1
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

In a $$\triangle A B C$$, suppose $$y=x$$ is the equation of the bisector of the angle $$B$$ and the equation of the side $$A C$$ is $$2 x-y=2$$. If $$2 A B=B C$$ and the points $$A$$ and $$B$$ are respectively $$(4,6)$$ and $$(\alpha, \beta)$$, then $$\alpha+2 \beta$$ is equal to

A
42
B
39
C
48
D
45
2
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Let $$\mathrm{R}$$ be the interior region between the lines $$3 x-y+1=0$$ and $$x+2 y-5=0$$ containing the origin. The set of all values of $$a$$, for which the points $$\left(a^2, a+1\right)$$ lie in $$R$$, is :

A
$$(-3,0) \cup\left(\frac{2}{3}, 1\right)$$
B
$$(-3,0) \cup\left(\frac{1}{3}, 1\right)$$
C
$$(-3,-1) \cup\left(\frac{1}{3}, 1\right)$$
D
$$(-3,-1) \cup\left(-\frac{1}{3}, 1\right)$$
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ passing through the origin. The tangent of an angle between the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ is :
A
$\frac{30}{41}$
B
$\frac{8}{5}$
C
$\frac{2}{5}$
D
$\frac{25}{41}$
4
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
If $(\alpha, \beta)$ is the orthocenter of the triangle $\mathrm{ABC}$ with vertices $A(3,-7), B(-1,2)$ and $C(4,5)$, then $9 \alpha-6 \beta+60$ is equal to :
A
30
B
40
C
25
D
35
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