Statement I If `a, b,c, in R` and not all equal, then `sec theta=((bc+ca+ab))/((a^(2)+b^(2)+c^(2)))`,
Statement II `sec theta le -1` and `sec theta ge 1`

Options

A. A
B. B
C. C
D. D

Answer

To solve the problem, we need to analyze both statements and determine their validity.

### Step 1: Analyze Statement I
Statement I claims that if \(a, b, c \in \mathbb{R}\) and not all equal, then:
\[ \sec \theta = \frac{bc + ca + ab}{a^2 + b^2 + c^2} \]

### Step 2: Use the identity for squares
Recall the identity:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \]
From this, we can rearrange to find:
\[ ab + bc + ca = \frac{(a + b + c)^2 - (a^2 + b^2 + c^2)}{2} \]

### Step 3: Substitute into the equation
Now substituting \(ab + bc + ca\) into the equation for \(\sec \theta\):
\[ \sec \theta = \frac{bc + ca + ab}{a^2 + b^2 + c^2} = \frac{\frac{(a + b + c)^2 - (a^2 + b^2 + c^2)}{2}}{a^2 + b^2 + c^2} \]

### Step 4: Simplify the expression
This simplifies to:
\[ \sec \theta = \frac{(a + b + c)^2 - (a^2 + b^2 + c^2)}{2(a^2 + b^2 + c^2)} \]
This expression needs to be evaluated to determine if it can take values outside the range of \(\sec \theta\).

### Step 5: Analyze the range of \(\sec \theta\)
The range of \(\sec \theta\) is:
\[ \sec \theta \leq -1 \quad \text{or} \quad \sec \theta \geq 1 \]
This means \(\sec \theta\) cannot take values between -1 and 1.

### Step 6: Determine if Statement I is true or false
From our analysis, we see that the expression for \(\sec \theta\) derived from \(ab + bc + ca\) does not restrict itself to the values of \(\sec \theta\). Thus, Statement I is **false**.

### Step 7: Analyze Statement II
Statement II states:
\[ \sec \theta \leq -1 \quad \text{and} \quad \sec \theta \geq 1 \]
This statement is true as it correctly describes the range of \(\sec \theta\).

### Conclusion
- Statement I is **false**.
- Statement II is **true**.