## Sequence and Series Questions and Answers Part-13

1. The value of $\frac{\log_{2}24}{\log_{96}2}-\frac{\log_{2}192}{\log_{12}2}$
is
a) 3
b) 0
c) 2
d) 1

Explanation:

2. The set of all value of x satisfying $x^{\log_{x}\left(1-x\right)^{2}}=9$
is
a) a subset of R containing N
b) a subset of R containing 1
c) is a finite set containg at least two elements
d) a finite set .

Explanation: Note that x > 0, x $\neq$ 1

3. The set $\left\{x: \mid 1-\log_{1/5}x\mid +2=\mid 3-\log_{1/5}x\mid\right\}$       is equal to
a) $\left( 0,\infty\right)$
b) $\left[ 1/5,\infty\right)$
c) $\left[ 1/5,5\right)$
d) $\left(0,1/5\right]$

Explanation:

4. The set of $\left\{x: \log_{1/3}\log_{4}\left(x^{2}-5\right)>0\right\}$       is equal to
a) $\left(3,\infty\right)$
b) $\left(\sqrt{6},3\right)$
c) $\left(-3,-\sqrt{6}\right)\cup\left(\sqrt{6},3\right)$
d) $\left(\sqrt{6},\infty\right)$

Explanation:

5. The number of solutions of $\log_{4}\left(x-1\right)=\log_{2}\left(x-3\right)$
is
a) 3
b) 1
c) 2
d) 0

Explanation:

6. Solutions set of the inequality
$\log_{3}\left(x+2\right)\left(x+4\right)+\log_{1/3}\left(x+2\right)<\frac{1}{2}\log_{\sqrt{3}}7$
is
a) $\left(-2,-1\right)$
b) $\left(-2,3\right)$
c) $\left(-1,3\right)$
d) $\left(3,\infty\right)$

Explanation:

7. The solution set of $\log_{\mid\sin x\mid}\left(x^{2}-8x+23\right)>\frac{3}{\log_2{\mid\sin x\mid}}$        contains
a) $x\epsilon \left(3,\pi\right)\cup\left(\pi,\frac{3\pi}{2}\right)\cup\left(\frac{3\pi}{2},5\right)$
b) $x\epsilon \left(3,\pi\right)\cup\left(\pi,5\right)$
c) $x\epsilon \left(3,\frac{5\pi}{2}\right)$
d) $x\epsilon \left(2,5\pi/2\right)$

Explanation:

8. If n>1, the value of $\frac{1}{\log_{2}n}+\frac{1}{\log_{3}n}+....+\frac{1}{\log_{53}n}$
is
a) $\frac{1}{\log_{53!}n}$
b) 1
c) $\frac{1}{\log_{n!}53}$
d) $\frac{1}{53}$

Explanation:

9. The number of values of $x \epsilon \left[0,n\pi\right],n \epsilon I$     that satisfy $\log_{\mid \sin x\mid}\left(1+\cos x\right)=2$      is
a) 0
b) n
c) 2n
d) 1

10. Let $\triangle=\begin{bmatrix}\log \left(7/15\right) & \log \left(49\right) & \log \left(49\right) \\\log \left(25\right) &\log \left(5/21\right) & \log \left(25\right) \\\log \left(9\right) & \log \left(9\right) &\log \left(3/35\right)\end{bmatrix}$
then $\triangle$ is equal to
a) $\left(\log \left(105\right)\right)^{2}$
b) $\left(\log \left(105\right)\right)^{3}$
c) $\left(\log \left(35/3\right)\right)^{3}$
d) $\left(\log \left(21/5\right)\right)^{3}$