The number of solution of equation ` 8[x^(2)-x]+4[x]=13+12[sinx],[.]` denotes GIF is

Options

A. (a)0
B. (b)2
C. (c)4
D. (d)6

Answer

To solve the equation \( 8[x^2 - x] + 4[x] = 13 + 12[\sin x] \), where \([.]\) denotes the greatest integer function (GIF), we will analyze both sides of the equation step by step.

### Step 1: Understand the components of the equation
The left-hand side (LHS) is \( 8[x^2 - x] + 4[x] \) and the right-hand side (RHS) is \( 13 + 12[\sin x] \). We need to analyze the behavior of both sides.

### Step 2: Analyze the right-hand side
The term \( [\sin x] \) can take values from \(-1\) to \(0\) because the sine function oscillates between \(-1\) and \(1\). Therefore, we can evaluate the RHS for these extreme values:
- If \( [\sin x] = -1 \):
\[ \text{RHS} = 13 + 12(-1) = 13 - 12 = 1 \]
- If \( [\sin x] = 0 \):
\[ \text{RHS} = 13 + 12(0) = 13 \]
- If \( [\sin x] = 1 \):
\[ \text{RHS} = 13 + 12(1) = 13 + 12 = 25 \]

Thus, the RHS can take values in the range from \(1\) to \(25\).

### Step 3: Analyze the left-hand side
The term \( [x^2 - x] \) is a quadratic function, and the term \( [x] \) is the greatest integer function of \(x\). The LHS can be expressed as:
\[ \text{LHS} = 8[x^2 - x] + 4[x] \]
Since both \( [x^2 - x] \) and \( [x] \) are integers, the LHS will always be an even integer.

### Step 4: Determine the parity of both sides
- The LHS is even because it is a sum of even multiples (8 and 4) of integers.
- The RHS, however, can take on odd or even values depending on the value of \( [\sin x] \).

### Step 5: Compare LHS and RHS
Since the LHS is always even, we need to check if the RHS can ever be even:
- The values of the RHS we calculated are \(1\), \(13\), and \(25\). All of these values are odd.

### Conclusion
Since the LHS is even and the RHS can only take odd values, there are no solutions to the equation:
\[ \text{Number of solutions} = 0 \]

### Final Answer
The number of solutions of the equation is \(0\).