Insertion of HP's between 2 Numbers

Q.

Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that

1) all vowels occur together

2) all vowels do not occur together.

Q. Let $A,G$ and $H$ be the arithmetic mean, geometric mean and harmonic mean, respetively of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G–H)x^2+G(H–A)x+H(A–G)=0$, then

1. $-2<\alpha <-1$
2. $0<\alpha <1$
3. $-1<\alpha <0$
4. $1<\alpha <2$

Q. x is one of the four harmonic means inserted between 2/3 and 2/13. The value/s of x can be:

1. $\frac{2}{5}$
2. $\frac{2}{11}$
3. $\frac{2}{13}$
4. $\frac{2}{15}$

Q.

If three real numbers $\mathrm{a},\mathrm{b}$ and $\mathrm{c}$ are in harmonic progression, then which of the following is true?

1. $\frac{1}{\mathrm{a}},\mathrm{b},\frac{1}{\mathrm{c}}$ are in AP

2. $\frac{1}{\mathrm{bc}},\frac{1}{\mathrm{ca}},\frac{1}{\mathrm{ab}}$ in HP

3. $\mathrm{ab},\mathrm{bc},\mathrm{ac}$ are in HP

4. $\frac{\mathrm{a}}{\mathrm{c}},\frac{\mathrm{b}}{\mathrm{c}},\frac{\mathrm{c}}{\mathrm{a}}$ are in HP

Q.

5 harmonic means are inserted between 5 and 10. The common difference of corresponding A.P is.

1. $\frac{1}{50}$

2. $\frac{-1}{60}$

3. $\frac{-1}{50}$

4. $\frac{1}{60}$

Q. Insert 2 no.s between 1 & $\frac{1}{3}$ so that the sequence becomes an Harmonic progression --

1. $\frac{3}{5},\frac{3}{7}$
2. $\frac{1}{7},\frac{2}{7}$
3. $\frac{5}{7},1$
4. $\frac{5}{3},\frac{7}{3}$

Q.

What is the nth term of H.P with first 2 terms as $\frac{4}{11}$ and $\frac{2}{3}$ ?

1. 

2. 

3. 

4. 

Q. If m is a root of the given equation $(1-ab)x^2-(a^2+b^2)x-(1+ab)=0$ and m harmonic means are inserted between a and b, then the difference between the last and the first of the means equals

1. ab(b - a)
2. a(b - a)
3. ab(a - b)
4. b - a

Q. Let $a,b$ be positive real numbers. If $a,G_1,G_2,b$ are in geometric progression and $a,H_1,H_2,b$ are in harmonic progression, show that
$\frac{G_1{G}_2}{H_1{H}_2}==\frac{(2a+b)(a+2b)}{9ab}$

Q.

If m is a root of the given equation $(1-ab)x^2-(a^2+b^2)x-(1+ab)=0$ and m harmonic means are inserted between a and b, then the difference between the last and the first of the means equals

1. 

2. 

3. 

4. 

Q. The vector from the point $-\hat{i}+2\hat{j}+6\hat{k}$ to the straight line through the point $(2,3,-4)$ and parallel to the vector $6\hat{i}+3\hat{j}+4\hat{k}$, is

1. $\frac{297}{61}i+\frac{118}{61}j-\frac{534}{61}k$
   
2. $\frac{297}{61}i+\frac{118}{61}j-+\frac{534}{61}k$
3. $\frac{297}{61}i-\frac{118}{61}j-\frac{534}{61}k$
4. none of these

Q. x is one of the four harmonic means inserted between 2/3 and 2/13. The value/s of x can be:

1. $\frac{2}{5}$
2. $\frac{2}{11}$
3. $\frac{2}{13}$
4. $\frac{2}{15}$

Q. x is one of the four harmonic means inserted between 2/3 and 2/13. The value/s of x can be:

1. $\frac{2}{5}$
2. $\frac{2}{11}$
3. $\frac{2}{13}$
4. $\frac{2}{15}$

Q.

5 harmonic means are inserted between 5 and 10. The common difference of corresponding A.P is.

1. 

2. 

3. 

4.