Question : If $2x+3y = 4$ and $4x^{2}+9y^{2}= 64$, then what is the value of $8x^{3}+27y^{3}$:
Option 1: 253
Option 2: 235
Option 3: 352
Option 4: 325
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Correct Answer: 352
Solution : Given: $2x+3y=4$ and $4x^2+9x^2=64$ To find: $8x^3+27y^3$ Now, $2x+3y=4$ Squaring both sides, we get: ⇒ $4x^2+9y^2+12xy=16$ Putting the values, we get: ⇒ $64+12xy=16$ ⇒ $12xy=-48$ ⇒ $xy=-4$ Now again, $2x+3y=4$ Cubing both sides, we get: ⇒ $8x^3+27y^3+3×2x×3y(2x+3y)=64$ Putting the values, we get: ⇒ $8x^3+27y^3+18×(-4)×4=64$ $\therefore8x^3+27y^3=64+288= 352$ Hence, the correct answer is 352.
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Question : If $\sin^4x + \sin^2x = 1$, then what is the value of $\cot^4x+ \cot^2x$?
Option 1: –2
Option 2: 2
Option 3: –1
Option 4: 1
Question : If $2x-3(4-2x)<4x-5<4x+\frac{2x}{3}$, then $x$ can take which of the following values?
Option 1: 2
Option 2: 8
Option 3: 0
Option 4: –8
Question : If $4x^2+y^2 = 40$ and $xy=6$, then the value of $2x-y$ is:
Option 1: 1
Option 2: 3
Option 3: 4
Option 4: 2
Question : If $2x+\frac{1}{4x}=1$, then the value of $x^{2}+\frac{1}{64x^{2}}$ is:
Option 1: $0$
Option 2: $1$
Option 3: $\frac{1}{4}$
Option 4: $2$
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