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41). If one root of the equation $$\Large x^{2}+px+12=0$$ is 4, while the equation $$\Large x^{2}-7x+q=0$$ has equal roots, then the value of 'g' is:
 A). $$\Large \frac{49}{4}$$ B). 12 C). 3 D). 4
42). Let $$\Large 2\sin^{2}+3\sin x-2>0$$ and $$\Large x^{2}-x-2<0$$ (x is measured in radians). Then x lies in the interval:
 A). $$\Large \left(\frac{ \pi }{6},\frac{5 \pi }{6}\right)$$ B). $$\Large \left(-1, \frac{5 \pi }{6}\right)$$ C). $$\Large \left(-1, 2\right)$$ D). $$\Large \left(\frac{ \pi }{6}, 2\right)$$
43). If at least one root of $$\Large 2x^{2}+3x+5=0$$ and $$\Large ax^{2}+bx+c=0$$, a, b, c, belongs to N is common, then the maximum value of a + b + c is:
 A). 10 B). 0 C). does not exist D). none of these
44). If the roots of the quadratic equation $$\Large x^{2}+px+q=0$$ are $$\Large \tan 30 ^{\circ} and\ \tan 15 ^{\circ}$$ respectively, then the value of$$\Large 2+q-p$$ is
 A). 3 B). 0 C). 1 D). 2
45). If $$\Large P \left(x\right)=ax^{2}+bx+c$$ and $$\Large Q \left(x\right)=-ax^{2}+dx+c$$ where $$\Large ac \ne 0$$ then $$\Large P \left(x\right) Q \left(x\right) = 0$$ has at least:
 A). four real roots B). two real roots C). four imaginary roots D). none of these

46). The coefficient of x in the equation $$\Large x^{2}+px+q=0$$ was taken as 17 in place of 13 its roots Were found to be -2 and -15. The roots of the original equation are:
 A). 3,10 B). -3 , -10 C). -5, -8 D). None of these
47). The number which exceeds its positive Square roots by 12 is:
 A). 9 B). 16 C). 25 D). none of these
48). Let a, b, c be real numbers a $$\ne$$ 0. If $$\Large \alpha$$ is a root of $$\Large a^{2}x^{2}+bx+c=0,$$,$$\Large \beta$$ is a root of $$\Large a^{2}x^{2}-bx-c=0$$ and $$\Large 0< \alpha < \beta$$ then the equation $$\Large a^{2}x^{2}+2bx+2c=0$$ has a root of $$\gamma$$ that always satisfies:
 A). $$\Large \gamma = \frac{ \alpha + \beta }{2}$$ B). $$\Large \gamma = \alpha + \frac{ \beta }{2}$$ C). $$\Large \gamma = \alpha$$ D). $$\Large \alpha < \gamma < \beta$$
49). The equation $$\Large x \left(\frac{3}{4}log_{2}x\right)^{2}+ \left(log_{2}x\right) -\frac{5}{4}=\sqrt{2}$$ has

 A). at least one real solution B). exactly three real solution C). exactly one irrational solution. D). all of the above
50). The solution of set of the equation $$\Large x log x \left(1-x\right)^{2}=9$$ is
 A). $$\Large \{ -2, 4 \}$$ B). $$\Large \{ 4 \}$$ C). $$\Large \{ 0, -2, 4 \}$$ D). none of these
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