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51). If x is real the expression \( \Large \frac{x+2}{2x^{2}+3x+6} \) takes all values in the interval:
A). \( \Large \left(\frac{1}{13}, \frac{1}{3}\right) \)
B). \( \Large \left(- \frac{1}{13}, \frac{1}{3}\right) \)
C). \( \Large \left(- \frac{1}{3}, \frac{1}{13}\right) \)
D). none of these.
52). If x is real, then the maximum and minimum values of the expression \( \Large \frac{x^{2}
-3x+4}{x^{2}+3x+4} \) will be:

A). 2,1
B). \( \Large 5, \frac{1}{5} \)
C). \( \Large 7, \frac{1}{7} \)
D). none of these.
53). The number of real solutions of the equation \( \Large |x^{2}+4x+3|+2x+5=0 \) are:
A). 1
B). 2
C). 3
D). 4
54). If the roots of the given equation:
\( \Large \left(\cos p-1\right)x^{2}+ \left(\cos p\right)x+\sin p = 0 \) are real, then:

A). \( \Large P \epsilon \left(- \pi ,0\right) \)
B). \( \Large P \epsilon \left(- \frac{ \pi }{2}, \frac{ \pi }{2} \right) \)
C). \( \Large P \epsilon \left(0, \pi \right) \)
D). \( \Large P \epsilon \left(0, 2 \pi \right) \)
55). The solution of the quadratic equation \( \Large \left(3|x|-3\right)^{2}=|x|+7 \) which belongs to the domain of definition of function \( \Large \gamma = \sqrt{x \left(x-3\right) } \) are given by:
A). \( \Large \pm \frac{1}{9}, \pm 2 \)
B). \( \Large -\frac{1}{9}, 2 \)
C). \( \Large \frac{1}{9}, -2 \)
D). \( \Large -\frac{1}{9}, -2 \)


56). The number of solution of \( \Large \frac{log 5 + log \left(x^{2}+1\right) }{log \left(x-2\right) }=2 \)
A). 2
B). 3
C). 1
D). none of these
57). If the expression \( \Large \left(mx-1+\frac{1}{x}\right) \) is always nonnegative, then the minimum value of m must be:
A). \( \Large -\frac{1}{2} \)
B). 0
C). \( \Large \frac{1}{4} \)
D). \( \Large \frac{1}{2} \)
58). The value of x in the given equation \( \Large 4^{x}-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1} \) is:
A). \( \Large \frac{4}{3} \)
B). \( \Large \frac{3}{2} \)
C). \( \Large \frac{2}{1} \)
D). \( \Large \frac{5}{3} \)
59). The harmonic mean of the roots of equation \( \Large \left(5+\sqrt{2}x^{2}-14+\sqrt{5}\right)x+8+2\sqrt{5}=0 \) is:
A). 2
B). 4
C). 6
D). 8
60). For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?
A). \( \Large \frac{3}{2} \)
B). 1
C). \( \Large \frac{1}{2} \)
D). \( \Large \frac{11}{4} \)
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