Home
Class 12
MATHS
Let x,y,z be three positive prime number...

Let x,y,z be three positive prime numbers. The progression in which `sqrt( x) , sqrt( y ) , sqrt( z)` can be three terms ( not necessarily consecutive ) is `:`

A

AP

B

GP

C

HP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
`:.a,b,c` are positive prime numbers.
Let `sqrt(a),sqrt(b),sqrt(c)` are 3 terms of AP. [not necessarily consecutive ]
Then, `sqrt(a)=A+(p-1)D " " "....(i)"`
`sqrt(b)=A+(q-1)D " " "....(ii)"`
`sqrt(c)=A+(r-1)D " " "....(iii)"`
[A and D be the first term and common difference of AP]
`sqrt(a)-sqrt(b)=(p-q)D " " "....(iv)"`
`sqrt(b)-sqrt(c )=(q-r)D " " "....(v)"`
`sqrt(c)-sqrt(a)=(r-p)D " " "....(vi)"`
On dividing Eq.(v), we get
`(sqrt(a)-sqrt(b))/(sqrt(b)-sqrt(c))=(p-q)/(q-r)" " "....(vii)"`
Since,p,q,r are natural numbers and a,b,c are positive prime numbers, so
Eq. (vii) does not hold.
So, `sqrt(a),sqrt(b),sqrt(c)` cannot be the 3 terms of AP. [not necessarily consecutive ]
Similarly, we can show that `sqrt(a),sqrt(b),sqrt(c)` cannot be any 3 terms of GP andHP. [ not necessarily, consecutive].
Promotional Banner

Similar Questions

Explore conceptually related problems

The numbers 1,4,16 can be three terms ( not necessarily consecutive ) of :

If sqrt(x) + sqrt(y) = sqrt(a) , prove that (dy)/(dx) = -sqrt(y/x) .

If x, y, z are positive integers then value of expression (x+y)(y+z)(z+x) is

The sum of three consecutive terms in an arithmetic progression is 6 and their product is 120. Find the three numbers.

Find the sum to indicated number of terms in each of the geometric progressions in sqrt7, sqrt(21)3sqrt7 , ........n terms

The domain of the function y=sqrt(x-2)+sqrt(1-x) is

Let p, q, r be distinct positive numbers. Three lines p x+q y+r=0, q x+r y+p=0 and r x+p y+q=0 are concurrent, if

The number of three digit numbers of the form xyz such that x lt y , z le y and x ne0 , is

If sqrt(x)+sqrt(y)=sqrt(10) , show that (dy)/(dx)+sqrt(y/x)=0