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Question 1:

The sum, of the squares of all the roots of the equation \( x^{2}+|2 x-3|-4=0 \), is

\(3(3-\sqrt{2})\)

\(6(3-\sqrt{2})\)

\(6(2-\sqrt{2})\)

\(3(2-\sqrt{2})\)

Question 2:

If the set of all \(a \in \mathbf{R}\), for which the equation \(2x^{2} + (a-5)x + 15 = 3a\) has no real root, is the interval \((\alpha, \beta)\), and \(X = \{x \in \mathbf{Z} : \alpha < x < \beta\}\), then \(\sum_{x \in X} x^{2}\) is equal to :

2109

2129

2119

2139

Question 3:

The product of all the rational roots of the equation \( (x^{2}-9 x+11)^{2}-(x-4)(x-5)=3 \), is equal to

14

21

28

7

Question 4:

Consider the equation \( x^{2} + |2x - 3| - 4 = 0 \). How many distinct real roots does this equation have?

One

Two

Three

Four

Question 5:

If \( x_1 \) and \( x_2 \) are the roots of the equation \( x^{2} + |2x - 3| - 4 = 0 \), what is the value of \( x_1^2 + x_2^2 \)?

\(3(3-\sqrt{2})\)

\(6(3-\sqrt{2})\)

\(6(2-\sqrt{2})\)

\(3(2-\sqrt{2})\)

Question 6:

For the equation \( x^{2} + |2x - 3| - 4 = 0 \), which of the following statements is true?

The roots are both positive.

The roots are both negative.

One root is positive and one is negative.

Both roots are zero.

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