GATE 2026 CE (CE1) – Question 49
Values of $y$ for different values of $x$ are tabulated below.
| $x$ | -2 | 1 | 2 |
|---|---|---|---|
| $y$ | 28 | 4 | 16 |
If a second-degree interpolating polynomial $P_2(x)$ is used to represent $y$, the value of $P_2(0)$ is __________ (*rounded off to the nearest integer*).
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Correct answer: 2
Explanation
Use Lagrange interpolation at $x = 0$. The weights are $\frac{(0 - 1)(0 - 2)}{(-2 - 1)(-2 - 2)} = \frac{1}{6}$ for $x = -2$, $\frac{(0 + 2)(0 - 2)}{(1 + 2)(1 - 2)} = \frac{4}{3}$ for $x = 1$, and $\frac{(0 + 2)(0 - 1)}{(2 + 2)(2 - 1)} = -\frac{1}{2}$ for $x = 2$. So $P_2(0) = 28 \times \frac{1}{6} + 4 \times \frac{4}{3} + 16 \times \left(-\frac{1}{2}\right) = 4.667 + 5.333 - 8 = 2$.