1
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

Let $$\alpha, \beta$$ be the roots of the equation $$x^2-\sqrt{6} x+3=0$$ such that $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by 3 and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+i b), i=\sqrt{-1}$$. Then $$n+a+b$$ is equal to __________.

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2
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations

$$\begin{aligned} & x+y+z=6, \\ & 2 x+5 y+\alpha z=\beta \text { and } \end{aligned}$$

$$x+2 y+3 z=4$$, has infinitely many solutions. Then $$(P Q)^2$$ is equal to _________.

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3
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2=4 x$$. If $$P$$ also lies on the chord of the parabola $$x^2=8 y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha-28)(\beta-8)$$ is equal to _________.

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4
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1
Change Language

Remainder when $$64^{32^{32}}$$ is divided by 9 is equal to ________.

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