Advertisements
Advertisements
प्रश्न
Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.
Advertisements
उत्तर
\[\text { Let the roots of the quadratic equation be a and b } . \]
\[ A = \frac{a + b}{2}\]
\[ \therefore a + b = 2A . . . . . . . . (i)\]
\[\text { Also, } G^2 = ab . . . . . . . (ii)\]
\[\text { The quadratic equation having roots a and b is given by } x^2 - (a + b)x + ab = 0 . \]
\[ \therefore x^2 - 2Ax + G^2 = 0 \left[ \text { Using (i) and (ii) } \right]\]
APPEARS IN
संबंधित प्रश्न
If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are `A+- sqrt((A+G)(A-G))`.
What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?
If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation.
The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that `a:b = (m + sqrt(m^2 - n^2)):(m - sqrt(m^2 - n^2))`.
Find the A.M. between:
7 and 13
Find the A.M. between:
12 and −8
Find the A.M. between:
(x − y) and (x + y).
Insert 4 A.M.s between 4 and 19.
Insert 7 A.M.s between 2 and 17.
Insert six A.M.s between 15 and −13.
There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.
Insert A.M.s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.s.
If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.
If a is the G.M. of 2 and \[\frac{1}{4}\] , find a.
Find the two numbers whose A.M. is 25 and GM is 20.
If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.
If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.
Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.
If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that:
\[a : b = (2 + \sqrt{3}) : (2 - \sqrt{3}) .\]
If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that \[\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A\]
If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. Then prove that xb – c. yc – a . za – b = 1
If x, y, z are positive integers then value of expression (x + y)(y + z)(z + x) is ______.
