If 4x-22-x-5-b-1-3-sin y- where x-y-b - has a real solution- then the maximum possible value of b is

4^x 2^2+x+5+||b 1| 3|=|sin y| ⇒ 2^2x 4· 2^x+4+1+||b 1| 3|=|sin y| ⇒ (2^x 2)^2+1+||b 1| 3|=|sin y| We know that |sin y | ∈ [0, 1] Also, nbsp; (2^x 2)^2+1+||b 1 ...

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Standard XII

Mathematics

Properties Derived from Trigonometric Identities

If 4x-22+x+5+...

Question

If $4^{x} - 2^{2 + x} + 5 + | | b - 1 | - 3 | = | sin y |$ , where $x, y, b \in R$ has a real solution, then the maximum possible value of $b$ is

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Solution

The correct option is D $4$
$4^{x} - 2^{2 + x} + 5 + | | b - 1 | - 3 | = | sin y | \Rightarrow 2^{2 x} - 4 \cdot 2^{x} + 4 + 1 + | | b - 1 | - 3 | = | sin y | \Rightarrow (2^{x} - 2)^{2} + 1 + | | b - 1 | - 3 | = | sin y |$
We know that $| sin y | \in [0, 1]$
Also,
$(2^{x} - 2)^{2} + 1 + | | b - 1 | - 3 | \geq 1$
So equality is possible only when
$(2^{x} - 2)^{2} = 0 and | | b - 1 | - 3 | = 0 \Rightarrow sin y = \pm 1 ∴ y = \frac{(2 n + 1) π}{2} and x = 1 and b = - 2, 4$

Hence, the maximum value of $b$ is $4.$

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If 4x−22+x+5+||b−1|−3|=|siny|, where x,y,b∈R has a real solution, then the maximum possible value of b is

If $4^{x} - 2^{2 + x} + 5 + | | b - 1 | - 3 | = | sin y |$ , where $x, y, b \in R$ has a real solution, then the maximum possible value of $b$ is