JEE Advanced 2026 Paper-2

Let $$\mathbb{R}$$ denote the set of all real numbers. Consider the polynomial function $$f:\mathbb{R}\to\mathbb{R}$$ defined by

$$f(x)=\dfrac{d^{10}}{dx^{10}}\big((x^2-1)^{10}\big),\qquad\text{for all }x\in\mathbb{R}.$$

Here $$\dfrac{d^{10}}{dx^{10}}\big((x^2-1)^{10}\big)$$ is the $$10^{\text{th}}$$ order derivative of the function $$(x^2-1)^{10}$$.

Then which of the following statements is (are) TRUE?

Let $$a,\,b,\,c$$ be positive integers in arithmetic progression such that the equation

$$ax^2+bx+c=0$$

has only integer solutions.

Then which of the following statements is (are) TRUE?

Let $$L$$ be the straight line joining the points $$P(1,2,-1)$$ and $$Q(2,3,1)$$. Let $$S$$ be the foot of the perpendicular drawn from the point $$R(4,-1,5)$$ to the line $$L$$. Another line passing through $$R$$ intersects $$L$$ at a point $$T$$ such that the point $$S$$ divides the line segment $$PT$$ internally in the ratio $$|PS|:|ST|=1:2$$, where $$|PS|$$ and $$|ST|$$ are the lengths of the line segments $$PS$$ and $$ST$$, respectively.

Then which of the following statements is (are) TRUE?

Let $$y=f(x)$$ be the real valued function defined on the interval $$(0,\infty)$$, satisfying $$y(1)=0$$ and the differential equation

$$x\dfrac{dy}{dx}=y-x^3.$$

Then which of the following statements is (are) TRUE?

Let $$\mathbb{R}$$ denote the set of all real numbers and let $$i=\sqrt{-1}$$. Consider the matrices

$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$

Let $$a,b,c,d$$ be real numbers such that

$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$

Let

$$H=\{\,x+iy:\;x,y\in\mathbb{R}\;\text{and}\;y>0\,\}.$$

Then which of the following statements is (are) TRUE?

Let $$\mathbb{N}$$ denote the set of all positive integers. Consider the sets

$$A=\{1,2,3,4,5\}\quad\text{and}\quad B=\{1,2,3,4,5,6,7\}.$$

Let $$S$$ be the set of all functions $$f:A\to B$$ such that $$f(2)\neq 2$$ and $$f(4)\neq 4$$. Consider the set

$$T=\big\{f\in S:\text{there exists a function }g:B\to\mathbb{N}\text{ such that }g\big(f(x)\big)=2^x\text{ for all }x\in A\big\}.$$

Then the number of elements in the set $$T$$ is ___.

A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $$X$$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $$\alpha$$ is the mean of the random variable $$X$$, then the value of $$77\alpha$$ is ___.

Consider a data consisting of 10 observations $$x_1,x_2,\dots,x_{10}$$, whose mean is $$5$$ and variance is $$7$$. If the mean and the variance of the first 8 observations $$x_1,x_2,\dots,x_8$$ are $$4$$ and $$3.5$$, respectively, and $$x_9 < x_{10}$$, then the value of $$3x_9 + 2x_{10}$$ is ___________.

Consider the ellipse $$E$$ given by $$\dfrac{x^2}{18}+\dfrac{y^2}{12}=1$$. Let $$H$$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $$E$$ and whose foci are the same as that of $$E$$. Let $$P$$ and $$Q$$ be the points of intersection of $$H$$ and the parabola $$\sqrt{5}\,y=x^2$$ in the first quadrant. Let $$d$$ be the distance between $$P$$ and $$Q$$.

If $$a$$ and $$b$$ are the integers such that $$d^2=a+b\sqrt{5}$$, then the value of $$a-b$$ is ___.

For a real number $$\alpha$$, let $$[\alpha]$$ denote the greatest integer less than or equal to $$\alpha$$. For a finite set $$S$$, let $$|S|$$ denote the number of elements in the set $$S$$.

Consider the functions $$f:(-3,3)\to(-\infty,\,\infty)$$ and $$g:(-3,3)\to(-\infty,\,\infty)$$ defined by

$$f(x)=[x^3]\log_e\big(1+\sin^2(\pi(x-[x])))\big)$$

and

$$g(x)=x^3\sin^2(\pi\log_e(1+x-[x])).$$

Let

$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$

and

$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$

Then the value of $$|A|+2|B|-|A\cap B|$$ is ___.