## Sequence and Series Questions and Answers Part-14

1. Suppose p,q,r,$s\epsilon N$ and $p\geq q\geq r$    . If $5^{\log_{5}\left(2^{p}\right)}+7^{\log_{7}\left(2^{q}\right)}+11^{\log_{11}\left(2^{r}\right)}=2^{s}$
a) q=p-1
b) r=p-1
c) s=p+1
d) All of the Above

Explanation: As for a > 0, a $\neq$ 1,

2. Suppose a,b,c,$d\epsilon \left(1,\infty\right)$   and let
$\alpha =\log_{abc}\left(10\right)+\log_{acd}\left(10\right)+\log_{abd}\left(10\right)+\log_{bcd }\left(10\right)$
and $\beta =\log_{a}\left(10\right)+\log_{b}\left(10\right)+\log_{c}\left(10\right)+\log_{d }\left(10\right)$
a) $\alpha\leq\beta$
b) $2\alpha\leq\beta$
c) $3\alpha\leq\beta$
d) All of the Above

Explanation:

3. The set of all x satisfying $4^{x^{2}+2}-\left(9\right)2^{x^{2}+2}+8=0$         consists of
a) infinitely many points
b) exactly two integers
c) finitely many points from the set of all integers
d) Both b and c

Explanation:

4. The equations $x^{\left[\left(\log_{5}x\right)^{2}-\frac{9}{2}\log_{5}x+5\right]}=5\sqrt{5}$
has
a) exactly three real solution
b) at least one real solution
c) exactly one irrational solution
d) All of the Above

Explanation: Taking logarithm of both sides, we get

5. if $\log_{2}\left(3^{2x-2}+7\right)=2+\log_{2}\left(3^{x-1}+1\right)$
then x equals
a) 0
b) 1
c) 2
d) Both b and c

Explanation:

6. Solution of $\log_{x^{2}+6x+8}\log_{2x^{2}+2x+3}\left(x^{2}-2x\right)=0$
is
a) a natural number
b) a negative integer
c) -1
d) Both b and c

Explanation:

7. $x^{\log_{5}x}>5$   implies
a) $x\epsilon \left(0, \infty\right)$
b) $x\epsilon \left(0, 1/5\right) \cup \left(5,\infty\right)$
c) $x\epsilon \left(1, \infty\right)$
d) $x\epsilon \left(1, 2\right)$

Explanation: Clearly x > 0, Taking logarithm with base 5,

8. If $\log x^{2} -\log 2x=3\log3-\log6$       then x equals
a) 9
b) 3
c) 4
d) 5

Explanation: Clearly x > 0

9. If $\log_{0.5}\left(x-1\right)<\log_{0.25}\left(x-1\right)$       then x lies in the interval
a) $\left(2,\infty\right)$
b) $\left(3,\infty\right)$
c) $\left(-\infty,0\right)$
d) $\left(0,3\right)$

10. If $N=n!\left(n \epsilon N,n>2\right)$     then $\lim_{N \rightarrow\infty}\left[\left(\log_{2}N\right)^{-1}+\left(\log_{3}N\right)^{-1}+....+\left(\log_{n}N\right)^{-1}\right]$