Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7thterm.
Which term of the following sequence:
`sqrt3, 3, 3sqrt3`, .... is 729?
Which term of the following sequence:
`1/3, 1/9, 1/27`, ...., is `1/19683`?
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
Evaluate `sum_(k=1)^11 (2+3^k )`
Find four numbers forming a geometric progression in which third term is greater than the first term by 9, and the second term is greater than the 4th by 18.
If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that f(1) = 3 and `sum_(x = 1)^n` f(x) = 120, find the value of n.
if `(a+ bx)/(a - bx) = (b +cx)/(b - cx) = (c + dx)/(c- dx) (x != 0)` then show that a, b, c and d are in G.P.
If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P.
Show that one of the following progression is a G.P. Also, find the common ratio in case:
4, −2, 1, −1/2, ...
The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is \[87\frac{1}{2}\] . Find them.
The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.
Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.
The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.
Find the sum of the following serie to infinity:
`2/5 + 3/5^2 +2/5^3 + 3/5^4 + ... ∞.`
Find the rational numbers having the following decimal expansion:
\[3 . 5\overline 2\]
If a, b, c are in G.P., prove that:
\[\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}\]
If (p + q)th and (p − q)th terms of a G.P. are m and n respectively, then write is pth term.
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 the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
Given that x > 0, the sum \[\sum^\infty_{n = 1} \left( \frac{x}{x + 1} \right)^{n - 1}\] equals
The two geometric means between the numbers 1 and 64 are
Find three numbers in G.P. such that their sum is 21 and sum of their squares is 189.
A ball is dropped from a height of 80 ft. The ball is such that it rebounds `(3/4)^"th"` of the height it has fallen. How high does the ball rebound on 6th bounce? How high does the ball rebound on nth bounce?
The numbers 3, x, and x + 6 form are in G.P. Find x
For a G.P. If t4 = 16, t9 = 512, find S10
Determine whether the sum to infinity of the following G.P.s exist, if exists find them:
9, 8.1, 7.29, ...
Insert two numbers between 1 and −27 so that the resulting sequence is a G.P.
Select the correct answer from the given alternative.
The tenth term of the geometric sequence `1/4, (-1)/2, 1, -2,` ... is –
Select the correct answer from the given alternative.
Sum to infinity of a G.P. 5, `-5/2, 5/4, -5/8, 5/16,...` is –
Answer the following:
Find the sum of the first 5 terms of the G.P. whose first term is 1 and common ratio is `2/3`
Answer the following:
For a G.P. a = `4/3` and t7 = `243/1024`, find the value of r
Answer the following:
If for a G.P. t3 = `1/3`, t6 = `1/81` find r
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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 pth, qth, and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ab–c . bc – a . ca – b = 1
The third term of a G.P. is 4, the product of the first five terms is ______.
The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in ______.
