51). \( \Large \left(x+\frac{1}{x}\right) \left(x-\frac{1}{x}\right) \left(x^{2}+\frac{1}{x^{2}}-1\right) \left(x^{2}+\frac{1}{x^{2}+1}\right) \) is equal to:
A). \( \Large x^{6}+\frac{1}{x^{6}} \) |
B). \( \Large x^{8}+\frac{1}{x^{8}} \) |
C). \( \Large x^{8}-\frac{1}{x^{8}} \) |
D). \( \Large x^{6}-\frac{1}{x^{6}} \) |
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52). If \( \Large a^{2x+2}=1 \), where a is a positive real number other than 1, then x is equal to:
A). -2 |
B). -1 |
C). 0 |
D). 1 |
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53). If x is real then the minimum value of \( \Large \left(x^{2}-x-1\right) \) is
A). \( \Large \frac{3}{4} \) |
B). 0 |
C). 1 |
D). \( \Large -\frac{5}{4} \) |
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54). If \( \Large \frac{\sqrt{7}-2}{\sqrt{7}+2}=a\sqrt{7}+b \), then the value of a is:
A). \( \Large \frac{11}{3} \) |
B). \( \Large -\frac{4}{3} \) |
C). \( \Large \frac{4}{3} \) |
D). \( \Large \frac{-4\sqrt{7}}{3} \) |
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55). If \( \Large a+\frac{1}{b}=1\ and\ b+\frac{1}{c}=1\ then\ c+\frac{1}{a} \) is equal to
A). 0 |
B). \( \Large \frac{1}{2} \) |
C). 1 |
D). 2 |
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56). If \( \Large x=\sqrt{3}+\sqrt{2} \), then the value of \( \Large \left(x^{3}+\frac{1}{x^{3}}\right) \) is
A). \( \Large 6\sqrt{3} \) |
B). \( \Large 12\sqrt{3} \) |
C). \( \Large 18\sqrt{3} \) |
D). \( \Large 24\sqrt{3} \) |
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57). If \( \Large x+y=7 \), then the value of \( \Large x^{3}+y^{3}+21xy \) is
A). 243 |
B). 143 |
C). 343 |
D). 443 |
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58). If \( \Large \otimes \) is an operation such that \( \Large a \otimes b=2a \) when a>b, a+b when a
A). \( \Large \frac{1}{3} \) |
B). \( \Large \frac{14}{23} \) |
C). \( \Large \frac{2}{3} \) |
D). \( \Large \frac{14}{13} \) |
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59). If \( \Large \left(125\right)^{x}=3125 \), then the value of x is
A). \( \Large \frac{1}{5} \) |
B). \( \Large \frac{3}{5} \) |
C). \( \Large \frac{5}{3} \) |
D). \( \Large \frac{5}{7} \) |
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60). If \( \Large 5^{\sqrt{x}}+12^{\sqrt{x}}=13^{\sqrt{x}} \), then x is equal to
A). \( \Large \frac{25}{4} \) |
B). 4 |
C). 9 |
D). 16 |
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