Number of irrational terms in the expansion of `(root(5) (2) + root(10) (3) )^(60)` are

Options

A. 54
B. 61
C. 30
D. 31

Answer

To find the number of irrational terms in the expansion of \((\sqrt{5} \cdot 2 + \sqrt{10} \cdot 3)^{60}\), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression in a more manageable form:
\[ (\sqrt{5} \cdot 2 + \sqrt{10} \cdot 3)^{60} = (2\sqrt{5} + 3\sqrt{10})^{60} \]

### Step 2: Identify the general term
Using the Binomial Theorem, the general term \(T_{r+1}\) in the expansion of \((a + b)^n\) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \(a = 2\sqrt{5}\), \(b = 3\sqrt{10}\), and \(n = 60\). Therefore, the general term becomes:
\[ T_{r+1} = \binom{60}{r} (2\sqrt{5})^{60-r} (3\sqrt{10})^r \]

### Step 3: Simplify the general term
Now, we can simplify this term:
\[ T_{r+1} = \binom{60}{r} \cdot 2^{60-r} \cdot 5^{(60-r)/2} \cdot 3^r \cdot 10^{r/2} \]
Since \(10 = 2 \cdot 5\), we can rewrite \(10^{r/2}\) as:
\[ 10^{r/2} = (2 \cdot 5)^{r/2} = 2^{r/2} \cdot 5^{r/2} \]
Thus, the general term becomes:
\[ T_{r+1} = \binom{60}{r} \cdot 2^{60-r + r/2} \cdot 5^{(60-r)/2 + r/2} \cdot 3^r \]
This simplifies to:
\[ T_{r+1} = \binom{60}{r} \cdot 2^{60 - r/2} \cdot 5^{30} \cdot 3^r \]

### Step 4: Determine conditions for rationality
For \(T_{r+1}\) to be rational, both exponents of \(2\) and \(5\) must be integers. This means:
1. \(60 - \frac{r}{2}\) must be an integer, which implies \(r\) must be even.
2. \(30\) is already an integer, so no condition is needed for \(5\).

Let \(r = 2k\) where \(k\) is an integer. Now we can find the possible values of \(r\):
- Since \(r\) can take values from \(0\) to \(60\) (inclusive), \(k\) can take values from \(0\) to \(30\) (inclusive).

### Step 5: Count rational terms
The possible values of \(k\) are \(0, 1, 2, \ldots, 30\), giving us \(31\) rational terms.

### Step 6: Calculate total terms
The total number of terms in the expansion is \(n + 1 = 60 + 1 = 61\).

### Step 7: Find irrational terms
To find the number of irrational terms, we subtract the number of rational terms from the total number of terms:
\[ \text{Number of irrational terms} = 61 - 31 = 30 \]

### Final Answer
Thus, the number of irrational terms in the expansion is:
\[ \boxed{30} \]