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>> Elementary Mathematics >> Quadratic Equations
61). Given that \( \Large \tan \alpha \) and \( \Large \tan \beta \) are the roots of \( \Large x^{2}-px+q=0 \), then the value of \( \Large \sin^{2} \left(\alpha + \beta \right) \) is equal to:
A). \( \Large \frac{p^{2}}{p^{2}+ \left(1-q\right)^{2} } \)
B). \( \Large \frac{p^{2}}{p^{2}+q^{2}} \)
C). \( \Large \frac{q^{2}}{p^{2} \left(1-q\right)^{2} } \)
D). \( \Large \frac{p^{2}}{ \left(p+q\right)^{2} } \)
62). The number of values of K for which the system of equations \( \Large \left(K+1\right)x+84=4K \) and \( \Large Kx+ \left(K+3\right)y=3 K-1 \) has infinitely many solution, is:
A). 0
B). 1
C). 2
D). infinite
63). The set of all real numbers x for which \( \Large x^{2}-|x+2|+x>0 \) is:
A). \( \Large \left(-\infty, -2\right) \ \left(2, \infty\right) \)
B). \( \Large \left(-\infty, -\sqrt{2}\right) \ \left(\sqrt{2}, \infty\right) \)
C). \( \Large \left(-\infty, -1\right) \ \left(1, \infty\right) \)
D). \( \Large \left(\sqrt{2}, \infty\right) \)
64). Let a, b, c be positive numbers, the following systems of equations in x, y and z \( \Large \frac{x^{2}}{a^{2}} +\ \frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1; \ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} +\ \frac{z^{2}}{c^{2}}=1 and\ \frac{-x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \) has;
A). no solutions
B). unique solution
C). infinitely many solution
D). finitely many solution.
65). If \( \Large 2 \sin^{2}\frac{ \pi }{8} \) is a root of equation\( \Large x^{2}+ax+b=0 \), where a and b are rational numbers, then a - b is equal to
A). \( \Large -\frac{5}{2} \)
B). \( \Large -\frac{3}{2} \)
C). \( \Large -\frac{1}{2} \)
D). \( \Large \frac{1}{2} \)


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