Sequence and Series Questions and Answers Part-11

1. Let N = 2 + 22 + 222 + ... upto 30000 terms.
a) last digit of N is 0
b) last four digits of N are 580
c) last four digits of N are 9580
d) All of the Above

Explanation:

2. Let $\left(1+x\right)^{2}\left(1+\frac{x}{2}\right)^{2}\left(1+\frac{x}{2^{2}}\right)^{2}\left(1+\frac{x}{2^{3}}\right)^{2}... \infty$
$=a_{0}+a_{1}x+a_{2}x_{2}+... \infty$
then
a) $a_{1}=4$
b) $a_{2}=20/3$
c) $a_{2}=16/3$
d) Both a and b

Explanation:

3. If a, b, c, d are in G.P., then value of $\left(a – c\right)^{2} + \left(c – b\right)^{2} + \left(b – d\right)^{2} –\left(d – a\right)^{2}$
is independent of
a) a
b) b
c) c
d) All of the Above

Explanation: Let common ratio be r. Given expression equals

4. If sum of the infinite G.P. p, 1, 1/p, 1/p2, .... is 9/2, then value of then value of p is
a) 2
b) 3/2
c) 3
d) Both b and c

Explanation:

5. If a, b, c are in A.P., and $a^{2} , b^{2} , c^{2}$  are in H.P., then
a) a = b = c
b) – a, 2b, 2c are in G.P.
c) a, b, c are in G.P.
d) All of the Above

Explanation:

6. Let $\log x=\log_{10}x\forall x>0$
suppose x,y,z>1, then least value of
$E=\log\left(xyz\right)=\left[\frac{\log x}{\log y \log z}+\frac{\log y}{\log z \log x}+\frac{\log z}{\log x \log y}\right]$
is
a) 3
b) 6
c) 9
d) 18

Explanation:

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

Explanation: It is defined for x > 0

It has exactly one solution

8. If a,b $\geq$ 1, a+b=10 ,then minimum value of $\log_{3}a+\log_{3}b$     is
a) 2
b) $2\log_{3}5$
c) $\frac{1}{2}\log_{3}5$
d) 1

Explanation:

9. If $5^{x}=7^{x+1}$ , then x is equal to
a) $\frac{1}{\log_{5}7+1}$
b) $\frac{1}{\log_{7}5-1}$
c) $\log_{7}5$
d) $\log_{5}7$

10. Let $\left(x_{0},y_{0}\right)$   be the solution of the following system of equations
$\left(2x\right)^{ln2}=\left(3y\right)^{ln3}$
$3^{lnx}=2^{lny}$
then $x_{0}$ is equals to