If number of solution and sum of solution of the equation ` 3sin^(2)x-7sinx +2=0, x in [0,2pi]` are respectively N and S and ` f_(n)(theta)=sin^(n)theta +cos^(n) theta`. On the basis of above information , answer the following questions.
Value of N is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the value of \( N \) (the number of solutions) for the equation \( 3\sin^2 x - 7\sin x + 2 = 0 \) in the interval \( x \in [0, 2\pi] \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ 3\sin^2 x - 7\sin x + 2 = 0 \]
Let \( y = \sin x \). Then the equation becomes:
\[ 3y^2 - 7y + 2 = 0 \]

### Step 2: Factor the quadratic equation
We need to factor the quadratic equation \( 3y^2 - 7y + 2 \). We can look for two numbers that multiply to \( 3 \times 2 = 6 \) and add up to \( -7 \). The numbers are \( -6 \) and \( -1 \).
Thus, we can write:
\[ 3y^2 - 6y - y + 2 = 0 \]
Grouping the terms:
\[ 3y(y - 2) - 1(y - 2) = 0 \]
Factoring out \( (y - 2) \):
\[ (3y - 1)(y - 2) = 0 \]

### Step 3: Solve for \( y \)
Setting each factor to zero gives us:
1. \( 3y - 1 = 0 \) → \( y = \frac{1}{3} \)
2. \( y - 2 = 0 \) → \( y = 2 \)

### Step 4: Analyze the solutions
The value \( y = 2 \) is not valid because the sine function only takes values in the range \([-1, 1]\). Thus, we only consider \( y = \frac{1}{3} \).

### Step 5: Find the corresponding \( x \) values
Now we need to find \( x \) such that:
\[ \sin x = \frac{1}{3} \]
In the interval \( [0, 2\pi] \), the sine function is positive in the first and second quadrants. Therefore, we will have two solutions:
1. \( x = \arcsin\left(\frac{1}{3}\right) \)
2. \( x = \pi - \arcsin\left(\frac{1}{3}\right) \)

### Step 6: Count the number of solutions
Thus, the number of solutions \( N \) is:
\[ N = 2 \]

### Final Answer
The value of \( N \) is \( 2 \).

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