## Sequence and Series Questions and Answers Part-15

1. If x satisfies $\log_{3}\left(2x+1\right)<\log_{3}5$       then x contains the intervals (s)
a) $\left(-\frac{1}{2},0\right)$
b) $\left[0,2\right)$
c) $\left[1,2\right)$
d) All of the Above

Explanation:

2. If $x\epsilon R$ satisfies $\left(\log_{10}\left(100\right)x\right)^{2}+\left(\log_{10}10x\right)^{2}+\log_{10}x \leq 14$
then x contains the interval.
a) $\left(1,10\right]$
b) $\left[10^{-9/2},1\right)$
c) $\left(0,\infty\right)$
d) Both a and b

Explanation:

3. For x, y, z>1, let
$\alpha=\frac{ln\left(xy\right)}{ln\left(xye\right)}+\frac{ln\left(yz\right)}{ln\left(yze\right)}+\frac{ln\left(zx\right)}{ln\left(zxe\right)}$
and $\beta=\frac{ln\left(x\right)}{ln\left(xe\right)}+\frac{ln\left(y\right)}{ln\left(ye\right)}+\frac{ln\left(z\right)}{ln\left(ze\right)}$
a) $\alpha >\beta$
b) $\alpha < 2\beta$
c) $\alpha =\beta$
d) Both a and b

Explanation:

4. Let S be solution set of $\left(\frac{1}{2}\right)^{x+1}=3^x$     and T be solution set of $\left(\frac{1}{3}\right)^{x+1}=2^x$     then
a) S contains exactly one element
b) T contains exactly one element
c) $S\cap T=\phi$
d) All of the Above

Explanation:

5. Let $\log x=\log_{10}x$    and suppose x,y,z>1 , then least value of the expression $E=\log \left(xyz\right)\sum\left(\frac{\log x}{\log y\log z}+\frac{\log y}{\log x\log z}\right)$
a) 9
b) 18
c) 27
d) 36

Explanation:

6. Number of real value of x for which $2017^{x}+2018^{x}+2019^{x}=3018^{x}$       is
a) 0
b) 1
c) 2
d) infinite

Explanation:

7. Number of solutions of $e^{x} = x^{e}$   is
a) 0
b) 1
c) 2
d) infinite

Explanation:

8. If 0< x,y< 1 and $\log_{x}\left(a\right)+\log_{y}\left(a\right)=4\log_{xy}\left(a\right)$        for some a>0, $a\neq1$  then
a) x + y = 2
b) x + y = 1
c) x=y
d) $xy=a^{2}$

Explanation:

9. The value of x satisfying
$\log_{3}\left(5x-2\right)-2\log_{3}\sqrt{3x+1}=1-\log_{3}4$
is
a) 2
b) 1
c) 3
d) 4

10. If $S=\left\{x\epsilon R:\left(\log_{0.6}0.216\right)\log_{5}\left(5-2x\right)\leq 0\right\}$
a) $\left[2.5,\infty\right)$
b) $\left[2,2.5\right)$
c) $\left(2,2.5\right)$
d) $\left[0,2.5\right)$