Statement-I: The number of integral values of K for which the equation `(x^(2))/(3K-2) +(y^(2))/(K-10)=1` represents a hyperbola is 10.
Statement-II : The above equation represents a rectangular hyperbola for K =3.
Which of above statements is true

Options

A. only I
B. only II
C. both I and II
D. neither I nor II

Answer

To determine the validity of the statements regarding the hyperbola represented by the equation

\[ \frac{x^2}{3K-2} + \frac{y^2}{K-10} = 1, \]

we will analyze each statement step by step.

### Step 1: Identify Conditions for Hyperbola

For the given equation to represent a hyperbola, the denominators must be positive and the equation must fit the standard form of a hyperbola:

1. \(3K - 2 > 0\)
2. \(K - 10 < 0\)

### Step 2: Solve the Inequalities

1. From \(3K - 2 > 0\):
\[ 3K > 2 \implies K > \frac{2}{3} \]

2. From \(K - 10 < 0\):
\[ K < 10 \]

### Step 3: Combine the Inequalities

Combining both inequalities, we get:
\[ \frac{2}{3} < K < 10 \]

### Step 4: Determine Integral Values of K

Next, we need to find the integral values of \(K\) that satisfy the combined inequality. The integral values of \(K\) within this range are:
- \(1, 2, 3, 4, 5, 6, 7, 8, 9\)

Counting these values gives us a total of **9 integral values**.

### Step 5: Evaluate Statement-I

Statement-I claims that the number of integral values of \(K\) for which the equation represents a hyperbola is **10**. From our calculations, we found **9** integral values. Therefore, Statement-I is **false**.

### Step 6: Evaluate Statement-II

Now we check Statement-II, which states that the equation represents a rectangular hyperbola for \(K = 3\).

For the hyperbola to be rectangular, the following condition must hold:
\[ 3K - 2 = K - 10 \]

Substituting \(K = 3\):
\[ 3(3) - 2 = 3 - 10 \implies 9 - 2 = 3 - 10 \implies 7 \neq -7 \]

Thus, \(K = 3\) does not satisfy the rectangular hyperbola condition. Therefore, Statement-II is also **false**.

### Conclusion

Both statements are false.