JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Properties of Triangle

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{{\sin }^2\mathrm{C}}{{\sin }^2\mathrm{A}}=1,\mathrm{A},\mathrm{B},\mathrm{C}\in \left(0,\frac{\pi }{2}\right)$, then

Let $\mathrm{A},\mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha ,\alpha >0$ at a distance of $\alpha$ from their point of intersection P . The line segment AB meets the angle bisector of the given half-lines at the point Q . If $\mathrm{PQ}=\frac{9}{2}$ and R is …

In a triangle $ABC,BC=7,AC=8,AB=\alpha \in \mathbb{N}$ and $\cos A=\frac{2}{3}$. If $49\cos (3C)+42=\frac{m}{n}$, where $\gcd (m,n)=1$, then $m+n$ is equal to ______.

Let $\left(5,\frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a,-2)$, $B(a,6)$ and $C\left(\frac{a}{4},-2\right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then …

In the figure, ${\theta }_1+{\theta }_2=\frac{\pi }{2}$ and $\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})$. If the area of $△\mathrm{CAB}$ is $2\sqrt{3}-3$ unit ${}^2$, when $\frac{{\theta }_2}{{\theta }_1}$ is the largest, then the perimeter (in unit) of …

If the line $\mathrm{x}=\mathrm{y}=\mathrm{z}$ intersects the line $\mathrm{x}\sin \mathrm{A}+\mathrm{y}\sin \mathrm{B}+\mathrm{z}\sin \mathrm{C}-18=0=\mathrm{x}\sin 2\mathrm{A}+\mathrm{y}\sin 2\mathrm{B}+\mathrm{z}\sin 2\mathrm{C}-9$, where …