The number of real solutions of the equation 2x-2-2-1x-5-2 -2x-2 is

The correct option is B one2^x/2+(√(2)+1)^x =(5+2√(2))^x/2 ⇒ ( √(2)/√(5+2√(2)) )^x + ( √(2)+1/√(5+2√(2)) )^x =1 which is of the form cos^x α + sin^x α = 1 ...

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The number of real solutions of the equation $2^{\frac{x}{2}} + (\sqrt{2} + 1)^{x} = (5 + 2 \sqrt{2})^{\frac{x}{2}}$ is:

one

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four

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six

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infinite

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Solution

The correct option is A one
$2^{\frac{x}{2}} + (\sqrt{2} + 1)^{x} = (5 + 2 \sqrt{2})^{\frac{x}{2}} \Rightarrow {(\frac{\sqrt{2}}{\sqrt{5 + 2 \sqrt{2}}})}^{x} + {(\frac{\sqrt{2} + 1}{\sqrt{5 + 2 \sqrt{2}}})}^{x} = 1$
which is of the form
$c o s^{x} α + s i n^{x} α = 1$
$∴$ x = 2.

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The number of real solutions of the equation 2x2+(√2+1)x=(5+2√2)x2 is:

The correct option is A one2x2+(√2+1)x=(5+2√2)x2⇒ (√2√5+2√2)x+(√2+1√5+2√2)x=1 which is of the form cosxα+sinxα=1 ∴ x = 2.

The number of real solutions of the equation $2^{\frac{x}{2}} + (\sqrt{2} + 1)^{x} = (5 + 2 \sqrt{2})^{\frac{x}{2}}$ is:

The correct option is A one
$2^{\frac{x}{2}} + (\sqrt{2} + 1)^{x} = (5 + 2 \sqrt{2})^{\frac{x}{2}} \Rightarrow {(\frac{\sqrt{2}}{\sqrt{5 + 2 \sqrt{2}}})}^{x} + {(\frac{\sqrt{2} + 1}{\sqrt{5 + 2 \sqrt{2}}})}^{x} = 1$
which is of the form
$c o s^{x} α + s i n^{x} α = 1$
$∴$ x = 2.