1
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^{2}-\sqrt{2} x+\sqrt{6}=0$$ and $$\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1$$ be the roots of the equation $$x^{2}+a x+b=0$$. Then the roots of the equation $$x^{2}-(a+b-2) x+(a+b+2)=0$$ are :

A
non-real complex numbers
B
real and both negative
C
real and both positive
D
real and exactly one of them is positive
2
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

If $$\alpha, \beta$$ are the roots of the equation

$$x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0$$,

then the equation, whose roots are $$\alpha+\frac{1}{\beta}$$ and $$\beta+\frac{1}{\alpha}$$, is :

A
$$3 x^{2}-20 x-12=0$$
B
$$3 x^{2}-10 x-4=0$$
C
$$3 x^{2}-10 x+2=0$$
D
$$3 x^{2}-20 x+16=0$$
3
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

The minimum value of the sum of the squares of the roots of $$x^{2}+(3-a) x+1=2 a$$ is:

A
4
B
5
C
6
D
8
4
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Let S be the set of all integral values of $$\alpha$$ for which the sum of squares of two real roots of the quadratic equation $$3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0$$ is minimum. Then S :

A
is an empty set
B
is a singleton
C
contains exactly two elements
D
contains more than two elements
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