JEE Advanced 2026 Paper-1

Let $$S=\{1,2,3,\dots,10\}$$. Consider the set

$$X=\{R:R\text{ is an equivalence relation on the set }S\text{ such that }R\text{ has exactly 42 elements}\}.$$

Then the number of elements in $$X$$ is ___.

Consider the function $$f:\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)\to(-\infty,\infty)$$ defined by

$$f(x)=(|x|+|x-1|)\sin x+[x\sin x],$$

where $$[x\sin x]$$ is the greatest integer less than or equal to $$x\sin x$$.

Let $$\alpha$$ be the total number of points in the interval $$\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$$ at which $$f$$ is NOT continuous, and let $$\beta$$ be the total number of points in the interval $$\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$$ at which $$f$$ is NOT differentiable. Then the value of $$\alpha+\beta$$ is ___.

The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ___.

Let $$\alpha=\left(1-2\cos\tfrac{\pi}{11}\right)\left(1-2\cos\tfrac{3\pi}{11}\right)\left(1-2\cos\tfrac{9\pi}{11}\right)\left(1-2\cos\tfrac{27\pi}{11}\right)\left(1-2\cos\tfrac{81\pi}{11}\right).$$

Then the value of $$5-\alpha^2$$ is ___.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

For real numbers $$\alpha,\beta,\gamma,\delta$$ and $$\mu$$, consider the matrix $$M=\begin{bmatrix}\alpha&\tfrac{1}{\sqrt{2}}&-\tfrac{1}{\sqrt{2}}\\[2pt]\tfrac{1}{\sqrt{3}}&\beta&\tfrac{1}{\sqrt{3}}\\[2pt]\gamma&\delta&\mu\end{bmatrix}.$$

Suppose that $$MM^{T}=I$$, where $$M^{T}$$ is the transpose of $$M$$ and $$I$$ is the $$3\times 3$$ identity matrix. Let $$\vec{u}=\alpha\hat{i}+\tfrac{1}{\sqrt{3}}\hat{j}+\gamma\hat{k},\quad \vec{v}=\tfrac{1}{\sqrt{2}}\hat{i}+\beta\hat{j}+\delta\hat{k},\quad \vec{w}=-\tfrac{1}{\sqrt{2}}\hat{i}+\tfrac{1}{\sqrt{3}}\hat{j}+\mu\hat{k}.$$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Match each entry in List-I to the correct entry in List-II and choose the correct option.