JEE Advanced 2026 Paper-2

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

The value of $$n$$ is ___.

Question Stem for Question Nos. 15 and 16

Consider the curve $$C_1$$ given by $$y=e^{-x}$$ for $$x\in[0,10\pi]$$, and the curve $$C_2$$ given by $$y=e^{-x}(\sin x+\cos x)$$ for $$x\in[0,10\pi]$$.

Let $$n$$ be the total number of points of intersection of the curves $$C_1$$ and $$C_2$$.

Suppose that $$\alpha_1,\alpha_2,\dots,\alpha_n\in[0,10\pi]$$ are the $$x$$-coordinates of the points of intersection of the curves $$C_1$$ and $$C_2$$ such that $$\alpha_1<\alpha_2<\cdots<\alpha_n$$.

Let $$\beta$$ be the area of the region enclosed between the curves $$C_1$$, $$C_2$$, and the lines $$x=\alpha_1$$ and $$x=\alpha_4$$. Then the value of

$$-\dfrac{1}{\pi}\log_e\!\left(\beta-2\,e^{-\frac{\pi}{2}}\right)$$

is ___.

Question Stem for Question Nos. 17 and 18

Consider the ellipses given by $$x^2+4y^2=1$$ and $$4x^2+y^2=1$$.

Let $$P$$ be the point in the first quadrant where the given ellipses intersect. If $$\theta$$ is the acute angle between the tangents to the given ellipses at the point $$P$$, then the value of $$4\tan\theta$$ is ___.

Question Stem for Question Nos. 17 and 18

Consider the ellipses given by $$x^2+4y^2=1$$ and $$4x^2+y^2=1$$.

If $$\alpha$$ is the area of the common region that lies inside both the given ellipses, then the value of $$\cot\alpha$$ is ___.