Given that `px^(2)+qx+6` leaves the remainder as `1` on division by `2x=1` and `2qx^(2)+6x+p` leaves the remainder as `2` on division by `3x-1` . Find p and q .

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) based on the given conditions. Let's break it down step by step.

### Step 1: Analyze the first polynomial
We have the polynomial \( px^2 + qx + 6 \) which leaves a remainder of 1 when divided by \( 2x = 1 \).

From \( 2x = 1 \), we find:
\[ x = \frac{1}{2} \]

Now, substituting \( x = \frac{1}{2} \) into the polynomial:
\[ p\left(\frac{1}{2}\right)^2 + q\left(\frac{1}{2}\right) + 6 = 1 \]
This simplifies to:
\[ p\left(\frac{1}{4}\right) + q\left(\frac{1}{2}\right) + 6 = 1 \]
Multiplying through by 4 to eliminate the fractions:
\[ p + 2q + 24 = 4 \]
Rearranging gives us:
\[ p + 2q = 4 - 24 \]
\[ p + 2q = -20 \quad \text{(Equation 1)} \]

### Step 2: Analyze the second polynomial
Next, we consider the polynomial \( 2qx^2 + 6x + p \) which leaves a remainder of 2 when divided by \( 3x - 1 \).

From \( 3x - 1 = 0 \), we find:
\[ x = \frac{1}{3} \]

Now, substituting \( x = \frac{1}{3} \) into the polynomial:
\[ 2q\left(\frac{1}{3}\right)^2 + 6\left(\frac{1}{3}\right) + p = 2 \]
This simplifies to:
\[ 2q\left(\frac{1}{9}\right) + 2 + p = 2 \]
Multiplying through by 9 to eliminate the fractions:
\[ 2q + 18 + 9p = 18 \]
Rearranging gives us:
\[ 2q + 9p = 18 - 18 \]
\[ 2q + 9p = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( p + 2q = -20 \)
2. \( 2q + 9p = 0 \)

From Equation 1, we can express \( p \) in terms of \( q \):
\[ p = -20 - 2q \]

Substituting this expression for \( p \) into Equation 2:
\[ 2q + 9(-20 - 2q) = 0 \]
Expanding this gives:
\[ 2q - 180 - 18q = 0 \]
Combining like terms:
\[ -16q - 180 = 0 \]
Solving for \( q \):
\[ -16q = 180 \]
\[ q = -\frac{180}{16} = -\frac{45}{4} \]

### Step 4: Find \( p \)
Now substituting \( q = -\frac{45}{4} \) back into the expression for \( p \):
\[ p = -20 - 2\left(-\frac{45}{4}\right) \]
Calculating this gives:
\[ p = -20 + \frac{90}{4} = -20 + 22.5 = 2.5 = \frac{5}{2} \]

### Final Answers
Thus, the values of \( p \) and \( q \) are:
\[ p = \frac{5}{2}, \quad q = -\frac{45}{4} \]