A.M Greater than or Equal to G.M

Q. Two equal glasses are respectively $\frac{1}{4}$ and $\frac{1}{3}$ full of milk. They are then filled up with water and contents mixed in a tumbler. The ratio of milk and water in the tumbler is:

1. None of these
2. 3 : 11
3. 7 : 17
4. 9 : 23
5. 11 : 23

Q.

$\text{If 47.2506 = 4A}+\frac{7}{B}+2C+\frac{5}{D}+6E,\text{then what is the value of 5A +3B + 6C + D+ 3E?}$

1. 159.003

2. 53.6003

3. 153.6003

4. 213.0003

Q.

For the function $x+\left(\frac{1}{x}\right),x\in [1,3]$, the value of $\mathrm{c}$ for the mean value theorem is

1. $1$

2. $\sqrt{3}$

3. $2$

4. None of these

Q.

The least value of the expression $2{\log }_{10}x-{\log }_x(0.01)$ for $x>1$, is

1. $10$

2. $2$

3. $-0.01$

4. None of these

Q. Four geometric means are inserted between 5 and 160. Find the 2nd geometric mean.

1. 40
2. 80
3. 10
4. 20

Q.

If $p\text{and}q$ are positive real numbers such that $p^2+q^2=1$, then the maximum value of $p+q$ is

1. $2$

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

3. $\frac{1}{\sqrt{2}}$

4. $\sqrt{2}$

Q. If x + y + z = 1 and x, y, z are positive numbers such that $(1-x)(1-y)(1-z)\ge kxyz$, then k =

1. 6
2. 9
3. 4
4. 8

Q.

The arithmetic mean and harmonic mean of two numbers is x and y respectively, then the geometric mean of the two numbers is

1. 

2. 

3. 

4. 

Q. The function f(x) = |x - 2| + |2.5 - x| + |3.6 - x|, where x is a real number, attains minimum at

1. x = 2.5
2. x = 2.7
3. x = 2.3
4. None of these

Q.

If the arithmetic mean and harmonic mean of two distinct positive numbers is 16 and 4 respectively, then the geometric mean of the two numbers is

1. 12

2. 64

3. 8

4. 4

Q. The A.M of two positive numbers is 15 and their G.M is 12. What is the smaller number?

1. 8
2. 12
3. 24
4. 6

Q.

Find the minimum value of $lo{g}_{a}b+lo{g}_{b}a$ (where a and b are not equal to 1).

1. 1

2. 2

3. 3

4. 4

Q. The ratio of HM and GM for two positive numbers is 4 : 5. What is the ratio of the numbers?

1. 3 : 2
2. 4 : 1
3. 3 : 4
4. 2 : 3

Q.

The harmonic mean of sum of first n natural numbers and sum of cubes of first n natural numbers is

1. n²(n+1)² / (n² + n + 2)

2. n²(n+1)² / (n² + n + 1)

3. n²(n+1)² / (n² + 2n+ 3 )

4. n²(n+1)² / (n² + n + 4 )

Q.

$\text{Find x :}\frac{x-b-c}{a}+\frac{x-c-a}{b}+\frac{x-a-b}{c}=3\text{, if}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ne 0$

1. a + b + c

2. None of these

3. ab + bc + ca

4. a + bc

Q. If a, b, c are positive real numbers, the least value of $(a+b+c)(\frac{1}{a}+\frac{1}{b},\frac{1}{c})$ is :

1. 1
2. 9
3. 12
4. None of these

Q.

Which of the following is the correct relation between a (arithmetic mean), b (geometric mean) and c (harmonic mean)?

1. ac = b

2. a/c = b

3. ab = c

4. ac = b²

Q. If $x^2+y^2+z^2=1$. What is the range of xy+yz+zx?

1. [-1, 1/2]
2. [-1/2, 3]
3. [-1/2, 1]
4. [-1, 2]

Q.

$\sqrt[3]{(8+27)}=\sqrt[3]{\left(8\right)}+\sqrt[3]{\left(27\right)}$. State whether the statement is true (T) or false (F).

1. True
2. False

Q. m is the smallest positive integer such that for any integer n < m, the quantity n³ – 7n² + 11n – 5 is positive. What is the value of m?(CAT 2001)

1. None of these
2. 4
3. 5
4. 8

Q.

The least value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ for positive values of $x,y,z$ such that $x+y+z=9$ is

1. $0$

2. $1$

3. $2$

4. $4$

Q. Let u _n+1 = 2u_n +1, (n = 0, 1, 2, …) and u₀ = 0. Then u₁₀ would be nearest to: (CAT 1993)

1. 1023
2. 2047
3. 4095
4. 8195

Q.

If a, b, c >0, a + b + c =6 and f(x) = $\left(\frac{6}{x}\right)$ - 1, find the minimum value of f(a).f(b).f(c)?

1. 6

2. 24

3. 8

4. 12

Q.

A natural number, when added to its reciprocal gives a value which is one -- eighth of 34. Find is the product of the number and its square root?

1. 8

2. 27

3. 32

4. 45

5. 28

Q.

Find the value of `x' in the figure below.

1. 5

2. 8

3. None of these

4. 9

Q. If the A.M of two positive number a and b (a >b) is twice their G.M, then a : b is:

1. $2:\sqrt{3}$
2. $7+4\sqrt{3}:7-4\sqrt{3}$
3. $2+\sqrt{3}:2-\sqrt{3}$
4. $2:7+4\sqrt{3}$

Q. If x, y, z are distinct positive real numbers then the expression below would be

1. Greater than 4
2. Greater than 5
3. Greater than 6
4. Less than 6
5. Less than 2.

Q. For positive numbers a, b, c the least value of $(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$ is

1. $3$
2. $9$
3. $\frac{27}{4}$
4. $Noneofthese$

Q. If $\alpha \in (0,\frac{\pi }{2})$ then $\sqrt{x^2+x}+\frac{ta{n}^2\alpha }{\sqrt{x^2+x}}$ is always greater than or equal to

1. 
2. 
3. 1
4. 2

Q. If $\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}=r$ then, r cannot take any value except :(CAT 2004)

1. 
2. -1
3. ½ or -1
4. -1/2 or -1