1
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1

Let $$\alpha, \beta$$ be the roots of the equation $$x^{2}-\sqrt{2} x+2=0$$. Then $$\alpha^{14}+\beta^{14}$$ is equal to

A
$$-64$$
B
$$-64 \sqrt{2}$$
C
$$-128 \sqrt{2}$$
D
$$-128$$
2
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

The set of all $$a \in \mathbb{R}$$ for which the equation $$x|x-1|+|x+2|+a=0$$ has exactly one real root, is :

A
$$(-\infty, \infty)$$
B
$$(-6, \infty)$$
C
$$(-\infty,-3)$$
D
$$(-6,-3)$$
3
JEE Main 2023 (Online) 12th April Morning Shift
+4
-1

Let $$\alpha, \beta$$ be the roots of the quadratic equation $$x^{2}+\sqrt{6} x+3=0$$. Then $$\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}$$ is equal to :

A
72
B
9
C
729
D
81
4
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1

Let $$\alpha, \beta, \gamma$$ be the three roots of the equation $$x^{3}+b x+c=0$$. If $$\beta \gamma=1=-\alpha$$, then $$b^{3}+2 c^{3}-3 \alpha^{3}-6 \beta^{3}-8 \gamma^{3}$$ is equal to :

A
21
B
19
C
$$\frac{169}{8}$$
D
$$\frac{155}{8}$$
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