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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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