JEE MAIN 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 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 $A=\left[\begin{array}{ccc}1& 21& \alpha \end{array}\right]$ 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 $f(x)$ 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:

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 ______.

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 ____.

A circular current loop of radius $R$ is placed inside a square loop of side length $L$ ($L\gg R$) such that they are co-planar and their centers coincide. The permeability of free space is ${\mu }_0$. The mutual inductance between the circular loop and square loop is ________.

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:

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:

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:

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:

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:

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

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:

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

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: