JEE Main 2024 (Online) 1st February Evening Shift | Straight Lines and Pair of Straight Lines Question 12 | Mathematics

JEE Main 2024 (Online) 1st February Evening Shift

Numerical

-1

Let $A B C$ be an isosceles triangle in which $A$ is at $(-1,0), \angle A=\frac{2 \pi}{3}, A B=A C$ and $B$ is on the positve $x$-axis. If $\mathrm{BC}=4 \sqrt{3}$ and the line $\mathrm{BC}$ intersects the line $y=x+3$ at $(\alpha, \beta)$, then $\frac{\beta^4}{\alpha^2}$ is __________.

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JEE Main 2024 (Online) 1st February Evening Shift

Numerical

-1

The lines $\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to each other and all the lines $L_{2 n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\left\{\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{L}_{20}\right\}$ is equal to ___________.

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JEE Main 2024 (Online) 31st January Evening Shift

Numerical

-1

Let $$A(-2,-1), B(1,0), C(\alpha, \beta)$$ and $$D(\gamma, \delta)$$ be the vertices of a parallelogram $$A B C D$$. If the point $$C$$ lies on $$2 x-y=5$$ and the point $$D$$ lies on $$3 x-2 y=6$$, then the value of $$|\alpha+\beta+\gamma+\delta|$$ is equal to ___________.

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JEE Main 2024 (Online) 27th January Evening Shift

Numerical

-1

If the sum of squares of all real values of $$\alpha$$, for which the lines $$2 x-y+3=0,6 x+3 y+1=0$$ and $$\alpha x+2 y-2=0$$ do not form a triangle is $$p$$, then the greatest integer less than or equal to $$p$$ is _________.

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