Advertisements
Advertisements
Question
Answer the following:
If a, b, c are in G.P. and ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have common roots then verify that pb2 – 2qba + ra2 = 0
Advertisements
Solution
a, b, c are in G.P.
∴ b2 = ac
ax2 + 2bx + c = 0 becomes
`"a"x^2 + 2sqrt("ac")x + "c"` = 0
`(sqrt("a")x + sqrt("c"))^2` = 0
∴ x = `(-sqrt("c"))/sqrt("a")`
∴ ax2 + 2bx + c = 0 and px2 + 2qx + r = 0 have a common root, x = `(-sqrt("c"))/sqrt("a")` Satisfying px2 + 2qx + r = 0
∴ `"p"."c"/"a" + 2"q".((-sqrt("c"))/sqrt("a")) + r` = 0
`"pc" - 2"q"sqrt("ac") + "ra"` = 0
`"p"."b"^2/"a" - 2"qb" + "ra"` = 0 ...`[because "b"^2 = "ac", "c" = "b"^2/"a", sqrt("c") = "b"/sqrt("a"), sqrt("ac") = "b"]`
∴ pb2 – 2qba + ra2 = 0
APPEARS IN
RELATED QUESTIONS
The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.
Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...
Find the sum of the following geometric progression:
1, −1/2, 1/4, −1/8, ... to 9 terms;
Find the sum of the following geometric progression:
(a2 − b2), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;
Find the sum of the following geometric series:
`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` to 2n terms;
Find the sum of the following series:
9 + 99 + 999 + ... to n terms;
Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.
Find the sum of the following series to infinity:
10 − 9 + 8.1 − 7.29 + ... ∞
Find the rational numbers having the following decimal expansion:
\[0 .\overline {231 }\]
If a, b, c are in G.P., prove that log a, log b, log c are in A.P.
If a, b, c are in G.P., prove that \[\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}\] are in A.P.
If a, b, c are in G.P., prove that the following is also in G.P.:
a2, b2, c2
If a, b, c are in G.P., then prove that:
If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
Insert 5 geometric means between \[\frac{32}{9}\text{and}\frac{81}{2}\] .
The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.
If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.
If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is
If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
For what values of x, the terms `4/3`, x, `4/27` are in G.P.?
The number of bacteria in a culture doubles every hour. If there were 50 bacteria originally in the culture, how many bacteria will be there at the end of 5th hour?
The numbers 3, x, and x + 6 form are in G.P. Find 20th term.
Mosquitoes are growing at a rate of 10% a year. If there were 200 mosquitoes in the beginning. Write down the number of mosquitoes after 10 years.
Find: `sum_("r" = 1)^10(3 xx 2^"r")`
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
`1/5, (-2)/5, 4/5, (-8)/5, 16/5, ...`
Express the following recurring decimal as a rational number:
`51.0bar(2)`
If the first term of the G.P. is 16 and its sum to infinity is `96/17` find the common ratio.
The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15 find the G.P.
If the A.M. of two numbers exceeds their G.M. by 2 and their H.M. by `18/5`, find the numbers.
Answer the following:
In a G.P., the fourth term is 48 and the eighth term is 768. Find the tenth term
At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.
If the pth and qth terms of a G.P. are q and p respectively, show that its (p + q)th term is `(q^p/p^q)^(1/(p - q))`
For an increasing G.P. a1, a2 , a3 ........., an, if a6 = 4a4, a9 – a7 = 192, then the value of `sum_(i = 1)^∞ 1/a_i` is ______.
If in a geometric progression {an}, a1 = 3, an = 96 and Sn = 189, then the value of n is ______.
If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series `sqrt(x)(sqrt(a) + sqrt(x)) + sqrt(x)(sqrt(ab) + sqrt(xy)) + sqrt(x)(bsqrt(a) + ysqrt(x)) + ...` is ______.
