If the third term in expansion of `(1+x^(log_2x))^5` is `2560` then `x` is equal to (a) `2sqrt2` (b) `1/8` (c) `1/4` (d) `4sqrt2`

Options

A. `2sqrt(2)`
B. `(1)/(8)`
C. `4sqrt(2)`
D. `(1)/(4)`

Answer

To solve the problem, we need to find the value of \( x \) such that the third term in the expansion of \( (1 + x^{\log_2 x})^5 \) equals 2560. Let's break it down step by step.

### Step 1: Identify the third term in the binomial expansion
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1} \]
For our case, \( a = 1 \), \( b = x^{\log_2 x} \), and \( n = 5 \). We want the third term, which corresponds to \( k = 3 \):
\[ T_3 = \binom{5}{2} (1)^{5-2} (x^{\log_2 x})^2 \]

### Step 2: Calculate \( T_3 \)
Calculating \( T_3 \):
\[ T_3 = \binom{5}{2} (x^{\log_2 x})^2 \]
\[ T_3 = 10 (x^{\log_2 x})^2 \]

### Step 3: Set the equation equal to 2560
We know that \( T_3 = 2560 \):
\[ 10 (x^{\log_2 x})^2 = 2560 \]
Dividing both sides by 10:
\[ (x^{\log_2 x})^2 = 256 \]

### Step 4: Take the square root
Taking the square root of both sides:
\[ x^{\log_2 x} = 16 \]

### Step 5: Express 16 in terms of powers of 2
We know that \( 16 = 2^4 \). Therefore, we can write:
\[ x^{\log_2 x} = 2^4 \]

### Step 6: Take logarithm on both sides
Taking logarithm base 2 on both sides:
\[ \log_2(x^{\log_2 x}) = \log_2(2^4) \]
Using the property of logarithms:
\[ \log_2 x \cdot \log_2 x = 4 \]
Let \( y = \log_2 x \):
\[ y^2 = 4 \]

### Step 7: Solve for \( y \)
Taking the square root:
\[ y = 2 \quad \text{or} \quad y = -2 \]

### Step 8: Convert back to \( x \)
Now, converting back to \( x \):
1. If \( y = 2 \):
\[ \log_2 x = 2 \implies x = 2^2 = 4 \]
2. If \( y = -2 \):
\[ \log_2 x = -2 \implies x = 2^{-2} = \frac{1}{4} \]

### Step 9: Check the options
The possible values of \( x \) are \( 4 \) and \( \frac{1}{4} \). Among the options provided:
- (a) \( 2\sqrt{2} \)
- (b) \( \frac{1}{8} \)
- (c) \( \frac{1}{4} \)
- (d) \( 4\sqrt{2} \)

The correct answer is \( \frac{1}{4} \).

### Final Answer
Thus, the value of \( x \) is \( \frac{1}{4} \).

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