JEE Mathematics 2025 Question with Solution

Given: The interior angles of an $n$-sided polygon are in A.P. with common difference $6^\circ$, and the largest interior angle is $219^\circ$.

Find: The value of $n$.

Let the first angle be $a$. Then the largest angle is

$$a + (n-1) \times 6 = 219$$

So,

$$a = 219 - 6(n-1) = 225 - 6n$$

The sum of the interior angles of an $n$-sided polygon is also

$$(n-2) \times 180$$

Since the angles are in A.P., their sum is

$$\frac{n}{2}\left[2a + (n-1) \times 6\right] = (n-2) \times 180$$

Substitute $a = 225 - 6n$:

$$\frac{n}{2}\left[2(225 - 6n) + (n-1) \times 6\right] = (n-2) \times 180$$ $$\frac{n}{2}(450 - 12n + 6n - 6) = 180n - 360$$ $$\frac{n}{2}(444 - 6n) = 180n - 360$$

Multiply both sides by $2$:

$$n(444 - 6n) = 360n - 720$$ $$444n - 6n^2 = 360n - 720$$ $$6n^2 - 84n - 720 = 0$$

Divide by $6$:

$$n^2 - 14n - 120 = 0$$

Factorizing,

$$(n-20)(n+6) = 0$$

So,

$$n = 20 \text{ or } -6$$

Since the number of sides must be positive,

$$n = 20$$

Therefore, the correct option is A.

Given: The interior angles form an A.P. with common difference $6^\circ$ and largest angle $219^\circ$.

Find: The number of sides of the polygon.

Let the angles be

$$a,\; a+6,\; a+12,\; \ldots,\; a+(n-1)6$$

Because the largest angle is $219^\circ$,

$$a + (n-1)6 = 219$$ $$a = 225 - 6n$$

Now use the sum of $n$ terms of an A.P.:

$$\frac{n}{2}\left[2a + (n-1)6\right]$$

This must equal the sum of interior angles of the polygon:

$$(n-2)180$$

Hence,

$$\frac{n}{2}\left[2a + (n-1)6\right] = (n-2)180$$

Substituting $a = 225 - 6n$,

$$\frac{n}{2}\left[2(225 - 6n) + 6(n-1)\right] = (n-2)180$$ $$\frac{n}{2}\left[450 - 12n + 6n - 6\right] = 180n - 360$$ $$\frac{n}{2}(444 - 6n) = 180n - 360$$ $$222n - 3n^2 = 180n - 360$$ $$42n - 3n^2 = -360$$ $$3n^2 - 42n - 360 = 0$$ $$n^2 - 14n - 120 = 0$$

Using the quadratic formula,

$$n = \frac{14 \pm \sqrt{14^2 + 480}}{2}$$ $$n = \frac{14 \pm \sqrt{676}}{2} = \frac{14 \pm 26}{2}$$

So,

$$n = 20 \text{ or } -6$$

Reject the negative value.

Therefore, the number of sides is $20$.