Consider the curve $\mathrm{C}_{1}$ given by $$\mathrm{y}=\mathrm{e}^{-\mathrm{x}} \text { for } \mathrm{x} \in[0,10 \pi],$$ and the curve $\mathrm{C}_{2}$ given by $$\mathrm{y}=\mathrm{e}^{-\mathrm{x}}(\sin \mathrm{x}+\cos \mathrm{x}) \text { for } \mathrm{x} \in[0,10 \pi] .$$ Let $n$ be the total number of points of intersection of the curves $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$. Suppose that $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}, \in[0,10 \pi]$ are the $x$ - coordinates of the points of intersection of the curves $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ such that $$\alpha_{1}<\alpha_{2}<\ldots<\alpha_{n} .$$ The value of $n$ is ____.

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Trigonometric Equations

Consider the curve

C_{1}

given by

y = e^{- x} for x \in [0, 10 π],

and the curve

C_{2}

given by

y = e^{- x} (sin x + cos x) for x \in [0, 10 π] .

Let

n

be the total number of points of intersection of the curves

C_{1}

and

C_{2}

.
Suppose that

α_{1}, α_{2}, \dots, α_{n}, \in [0, 10 π]

are the

x

- coordinates of the points of intersection of the curves

C_{1}

and

C_{2}

such that

α_{1} < α_{2} < \dots < α_{n} .

The value of

n

is ____.

Answer: 11

Setting

e^{- x} = e^{- x} (sin x + cos x)

gives

sin x + cos x = 1

, or

2 sin (x + \frac{π}{4}) = 1

, so

sin (x + \frac{π}{4}) = \frac{1}{2}

. This gives

x + \frac{π}{4} = \frac{π}{4} + 2 k π

\frac{3 π}{4} + 2 k π

, i.e.,

x = 2 k π

x = \frac{π}{2} + 2 k π

. In

[0, 10 π]

x \in {0, \frac{π}{2}, 2 π, \frac{5 π}{2}, 4 π, \frac{9 π}{2}, 6 π, \frac{13 π}{2}, 8 π, \frac{17 π}{2}, 10 π}

, giving

n = 11

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Numerical answer