Geometric Mean

Q. The number of real roots of the equation $e^{4x}+{2e}^{3x}-e^x-6=0$ is

Q. The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation $x^2+ax+b=0,{\alpha }^2-2$ is also a root of this equation, is

1. $8$
2. $4$
3. $6$
4. $2$

Q.

Can a geometric mean be negative?

Q.

If the conjugate of $(x+iy)(1-2i)$ be $1+i$, then:

1. $x-iy=\frac{1-i}{1+2i}$

2. $x+iy=\frac{1-i}{1-2i}$

3. $x=\frac{3}{5}$

4. $y=\frac{1}{5}$

Q. The sum of two positive numbers is $6$ times their geometric mean. Then the ratio of these two numbers is

1. $(6+\sqrt{2}):(6-\sqrt{2})$
2. $(3+4\sqrt{2}):(3-4\sqrt{2})$
3. $(3+2\sqrt{2}):(3-2\sqrt{2})$
4. $(3+\sqrt{2}):(3-\sqrt{2})$

Q.

If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

Q.

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

Q.

The difference between two numbers is $48$ and the difference between their arithmetic mean and their geometric mean is $18$. Then, the greater of two numbers is

1. $96$

2. $60$

3. $54$

4. $49$

Q. $Thegeometricmeanofnumbers7,7^2,7^3,......{.7}^{n}is:$

1. 
2. 
3. 
4. 

Q. If the A.M. and G.M. between two numbers are in ratio of $m:n,$ then the ratio of numbers can be

1. $\frac{m+\sqrt{m^2-n^2}}{m-\sqrt{m^2-n^2}}$
2. $\frac{m+\sqrt{m^2-n^2}}{n-\sqrt{m^2-n^2}}$
3. $\frac{m+\sqrt{m^2+n^2}}{m-\sqrt{m^2+n^2}}$
4. $\frac{n+\sqrt{m^2-n^2}}{m-\sqrt{m^2-n^2}}$

Q.

If A be one A.M. and p, q be two G.M.'s between two numbers, then 2A is equal to

1. $\frac{p^3+q^3}{pq}$

2. $\frac{p^3-q^3}{pq}$

3. $\frac{p^2+q^2}{2}$

4. $\frac{pq}{2}$

Q. If $n=10,\sum x=50,\sum y=-30,\sum x^2=290,\sum y^2=300,\sum xy=-115$, then find the coefficient of correlation.

Q. The A.M. between two positive numbers a and b is twice the G. M. between them. The ratio of the numbers is

1. $(\sqrt{3}+1):(\sqrt{3}-1)$
2. $(\sqrt{2}+3):(\sqrt{2}-3)$
3. $(2+\sqrt{3}):(2-\sqrt{3})$
4. None of these

Q.

Find the mean and variance for the data

xi	92	93	97	98	102	104	109
f i	3	2	3	2	6	3	3

Q.

Find the mean and variance for the data

xi	6	10	14	18	24	28	30
f i	2	4	7	12	8	4	3

Q. Two sequences cannot be in both A.P. and G.P. together.

1. True
2. False

Q.

If p, q be two A.M.'s and G be one G.M. between two numbers, then $G^2=$

Q. If $x$ be the arithmetic mean and $y,z$ be two geometric means between two positive numbers, then the value of $\frac{y^3+z^3}{xyz}$ is

Q.

If a is the G.M. of 2 and $\frac{1}{4},$ find a.

Q.

If $a$ is a positive number such that the arithmetic mean of $a$ and $2$ exceeds their geometric mean by $1$. Then, the value of $a$ is

1. $3$

2. $5$

3. $9$

4. $8$

Q. If n geometric means be inserted between a and b then the $n^{th}$ geometric mean will be

1. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n-1)}}$
2. $a{\left(\frac{b}{a}\right)}^{\frac{n-1}{\left(n\right)}}$
3. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n+1)}}$
4. $a{\left(\frac{b}{a}\right)}^{\frac{1}{\left(n\right)}}$

Q. If the arithmetic and geometric mean of two positive real numbers $a$ and $b(a>b)$ are $13$ and $12$ respectively, then the value of $a-b$ is

1. $13$
2. $10$
3. $12$
4. $8$

Q. If $2x^3+a{x}^2+bx+4=0,a,b>0$ has $3$ real roots and the minimum value of $a+b$ is $m(x^{1/3}+4^{1/3})$, then the value of $m+x$ is

Q.

If a, b, c are in G.P., prove that $\frac{1}{lo{g}_{a}m},\frac{1}{lo{g}_{b}m},\frac{1}{lo{g}_{c}m}$ are in A.P.,

Q.

Find the geometric means of the following pairs of numbers.

(i) 2 and 8

(ii) $a^{3}b$ and $a{b}^3$

(iii) - 8 and - 2

Q. The A.M. between two positive numbers a and b is twice the G. M. between them. The ratio of the numbers is

1. $(\sqrt{2}+3):(\sqrt{2}-3)$
2. $(2+\sqrt{3}):(2-\sqrt{3})$
3. $(\sqrt{3}+1):(\sqrt{3}-1)$
4. None of these

Q. If $t$ lies between real roots of the equation $2x^2-2(2t+1)x+t(t+1)=0,$ then $t$ cannot be

1. $1$
2. $-2$
3. $-\frac{1}{2}$
4. $\frac{1}{2}$

Q. If A.M. of $b$ and $c$ is $a$ and the two geometric means between them are $G_1$ and $G_2$, then the value of $G_1^3+G_2^3$ is

1. $abc$
2. $2abc$
3. $3abc$
4. $4abc$

Q.

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio $(3+2\sqrt{2}):(3-2\sqrt{2})$.

Q.

The arithmetic mean of nC0, nC1, nC2....nCn is
(A)2^n/n
(B)2^n-1/n
(C)2^n/n+1
(D)2^n-1/n+1