1
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1
If the value of real number $a>0$ for which $x^2-5 a x+1=0$ and $x^2-a x-5=0$

have a common real root is $\frac{3}{\sqrt{2 \beta}}$ then $\beta$ is equal to ___________.
2
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1

Let $$\alpha_1,\alpha_2,....,\alpha_7$$ be the roots of the equation $${x^7} + 3{x^5} - 13{x^3} - 15x = 0$$ and $$|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|$$. Then $$\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6$$ is equal to _________.

3
JEE Main 2023 (Online) 25th January Evening Shift
Numerical
+4
-1

Let $$\alpha \in\mathbb{R}$$ and let $$\alpha,\beta$$ be the roots of the equation $${x^2} + {60^{{1 \over 4}}}x + a = 0$$. If $${\alpha ^4} + {\beta ^4} = - 30$$, then the product of all possible values of $$a$$ is ____________.

4
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1

Let $$\lambda \in \mathbb{R}$$ and let the equation E be $$|x{|^2} - 2|x| + |\lambda - 3| = 0$$. Then the largest element in the set S = {$$x+\lambda:x$$ is an integer solution of E} is ______

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