There are trignometrical equations that are nonstandard in aspect in that the equation may contain dissimilar terms, say algebric as well as trignometric expressions, or three may be only one equation in more than unknowns. In such cases we make use of the extreme values of trignometric functions or algebric functions. In particular `|sinx| le 1, |cos x| le 1`, and maximum/minimaof quadratic expressions are widely made use of
The equation `sin(pi/6)=x^(2)-6x+10` holds for
i)Infintely many values of x
ii)Finitely many values of x
iii)Just one value of x
iv)No value of x

Options

A. Infintely many values of `x`
B. Finitely many values of `x`
C. Just one value of `x`
D. No value of x

Answer

To solve the equation \( \sin\left(\frac{\pi}{6}\right) = x^2 - 6x + 10 \), we will follow these steps:

### Step 1: Evaluate \( \sin\left(\frac{\pi}{6}\right) \)
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 2: Set up the equation
Now we can substitute this value into the equation:
\[ \frac{1}{2} = x^2 - 6x + 10 \]

### Step 3: Rearrange the equation
To make it a standard quadratic equation, we rearrange it:
\[ x^2 - 6x + 10 - \frac{1}{2} = 0 \]
This simplifies to:
\[ x^2 - 6x + \frac{19}{2} = 0 \]

### Step 4: Multiply through by 2 to eliminate the fraction
To eliminate the fraction, we multiply the entire equation by 2:
\[ 2x^2 - 12x + 19 = 0 \]

### Step 5: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -12 \), and \( c = 19 \). Plugging in these values:
\[ x = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 2 \cdot 19}}{2 \cdot 2} \]
Calculating the discriminant:
\[ x = \frac{12 \pm \sqrt{144 - 152}}{4} \]
\[ x = \frac{12 \pm \sqrt{-8}}{4} \]

### Step 6: Simplify the expression
Since the discriminant is negative, we have:
\[ \sqrt{-8} = 2\sqrt{2}i \]
Thus, the expression for \( x \) becomes:
\[ x = \frac{12 \pm 2\sqrt{2}i}{4} \]
This simplifies to:
\[ x = 3 \pm \frac{\sqrt{2}}{2}i \]

### Step 7: Conclusion about the values of \( x \)
Since the solutions involve imaginary numbers, there are no real values of \( x \) that satisfy the equation. Therefore, the answer to the question is:
**iv) No value of \( x \)**.

---