Home
Class 12
MATHS
If a b^2c^3, a^2b^3c^4,a^3b^4c^5 are in ...

If `a b^2c^3, a^2b^3c^4,a^3b^4c^5` are in A.P. `(a ,b ,c >0),` then the minimum value of `a+b+c` is (a) `1` (b) `3` (c) `5` (d) `9`

A

1

B

3

C

5

D

9

Text Solution

Verified by Experts

The correct Answer is:
B
A.M. `ge` G.M
`implies (a + b+ c)/(3) ge (abc)^(1//3)`
or `a + b + c ge 3 (abc)^(1//3)`
But given `ab^(2) c^(3), a^(2) b^(3) c^(4), a^(3) b^(4) c^(5)` are in A.P (`:' abc =! 0`) hence,
`2abc = 1 a^(2) b^(2) c^(2)`
`implies (abc - 1)^(2) = 0`
`:. abc = 1`
Now from Eq. (1) we, get
`a + b + c ge 3 (1)^(1//3)`
or `(a + b + c) ge 3`
Hence, minimum value of `a + b + c` is 3.
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Multiple correct answers type|5 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Linked comprehension type|6 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE ENGLISH|Exercise Concept Application Eexercises 6.4|4 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE ENGLISH|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE ENGLISH|Exercise All Questions|764 Videos

Similar Questions

Explore conceptually related problems

If three positive real numbers a,b ,c are in A.P such that a b c=4 , then the minimum value of b is a) 2^(1//3) b) 2^(2//3) c) 2^(1//2) d) 2^(3//23)

If a , b , c are real numbers forming an A.P. and 3+a , 2+b , 3+c are in G.P. , then minimum value of ac is

If |a|=1,|b|=3 and |c|=5 , then the value of [a-b" "b-c" "c-a] is

If a ,b and c are in A.P. and b-a ,c-b and a are in G.P., then a : b : c is (a). 1:2:3 (b). 1:3:5 (c). 2:3:4 (d). 1:2:4

If a^(2)+b^(2)+c^(2)=1 where, a,b, cin R , then the maximum value of (4a-3b)^(2) + (5b-4c)^(2)+(3c-5a)^(2) is

If a , ba n dc are the side of a triangle, then the minimum value of (2a)/(b+c-a)+(2b)/(c+a-b)+(2c)/(a+b-c)i s (a) 3 (b) 9 (c) 6 (d) 1

The minimum value of (b^2p_1)/c+(c^2p_2)/a+(a^2p_3)/b\ is

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

If a = 3, b = 2 and c = -4, find the values of: 2a+3b-5c

Suppose a ,b, and c are in A.P. and a^2, b^2 and c^2 are in G.P. If a

CENGAGE ENGLISH-INEQUALITIES INVOLVING MEANS -EXERCISES (Single Correct answer type)
  1. The minimum value of (x^(4)+y^(4)+z^(2))/(xyz) for positive real numbe...

    Text Solution

    |

  2. A rod of fixed length k slides along the coordinates axes, If it meets...

    Text Solution

    |

  3. The least value of 6tan^2varphi+54cot^2varphi+18 is 54 when A.M.geq GM...

    Text Solution

    |

  4. If a b^2c^3, a^2b^3c^4,a^3b^4c^5 are in A.P. (a ,b ,c >0), then the mi...

    Text Solution

    |

  5. If y=3^(x-1)+3^(-x-1) , then the least value of y is 2 6 2//3 3//2

    Text Solution

    |

  6. Minimum value of (b+c)//a+(c+a)//b+(a+b)//c (for real positive numbers...

    Text Solution

    |

  7. If the product of n positive numbers is n^n , then their sum is a posi...

    Text Solution

    |

  8. The minimum value of P=b c x+c a y+a b z , when x y z=a b c , is a. 3...

    Text Solution

    |

  9. If l ,m ,n are the three positive roots of the equation x^3-a x^2+b x-...

    Text Solution

    |

  10. If positive numbers a ,b ,c are in H.P., then equation x^2-k x+2b^(101...

    Text Solution

    |

  11. For x^2-(a+3)|x|-4=0 to have real solutions, the range of a is a)(-oo...

    Text Solution

    |

  12. If a ,b ,c are the sides of a triangle, then the minimum value of a/(b...

    Text Solution

    |

  13. If a,b,c,d in R^(+)-{1}, then the minimum value of log(d) a+ log(c)b+l...

    Text Solution

    |

  14. If a ,b ,c in R^+, t h e n(b c)/(b+c)+(a c)/(a+c)+(a b)/(a+b) is alwa...

    Text Solution

    |

  15. If a ,b ,c in R^+t h e n(a+b+c)(1/a+1/b+1/c) is always geq12 geq9 lt...

    Text Solution

    |

  16. If a ,b ,c in R^+ , then the minimum value of a(b^2+c^2)+b(c^2+a^2)+c...

    Text Solution

    |

  17. If a ,b ,c ,d in R^+a n da ,b ,c are in H.P., then a+d > b+c a+b > c+...

    Text Solution

    |

  18. If a ,b ,c ,d in R^+ such that a+b+c=18 , then the maximum value of a...

    Text Solution

    |

  19. f(x)=((x-2)(x-1))/(x-3), forall xgt3. The minimum value of f(x) is equ...

    Text Solution

    |

  20. If a >0, then least value of (a^3+a^2+a+1)^2 is (a)64 a^2 (b)16 a^4 ...

    Text Solution

    |