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
|
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 |
|
