Relation Between A.M., G.M., and H.M.

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JEE Main 2015

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

4 l m n^{2}

(B)

4 l^{2} m^{2} n^{2}

(C)

4 l^{2} mn

(D)

4 l m^{2} n

JEE Advanced 2016

LEVELJEE Main

Let $b_{i} > 1$ for $i = 1, 2, \dots, 101$ . Suppose $lo g_{e} b_{1}, lo g_{e} b_{2}, \dots, lo g_{e} b_{101}$ are in Arithmetic Progression (A.P.) with the common difference $lo g_{e} 2$ . Suppose $a_{1}, a_{2}, \dots, a_{101}$ are in A.P. such that $a_{1} = b_{1}$ and $a_{51} = b_{51}$ . If $t = b_{1} + b_{2} + \dots + b_{51}$ and $s = a_{1} + a_{2} + \dots + a_{51}$ , then

(A)

s > t

and

a_{101} > b_{101}

(B)

s > t

and

a_{101} < b_{101}

(C)

s < t

and

a_{101} > b_{101}

(D)

s < t

and

a_{101} < b_{101}

JEE Advanced 2003

LEVELJEE Main

If $α \in (0, π /2)$ , then $x^{2} + x + \frac{t a n ^{2} α}{x ^{2} + x}$ is always greater than or equal to

(A)

2 tan α

(B)

1

(C)

2

(D)

sec^{2} α

JEE Advanced 2002

LEVELBoard

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

(A)

n (2 c)^{1/ n}

(B)

(n + 1) c^{1/ n}

(C)

2 n c^{1/ n}

(D)

(n + 1) (2 c)^{1/ n}

JEE Advanced 2000

LEVELJEE Main

If $a, b, c, d$ are positive real numbers such that $a + b + c + d = 2$ , then $M = (a + b) (c + d)$ satisfies the relation

(A)

0 \leq M \leq 1

(B)

1 \leq M \leq 2

(C)

2 \leq M \leq 3

(D)

3 \leq M \leq 4

JEE Advanced 1994

LEVELJEE Main

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

(A)

a, b, c

are in A.P.

(B)

a^{2}, b^{2}, c^{2}

are in A.P.

(C)

a, b, c

are in G.P.

(D)

a, b, c

are in H.P.

JEE Advanced 1991

LEVELBoard

The product of $n$ positive numbers is unity Then their sum is

n

(C)

equal to

n + 1/ n

(D)

never less than

n

JEE Advanced 1988

LEVELJEE Main

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

* Multiple Correct Options

(A)

a = b = c

(B)

a \geq b \geq c

(C)

a + c = b

(D)

a c - b^{2} = 0

JEE Advanced 1986

LEVELJEE Main

If $a, b$ and $c$ are distinct positive numbers, then the expression $(b + c - a) (c + a - b) (a + b - c) - ab c$ is

If $x_{1}, x_{2}, \dots, x_{n}$ are any real numbers and $n$ is any positive integer, then

(A)

n \sum_{i = 1}^{n} x_{i}^{2} < (\sum_{i = 1}^{n} x_{i})^{2}

(B)

\sum_{i = 1}^{n} x_{i}^{2} \geq (\sum_{i = 1}^{n} x_{i})^{2}

(C)

\sum_{i = 1}^{n} x_{i}^{2} \geq n (\sum_{i = 1}^{n} x_{i})^{2}

Let $a, b, c$ be positive integers such that $b / a$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$ , then the value of $\frac{a ^{2} + a - 14}{a + 1}$ is \dots.

JEE Advanced 2011

LEVELBoard

The minimum value of the sum of real numbers $a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}$ and $a^{10}$ where $a > 0$ is .........

JEE Advanced 2001

LEVELJEE Main

Let $a_{1}, a_{2}, \dots a_{n}$ be positive real numbers in geometric progression. For each $n$ , let $A_{n}, G_{n}, H_{n}$ be respectively, the arithmetic mean, geometric mean, and harmonic mean of $a_{1}, a_{2}, \dots a_{n}$ . Find an expression for the geometric mean of $G_{1}, G_{2}, \dots G_{n}$ in terms of $A_{1}, A_{2}, \dots A_{n}, H_{1}, H_{2}, \dots H_{n}$ .

JEE Advanced 1992

LEVELBoard

Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

JEE Advanced 1979

LEVELBoard

The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

JEE Advanced 2004

LEVELJEE Main

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

JEE Advanced 2003

LEVELJEE Main

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

JEE Advanced 2002

LEVELJEE Main

Let $a, b$ be positive real numbers. If $a, A_{1}, A_{2}, b$ are in arithmetic progression, $a, G_{1}, G_{2}, b$ are in geometric progression and $a, H_{1}, H_{2}, b$ are in harmonic progression, show that $\frac{G _{1} G _{2}}{H _{1} H _{2}} = \frac{A _{1} + A _{2}}{H _{1} + H _{2}} = \frac{( 2 a + b ) ( a + 2 b )}{9 ab}$ .

JEE Advanced 1991

LEVELJEE Main

Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between $p$ and $(\frac{n + 1}{n - 1})^{2} p$ .

JEE Advanced 1984

LEVELBoard

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

JEE Advanced 1982

LEVELJEE Main

$mn$ squares of equal size are arranged to from a rectangle of dimension $m$ by $n$ , where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.

JEE Advanced 2007

LEVELJEE Main

Comprehension Passage

Let

A_{1}, G_{1}, H_{1}

denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For

n \geq 2

, Let

A_{n - 1}

and

H_{n - 1}

have arithmetic, geometric and harmonic means as

A_{n}, G_{n}, H_{n}

respectively.

Question 1:

Which one of the following statements is correct ?

(A)

G_{1} > G_{2} > G_{3} > \dots

(B)

G_{1} < G_{2} < G_{3} < \dots

(C)

G_{1} = G_{2} = G_{3} = \dots

(D)

G_{1} < G_{3} < G_{5} < \dots

and

G_{2} > G_{4} > G_{6} > \dots

Question 2:

Which one of the following statements is correct ?

(A)

A_{1} > A_{2} > A_{3} > \dots

(B)

A_{1} < A_{2} < A_{3} < \dots

(C)

A_{1} > A_{2} > A_{5} > \dots

and

A_{2} < A_{4} < A_{6} < \dots

(D)

A_{1} < A_{3} < A_{5} < \dots

and

A_{2} > A_{4} > A_{6} > \dots

Question 3:

Which one of the following statements is correct?

(A)

H_{1} > H_{2} > H_{3} > \dots

(B)

H_{1} < H_{2} < H_{3} < \dots

(C)

H_{1} > H_{3} > H_{5} > \dots

and

H_{2} < H_{4} < H_{6} < \dots

(D)

H_{1} < H_{3} < H_{5} < \dots

and

H_{2} > H_{4} > H_{6} > \dots

JEE Main 11 Jan 2019 (Evening)

LEVELBoard

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x ^{m} y ^{n}}{( 1 + x ^{2 m} ) ( 1 + y ^{2 n} )}$ is:

(A)

1/2

(B)

(C)

1/4

(D)

\frac{m + n}{6 mn}

JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

Let $f : R \to R$ be such that for all $x \in R$ , $(2^{1 + x} + 2^{1 - x})$ , $f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

JEE Main 2020 (8 Jan Morning)

LEVELJEE Main

Let $f : R \to R$ be such that for all $x \in R (2^{1 + x} + 2^{1 - x}), f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If m is the A.M. of two distinct real numbers l and n (l,n>1) and G1​,G2​ and G3​ are three geometric means between l and n, then G14​+2G24​+G34​ equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α∈(0,π/2), then x2+x​+x2+x​tan2α​ is always greater than or equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,…,an​ are positive real numbers whose product is a fixed number c, then the minimum value of a1​+a2​+⋯+an−1​+2an​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c,d are positive real numbers such that a+b+c+d=2, then M=(a+b)(c+d) satisfies the relation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ln(a+c),ln(a−c),ln(a−2b+c) are in A.P., then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The product of n positive numbers is unity Then their sum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the first and the (2n−1)st terms of an A.P., a G.P. and an H.P. are equal and their nth terms are a,b and c respectively, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b and c are distinct positive numbers, then the expression (b+c−a)(c+a−b)(a+b−c)−abc is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x1​,x2​,…,xn​ are any real numbers and n is any positive integer, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The minimum value of the sum of real numbers a−5,a−4,3a−3,1,a8 and a10 where a>0 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the harmonic mean and geometric mean of two positive numbers be the ratio 4:5. Then the two number are in the ratio \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The harmonic mean of two numbers is 4. Their arithmetic mean A and the geometric mean G satisfy the relation 2A+G2=27. Find the two numbers.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are positive real numbers. Then prove that (a+1)7(b+1)7(c+1)7>77a4b4c4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in A.P., a2,b2,c2 are in H.P., then prove that either a=b=c or a,b,−c/2 form a G.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let x,y be positive real numbers and m,n positive integers. The maximum value of the expression (1+x2m)(1+y2n)xmyn​ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be such that for all x∈R, (21+x+21−x), f(x) and (3x+3−x) are in A.P., then the minimum value of f(x) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be such that for all x∈R(21+x+21−x),f(x) and (3x+3−x) are in A.P., then the minimum value of f(x) is :

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

If $α \in (0, π /2)$ , then $x^{2} + x + \frac{t a n ^{2} α}{x ^{2} + x}$ is always greater than or equal to

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

If $a, b, c, d$ are positive real numbers such that $a + b + c + d = 2$ , then $M = (a + b) (c + d)$ satisfies the relation

If $ln (a + c), ln (a - c), ln (a - 2 b + c)$ are in A.P., then

The product of $n$ positive numbers is unity Then their sum is

If the first and the $(2 n - 1)$ st terms of an A.P., a G.P. and an H.P. are equal and their $n$ th terms are $a, b$ and $c$ respectively, then

If $a, b$ and $c$ are distinct positive numbers, then the expression $(b + c - a) (c + a - b) (a + b - c) - ab c$ is

If $x_{1}, x_{2}, \dots, x_{n}$ are any real numbers and $n$ is any positive integer, then

The minimum value of the sum of real numbers $a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}$ and $a^{10}$ where $a > 0$ is .........

Let the harmonic mean and geometric mean of two positive numbers be the ratio $4 : 5$ . Then the two number are in the ratio \dots.

The harmonic mean of two numbers is $4$ . Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation $2 A + G^{2} = 27$ . Find the two numbers.

If $a, b, c$ are positive real numbers. Then prove that $(a + 1)^{7} (b + 1)^{7} (c + 1)^{7} > 7^{7} a^{4} b^{4} c^{4}$

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x ^{m} y ^{n}}{( 1 + x ^{2 m} ) ( 1 + y ^{2 n} )}$ is:

Let $f : R \to R$ be such that for all $x \in R$ , $(2^{1 + x} + 2^{1 - x})$ , $f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :

Let $f : R \to R$ be such that for all $x \in R (2^{1 + x} + 2^{1 - x}), f (x)$ and $(3^{x} + 3^{- x})$ are in A.P., then the minimum value of $f (x)$ is :