Let m and n (m`<`n) be the roots of the equation `x^(2)-16x+39=0`. If four terms p,q,r and s are inserted between m and n to form an AP. Then what is the value of `p+q+r+s` ? (a) 29 (b) 30 (c) 32 (d) 35

Options

A. 29
B. 30
C. 32
D. 35

Answer

To solve the problem step by step, we need to find the roots of the quadratic equation given and then determine the values of \( p, q, r, \) and \( s \) that are inserted between the roots to form an arithmetic progression (AP).

### Step 1: Find the roots of the equation \( x^2 - 16x + 39 = 0 \).

To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -16, c = 39 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-16)^2 - 4 \cdot 1 \cdot 39 = 256 - 156 = 100 \]

Now substituting into the quadratic formula:
\[ x = \frac{16 \pm \sqrt{100}}{2 \cdot 1} = \frac{16 \pm 10}{2} \]
This gives us two roots:
\[ x_1 = \frac{26}{2} = 13 \quad \text{and} \quad x_2 = \frac{6}{2} = 3 \]
Thus, the roots are \( m = 3 \) and \( n = 13 \).

### Step 2: Insert four terms \( p, q, r, s \) between \( m \) and \( n \) to form an AP.

In an arithmetic progression, the difference between consecutive terms is constant. If we denote the common difference as \( d \), we can express the terms as:
- \( p = m + d \)
- \( q = m + 2d \)
- \( r = m + 3d \)
- \( s = m + 4d \)

The last term will be \( n = m + 5d \).

### Step 3: Set up the equation using the roots.

Since \( n = 13 \) and \( m = 3 \):
\[ n = m + 5d \implies 13 = 3 + 5d \]
Solving for \( d \):
\[ 13 - 3 = 5d \implies 10 = 5d \implies d = 2 \]

### Step 4: Calculate the values of \( p, q, r, \) and \( s \).

Now substituting \( d = 2 \) back into the expressions for \( p, q, r, \) and \( s \):
- \( p = 3 + 2 = 5 \)
- \( q = 3 + 4 = 7 \)
- \( r = 3 + 6 = 9 \)
- \( s = 3 + 8 = 11 \)

### Step 5: Calculate \( p + q + r + s \).

Now we can find the sum:
\[ p + q + r + s = 5 + 7 + 9 + 11 \]
Calculating this gives:
\[ p + q + r + s = 32 \]

### Final Answer:

Thus, the value of \( p + q + r + s \) is \( \boxed{32} \).