A homogeneous polynomial of the second degree in `n` variables i.e., the expression
`phi=sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j)` where `a_(ij)=a_(ji)` is called a quadratic form in `n` variables `x_(1),x_(2)`….`x_(n)` if `A=[a_(ij)]_(nxn)` is
a symmetric matrix and `x=[{:(x_(1)),(x_(2)),(x_(n)):}]` then
`X^(T)AX=[X_(1)X_(2)X_(3) . . . .X_(n)][{:(a_(11),a_(12) ....a_(1n)),(a_(21),a_(22)....a_(2n)),(a_(n1),a_(n2)....a_(n n)):}][{:(x_(1)),(x_(2)),(x_(n)):}]=sum_(i=1)^(n)sum_(j=1)^(n)a_(ij)x_(i)x_(j)=phi`
Matrix A is called matrix of quadratic form `phi`.
Q. If number of distinct terms in a quadratic form is 10 then number of variables in quadratic form is

Options

A. 4
B. 3
C. 5
D. can not found uniquely

Explanation

To solve the problem, we need to determine the number of variables \( n \) in a quadratic form given that the number of distinct terms in the quadratic form is 10.

### Step-by-Step Solution:

1. **Understanding the Quadratic Form**:
A quadratic form in \( n \) variables can be expressed as:
\[ \phi = \sum_{i=1}^{n} \sum_{j=1}^{n} a_{ij} x_i x_j \]
where \( a_{ij} = a_{ji} \) and \( A = [a_{ij}] \) is a symmetric matrix.

2. **Counting Distinct Terms**:
The total number of distinct terms in a quadratic form with \( n \) variables is given by the formula:
\[ \text{Total distinct terms} = n + \binom{n}{2} \]
Here, \( n \) accounts for the linear terms \( x_i^2 \) (for \( i = 1, 2, \ldots, n \)), and \( \binom{n}{2} \) accounts for the cross-product terms \( x_i x_j \) (for \( 1 \leq i < j \leq n \)).

3. **Setting Up the Equation**:
Given that the total number of distinct terms is 10, we can set up the equation:
\[ n + \binom{n}{2} = 10 \]
The binomial coefficient \( \binom{n}{2} \) can be expressed as:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
Therefore, the equation becomes:
\[ n + \frac{n(n-1)}{2} = 10 \]

4. **Clearing the Denominator**:
To eliminate the fraction, multiply the entire equation by 2:
\[ 2n + n(n-1) = 20 \]
This simplifies to:
\[ n^2 + n - 20 = 0 \]

5. **Factoring the Quadratic Equation**:
We can factor the quadratic equation:
\[ n^2 + n - 20 = (n - 4)(n + 5) = 0 \]

6. **Finding the Roots**:
Setting each factor to zero gives us:
\[ n - 4 = 0 \quad \Rightarrow \quad n = 4 \]
\[ n + 5 = 0 \quad \Rightarrow \quad n = -5 \]

7. **Selecting the Valid Solution**:
Since \( n \) represents the number of variables, it cannot be negative. Therefore, the only valid solution is:
\[ n = 4 \]

### Final Answer:
The number of variables in the quadratic form is \( \boxed{4} \).