
⇒ $(2 a)^{3} + (3 b)^{3} = 16$
⇒ $(2 a + 3 b) (4 a^{2} - 6 a b + 9 b^{2}) = 16$
⇒ $(4) (4 a^{2} - 6 a b + 9 b^{2}) = 16$
⇒ $4 a^{2} - 6 a b + 9 b^{2} = 4$ ......(1)
Now, consider $2 a + 3 b = 4$
Squaring both sides, we get,
⇒ $(2 a + 3 b)^{2} = 4^{2}$
⇒ $4 a^{2} + 12 a b + 9 b^{2} = 16$ ......(2)
Solving (1) and (2), we get,
$18 a b = 12$
⇒ $a b = \frac{2}{3}$
And, $4 a^{2} + 9 b^{2} = 4 + 6 \times \frac{2}{3}$
⇒ $4 a^{2} + 9 b^{2} = 4 + 4 = 8$
Now, $16 a^{4} + 81 b^{4}$
$= (4 a^{2})^{2} + (9 b^{2})^{2}$
$= (4 a^{2} + 9 b^{2})^{2} - 72 a^{2} b^{2}$
$= 8^{2} - 72 \times \frac{2}{3} \times \frac{2}{3}$
$= 64 - 32 = 32$
Hence, the correct answer is 32.

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Question : If 8a3+27b3=16 and 2a+3b=4, then find the value of 16a4+81b4.Option 1: 26Option 2: 30Option 3: 28Option 4: 32

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Question : If $8 a^{3} + 27 b^{3} = 16$ and $2 a + 3 b = 4$ , then find the value of $16 a^{4} + 81 b^{4}$ .
Option 1: 26
Option 2: 30
Option 3: 28
Option 4: 32