JEE Main 2024 (Online) 27th January Evening Shift | Straight Lines and Pair of Straight Lines Question 22 | Mathematics

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 :

$$(-3,0) \cup\left(\frac{2}{3}, 1\right)$$

$$(-3,0) \cup\left(\frac{1}{3}, 1\right)$$

$$(-3,-1) \cup\left(\frac{1}{3}, 1\right)$$

$$(-3,-1) \cup\left(-\frac{1}{3}, 1\right)$$

JEE Main 2024 (Online) 27th January Morning Shift

MCQ (Single Correct Answer)

-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 :

$\frac{30}{41}$

$\frac{8}{5}$

$\frac{2}{5}$

$\frac{25}{41}$

JEE Main 2023 (Online) 15th April Morning Shift

MCQ (Single Correct Answer)

-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 :

JEE Main 2023 (Online) 13th April Evening Shift

MCQ (Single Correct Answer)

-1

Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15 x-y=82,6 x-5 y=-4$$ and $$9 x+4 y=17$$. Then $$\alpha+2 \beta$$ and $$2 \alpha-\beta$$ are the roots of the equation :

$$x^{2}-7 x+12=0$$

$$x^{2}-13 x+42=0$$

$$x^{2}-14 x+48=0$$

$$x^{2}-10 x+25=0$$

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