JEE Mathematics 2025 Question with Solution

Given: The roots of $3x^2 - px + q = 0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an A.P. with common difference $d = \frac{3}{2}$. Also, the sum of the first $11$ terms is $88$.

Find: $q - 2p$.

Using the sum formula of an A.P.:

$$S_{11} = \frac{11}{2}(2a + 10d) = 88$$

So,

$$a + 5d = 8$$

Substitute $d = \frac{3}{2}$:

$$a + 5 \times \frac{3}{2} = 8$$ $$a + \frac{15}{2} = 8$$ $$a = \frac{1}{2}$$

Now the $10^{\text{th}}$ and $11^{\text{th}}$ terms are:

$$T_{10} = a + 9d = \frac{1}{2} + 9 \times \frac{3}{2} = 14$$ $$T_{11} = a + 10d = \frac{1}{2} + 10 \times \frac{3}{2} = \frac{31}{2}$$

Since these are the roots of $3x^2 - px + q = 0$, by Vieta's formulas:

$$\frac{p}{3} = T_{10} + T_{11}$$ $$\frac{q}{3} = T_{10}T_{11}$$

So,

$$\frac{p}{3} = 14 + \frac{31}{2} = \frac{59}{2}$$ $$p = \frac{177}{2}$$

And,

$$\frac{q}{3} = 14 \times \frac{31}{2} = 217$$ $$q = 651$$

Therefore,

$$q - 2p = 651 - 2 \times \frac{177}{2} = 651 - 177 = 474$$

Therefore, the required value is $474$.

Given: The required roots are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an A.P. with $d = \frac{3}{2}$ and $S_{11} = 88$.

Find: $q - 2p$.

Let the first term be $a$. Then

$$S_{11} = \frac{11}{2}\left(2a + 10d\right) = 88$$

Substituting $d = \frac{3}{2}$:

$$\frac{11}{2}\left(2a + 15\right) = 88$$ $$11(2a + 15) = 176$$ $$2a + 15 = 16$$ $$2a = 1$$ $$a = \frac{1}{2}$$

Hence,

$$T_{10} = a + 9d = \frac{1}{2} + \frac{27}{2} = 14$$ $$T_{11} = a + 10d = \frac{1}{2} + 15 = \frac{31}{2}$$

Now for $3x^2 - px + q = 0$:

$$\text{Sum of roots} = \frac{p}{3}, \qquad \text{Product of roots} = \frac{q}{3}$$

So,

$$\frac{p}{3} = 14 + \frac{31}{2} = \frac{59}{2}$$ $$p = \frac{177}{2}$$

Also,

$$\frac{q}{3} = 14 \cdot \frac{31}{2} = 217$$ $$q = 651$$

Thus,

$$q - 2p = 651 - 177 = 474$$

The correct numerical value is $474$.

The second provided approach contains an internal inconsistency in intermediate evaluation, but it also concludes with $474$. The consistent calculation above confirms the value.