Harmonic Progression

Q.

If sixth term of a H.P. is $\frac{1}{61}$ and its tenth term is $\frac{1}{105}$, then first term of that H.P. is

1. 

2. 

3. 

4. 

Q. If $x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$ where a, b, c, are in AP
and |a| <1, |b|<1, |c|<1 then x, y, z are in

1. GP
2. AP
3. H.P
4. Arithmetic- Geometric progression

Q. What is the formula for sum of harmonic progration ?

Q.

If $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c},$ then a, b, c, are in

1. G.P
2. A.P
3. H.P
4. none of these

Q. If $4^{{\log }_2\log x}=\log x-{\left(\log x\right)}^2+1$ (base is e), then find the value of $x$

1. $x=e$
2. $x=2e$
3. $x=3e$
4. none of these

Q. Let $a_1,a_2,a_3.$..... be in harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer n for which $a_n<0$ is

1. 24
2. 25
3. 22
4. 23

Q.

Subtract using column method : $-3\mathrm{ab}+4\mathrm{bc}-6\mathrm{ca}$from $2\mathrm{ab}+13\mathrm{bc}+6\mathrm{ca}$

Q.

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_2=25.$ The least positive integer n for which $a_n<0$ is

1. 22

2. 23

3. 24

4. 25

Q. If $a_1,a_2,a_3,\dots \dots$ are in a harmonic progression with $a_1=5$ and $a_{20}=25$. Then, the least positive integer n for which $a_n<0$, is

1. 24
2. 22
3. 23
4. 25

Q.

If a, b, c are in H.P then $4^{-a},4^{-b},4^{-c}$ are in

1. A.P
2. G.P
3. H.P
4. none of these

Q. If a, b, c, d are in H.P. , then ab +bc+cd =

1. ad
2. 3ad
3. 4ad
4. 2ad

Q. Find the numerical value of the following:
$\frac{\cos {60}^o}{{\sin }^2{45}^o}-3\cot {45}^o+2\sin {90}^o$

Q. Let a, b, c be the sides of a triangle. No two of them are equal and $\lambda \in R$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real then

1. $\lambda <\frac{4}{3}$
2. $\lambda >\frac{5}{3}$
3. $\lambda \in (\frac{1}{3},\frac{5}{3})$
4. $\lambda \in (\frac{4}{3},\frac{5}{3})$

Q. $\log 8=\log (x-1)-\log 20$. solve for $x$?

Q. If $a=\frac{1}{3-2\sqrt{2}},b=\frac{1}{3+2\sqrt{2}}$ then the value of $a^2+b^2$ is

1. 34
2. 35
3. 36
4. 37

Q.

Simplify the expression: $\frac{1}{2}a-\frac{2}{3}b-4c+\frac{2}{3}a+2b-\frac{1}{2}c+5b-\frac{1}{3}a$

Q. Find x if :
${\log }_7(2x^2-1)=2$