If roots of the equation `(m-2)x^2-(8-2m)x-(8-3m)=0` are opposite in sign then the number of integral values of m is/are :

Options

A. A) 0
B. B) 1
C. C) 2
D. D) more than 2

Answer

To solve the problem, we need to determine the integral values of \( m \) for which the roots of the quadratic equation

\[ (m-2)x^2 - (8-2m)x - (8-3m) = 0 \]

are opposite in sign.

### Step 1: Identify the coefficients
The given quadratic equation can be rewritten in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = m - 2 \)
- \( b = -(8 - 2m) = 2m - 8 \)
- \( c = -(8 - 3m) = 3m - 8 \)

### Step 2: Condition for roots to be opposite in sign
For the roots of the quadratic equation to be opposite in sign, the product of the roots must be negative. The product of the roots \( \alpha \) and \( \beta \) can be given by:

\[ \alpha \beta = \frac{c}{a} = \frac{3m - 8}{m - 2} \]

We need this product to be less than zero:

\[ \frac{3m - 8}{m - 2} < 0 \]

### Step 3: Analyze the inequality
The inequality \( \frac{3m - 8}{m - 2} < 0 \) holds true when the numerator and denominator have opposite signs. We will consider two cases:

#### Case 1:
1. \( 3m - 8 > 0 \) and \( m - 2 < 0 \)
- From \( 3m - 8 > 0 \):
\[ 3m > 8 \implies m > \frac{8}{3} \]
- From \( m - 2 < 0 \):
\[ m < 2 \]
- This case leads to \( m > \frac{8}{3} \) and \( m < 2 \), which is not possible since \( \frac{8}{3} \approx 2.67 \) is greater than 2.

#### Case 2:
2. \( 3m - 8 < 0 \) and \( m - 2 > 0 \)
- From \( 3m - 8 < 0 \):
\[ 3m < 8 \implies m < \frac{8}{3} \]
- From \( m - 2 > 0 \):
\[ m > 2 \]
- This case leads to \( 2 < m < \frac{8}{3} \).

### Step 4: Determine integral values of \( m \)
Now we need to find the integer values of \( m \) that satisfy \( 2 < m < \frac{8}{3} \).

Since \( \frac{8}{3} \approx 2.67 \), the only integer value that lies between 2 and \( \frac{8}{3} \) is:

- \( m = 3 \)

However, \( m = 3 \) does not satisfy \( m < \frac{8}{3} \).

### Conclusion
Since there are no integers that satisfy the condition \( 2 < m < \frac{8}{3} \), the number of integral values of \( m \) is:

\[ \text{Number of integral values of } m = 0 \]

### Final Answer
Thus, the answer is option (a) 0.

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