## Sequence and Series Questions and Answers Part-12

1. Solutons set of the inequality $\log_{x}\left(2x^{2}+x-1\right)>\log_{x}\left(2\right)-1$
is
a) $\left(1/2 ,1\right)$
b) $\left(1/2 ,1\right)\cup\left(1,\infty\right)$
c) $\left(1,\infty\right)$
d) $\left(0,1\right)$

Explanation: It holds for 1/2 < x < 1, x > 1

2. The value of of x stisfying $\log_{2}\left(3x-2\right)=\log_{1/2}x$
is
a) -1/3
b) 2
c) 1/2
d) 1

Explanation:

3. The solutions of the equations $4^{\log_{2}\log x}=\log x-\left(\log x\right)^{2}+1$
is
a) x = 1
b) x = 4
c) x = e
d) $x = e^{2}$

Explanation:

4. The set of all x satisfying the equations $x^{\log_{3}x^{2}+\left(\log_{3} x\right)^{2}-10} =1/x^{2}$
is
a) $\left\{ 1,9\right\}$
b) $\left\{ 1,9,1/81\right\}$
c) $\left\{ 1,4, 1/81\right\}$
d) $\left\{ 9,1/81 \right\}$

Explanation: Taking log of both the sides with base 3, we have

5.The set of all solutions of the inequality $\left(1/2\right)^{x^{2}-2x}< 1/4$     contains the sets
a) $\left(-\infty ,0\right)$
b) $\left(-\infty ,1\right)$
c) $\left(1,\infty \right)$
d) $\left(3,\infty \right)$

Explanation:

6. The set of all the solutions of the inequality $\log_{1-x}\left(x-2\right)\geq -1$       is
a) $\left(-\infty ,0\right)$
b) $\left(2,\infty \right)$
c) $\left(-\infty ,1\right)$
d) $\phi$

Explanation:

7. If $\log_{3}x+\log_{3}y = 2+\log_{3}2$      and $\log_{3}\left(x+y\right)=2$    then
a) x = 1, y = 8
b) x = 8, y = 1
c) x = 3, y = 6
d) x = 9, y = 3

Explanation:

8. The set of all solutions of the equations $\log_{3}x\log_{4}x\log_{5}x=\log_{3}x\log_{4}x+\log_{4}x\log_{5}x+\log_{5}x\log_{3}x$
is
a) {1}
b) {1,60}
c) {1,5,10,60}
d) {1,4,8,60}

Explanation:

9. If $\log_{30}3=c,\log_{30}5=d$      then the value of $\log_{30}8$
a) 2(1-c-d)
b) 3(1+c+d)
c) 3(1+c-d)
d) 3(1-c-d)

10. if $a=\log_{12}18,b=\log_{24}54$       then the value of ab+5(a-b) is