If 'head' means one and 'tail' means two,then coefficient of quadratic equation `ax^2 +bx+c=0` are chosen by tossing three fair coins. The probability that roots of the equating are imaginary is

Options

A. `5//8`
B. `3//8`
C. `7//8`
D. `1//8`.

Answer

To solve the problem, we need to determine the probability that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are imaginary, given that the coefficients \( a \), \( b \), and \( c \) are determined by tossing three fair coins, where 'head' represents 1 and 'tail' represents 2.

### Step-by-Step Solution:

1. **Determine Possible Outcomes from Coin Tosses**:
- Each coin can either be a head (1) or a tail (2).
- Since we are tossing three coins, the total number of outcomes is \( 2^3 = 8 \).
- The possible combinations of outcomes (H for head, T for tail) are:
- HHH (1, 1, 1)
- HHT (1, 1, 2)
- HTH (1, 2, 1)
- HTT (1, 2, 2)
- THH (2, 1, 1)
- THT (2, 1, 2)
- TTH (2, 2, 1)
- TTT (2, 2, 2)

2. **Assign Values to Coefficients**:
- From the outcomes, we can assign values to \( a \), \( b \), and \( c \):
- H corresponds to 1
- T corresponds to 2
- Thus, the coefficients can take values from the set {1, 2}.

3. **Condition for Imaginary Roots**:
- The roots of the quadratic equation are imaginary when the discriminant \( D < 0 \).
- The discriminant is given by \( D = b^2 - 4ac \).
- Therefore, we need to find the cases where \( b^2 < 4ac \).

4. **Evaluate Each Case**:
- We will evaluate the discriminant for each combination of \( (a, b, c) \):
- For \( (1, 1, 1) \): \( D = 1^2 - 4(1)(1) = 1 - 4 = -3 \) (Imaginary)
- For \( (1, 1, 2) \): \( D = 1^2 - 4(1)(2) = 1 - 8 = -7 \) (Imaginary)
- For \( (1, 2, 1) \): \( D = 2^2 - 4(1)(1) = 4 - 4 = 0 \) (Real)
- For \( (1, 2, 2) \): \( D = 2^2 - 4(1)(2) = 4 - 8 = -4 \) (Imaginary)
- For \( (2, 1, 1) \): \( D = 1^2 - 4(2)(1) = 1 - 8 = -7 \) (Imaginary)
- For \( (2, 1, 2) \): \( D = 1^2 - 4(2)(2) = 1 - 16 = -15 \) (Imaginary)
- For \( (2, 2, 1) \): \( D = 2^2 - 4(2)(1) = 4 - 8 = -4 \) (Imaginary)
- For \( (2, 2, 2) \): \( D = 2^2 - 4(2)(2) = 4 - 16 = -12 \) (Imaginary)

5. **Count Favorable Outcomes**:
- The cases that yield imaginary roots are:
- (1, 1, 1)
- (1, 1, 2)
- (1, 2, 2)
- (2, 1, 1)
- (2, 1, 2)
- (2, 2, 1)
- (2, 2, 2)
- This gives us a total of 7 favorable outcomes.

6. **Calculate Probability**:
- The probability \( P \) that the roots are imaginary is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{8} \]

### Final Answer:
The probability that the roots of the equation are imaginary is \( \frac{7}{8} \).