JEE MAIN 2026 Mathematics Hard Questions

If $\sin \left(\frac{\pi }{18}\right)\sin \left(\frac{5\pi }{18}\right)\sin \left(\frac{7\pi }{18}\right)=K$, then the value of $\sin \left(\frac{10K\pi }{3}\right)$ is:

If the domain of the function $f(x)=\sqrt{{\log }_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty ,a]\cup \{b\}\cup [c,d)\cup (e,\infty )$, then the value of $a+b+c+d+e$ is ____.

Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy=12$ such that the mid-point of the line segment $PQ$ is $\left(\frac{1}{2},-\frac{1}{2}\right)$. Then the area of the triangle $OPQ$ equals:

Let $A=\left[\begin{array}{ccc}1& 21& \alpha \end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 3\beta & 2\end{array}\right]$. If $A^2-4A+I=O$ and $B^2-5B-6I=O$, then among the two statements:
(S1): ${\left[(B-A)(B+A)\right]}^T=\left[\begin{array}{ccc}13& 157& 10\end{array}\right]$ (S2): $\det (\mathrm{adj}(A+B))=-5$
which one is correct?

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $x+(k-1)y+3=0$ and $2x+k^{2}y-4=0$. If the line $x-y+2=0$ intersects the circle at the points $A$ and $B$, then $(AB{)}^2$ is equal to:

Let $a_1,a_2,a_3,\dots$ be an A.P. and $g_1,g_2,g_3,\dots$ be an increasing G.P. If $g_1=a_1$, $a_2+g_2=1$ and $a_3+g_3=4$, then $a_{10}+g_5$ is equal to:

If the area of the region bounded by $16x^2-9y^2=144$ and $8x-3y=24$ is $A$, then $3(A+6{\log }_e(3))$ is equal to ____.

Let $x=x(y)$ be the solution of the differential equation $2y^2\frac{dx}{dy}-2xy+x^2=0$, $y>1$, $x(e)=e$. Then $x(e^2)$ is equal to:

Let $f(x)$ be a polynomial of degree $5$, and have extrema at $x=1$ and $x=-1$. If $\underset{x\to 0}{\lim }\left(\frac{f(x)}{x^3}\right)=-5$, then $f(2)-f(-2)$ is equal to:

Two adjacent sides of a parallelogram $PQRS$ are given by $\overrightarrow{PQ}=\hat{j}+\hat{k}$ and $\overrightarrow{PS}=\hat{i}-\hat{j}$. If the side $PS$ is rotated about the point $P$ by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side $PQ$, then ${\sin }^2\left(\frac{5\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)$ is equal to:

Let the point $A$ be the foot of perpendicular drawn from the point $P(a,b,0)$ on the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-\alpha }{3}$. If the midpoint of the line segment $PA$ is $\left(0,\frac{3}{4},-\frac{1}{4}\right)$, then the value of $a^2+b^2+{\alpha }^2$ is equal to:

Let $P=\{\theta \in [0,4\pi ]:{\tan }^2\theta \ne 1\}$ and $S=\{a\in \mathbb{Z}:2({\cos }^8\theta -{\sin }^8\theta )\sec 2\theta =a^2,\theta \in P\}$. Then $n(S)$ is:

If for $3\le r\le 30$, $\left(\genfrac{}{}{0ex}{}{30}{30-r}\right)+3\left(\genfrac{}{}{0ex}{}{30}{31-r}\right)+3\left(\genfrac{}{}{0ex}{}{30}{32-r}\right)+\left(\genfrac{}{}{0ex}{}{30}{33-r}\right)=\left(\genfrac{}{}{0ex}{}{m}{r}\right)$, then $m$ equals:

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points, and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x+y=1$ is $\sqrt{14}$, then the square of the radius of $C$ is ______.

Let $[\cdot ]$ denote the greatest integer function. Then the value of ${\int }_0^3\frac{e^x+e^{-x}}{[x]!}dx$ is:

The number of critical points of the function $f(x)=\{\begin{array}{llc}\frac{|\sin x|}{x},& x\ne 01,& x=0\end{array}$ in the interval $(-2\pi ,2\pi )$ is equal to:

If $\underset{x\to 2}{\lim }\frac{\sin \left(x^3-5x^2+ax+b\right)}{\left(\sqrt{x-1}-1\right){\log }_e(x-1)}=m$, then $a+b+m$ is equal to:

Let $S=\{x\in [-\pi ,\pi ]:\sin x(\sin x+\cos x)=a,a\in \mathbb{Z}\}$. Then $n(S)$ is equal to:

Let the midpoints of the sides of a triangle $ABC$ be $\left(\frac{5}{2},7\right)$, $\left(\frac{5}{2},3\right)$ and $(4,5)$. If its incenter is $(h,k)$, then $3h+k$ is equal to: