## Sequence and Series Questions and Answers Part-16

1. If $S=\left\{x\epsilon N :2+\log_{2}\sqrt{x+1}>1-\log_{1/2}\sqrt{4-x^{2}}\right\}$
then
a) $S=\left\{1\right\}$
b) $S=\phi$
c) S = N
d) none of these

Answer: a
Explanation:

2. If $\left(x+1\right)^{\log_{10}\left(x+1\right)}=100\left(x+1\right)$       then
a) all the roots of the equations are positive real number
b) all the roots lie in the interval (0,100)
c) all the roots lie in the interval [-9/10,99]
d) none of these

Answer: c
Explanation:

3. The value of $\left(\log_{3}11\right)\left(\log_{11}13\right)\left(\log_{13}15\right)\left(\log_{15}27\right)\left(\log_{27}81\right)$
is
a) 2
b) 1
c) 4
d) 3

Answer: c
Explanation: Use change of base.

4. The value of $x=25^{\left(1/2+\log_{1/5}27+\log_{125}81\right)}$       is
a) 4/81
b) 1
c) a rational number
d) a irrational number

Answer: d
Explanation:

5. If $\frac{\log_{1/2}x}{b-c}=\frac{\log_{1/2}y}{c-a}=\frac{\log_{1/2}z}{a-b}$       then the value of $x^{a}y^{b}z^{c}$   is
a) 1/2
b) xyz
c) $\left(1/2\right)^{abc}$
d) 1

Answer: d
Explanation:

6. If $\log_{10}x+\log_{10}y=2 , x-y=15$       then
a) (x, y) lies on the line y = 4x + 3
b) (x, y) lies on $y^{2}=4x$
c) (x, y) lies on x=4y
d) (x, y) lies on 4x=y

Answer: c
Explanation:

7. If $\frac{\left(\log_{e}x\right)^{2}-3\log_{e}x+3}{\log_{e}x-1}< 1$       then x belong to
a) (0,e)
b) (1,e)
c) (e,1)
d) none of these

Answer: a
Explanation:

8. The value of b for which the equation $2\log_{1/25}\left(bx+28\right)=-\log_{5}\left(12-4x-x^{2}\right)$         has coincident roots if
a) b=-12
b) b=4
c) b=4 or b=-12
d) b=-4 or b=12

Answer: c
Explanation:

9. If $0< x< \pi/2$   and $\cos x=\frac{3}{\sqrt{10}}$     then the value of $\log_{10}\sin x+\log_{10}\cos x+\log_{10}\tan x$         is equal to
a) 0
b) 1
c) -1
d) none of these

Answer: c
Explanation:

10. The solution set of $\log_{2}\mid 4-5x\mid>2$     is
a) $\left(8/5,\infty\right)$
b) $\left(4/5,8/5\right)$
c) $\left(-\infty,0\right)\cup\left(8/5,\infty\right)$
d) none of these

Answer: c
Explanation: