Advertisements
Advertisements
प्रश्न
Answer the following:
Find the term independent of x in the in expansion of `(1 - x^2) (x + 2/x)^6`
उत्तर १
`(1 - x^2) (x + 2/x)^6 = (x + 2/x)^6 - x^2(x + 2/x)^6` ...(1)
(r + 1)th term in the expansion of (a + b)n is given by
tr+1 = nCr, an–r br
For `(x + 2/x)^6`, a = x, b = `2/x`, n = 6
∴ tr+1 = `""^6"C"_"r" (x)^(6 - "r") (2/x)^"r"`
= 6Cr x6–r · 2r · x–r
= 6Cr · 2r · x6–2r ...(2)
For term independent of x in `(x + 2/x)^6`,
x6–2r = x0
∴ 6 – 2r = 0
∴ r = 3
For term independent of x in x2 `(x + 2/x)^6`,
x2 · x6–2r = x0
∴ x8–2r = x0
∴ 8 – 2r = 0
∴ r = 4
∴ from (1) and (2), the term independent of x in `(1 - x^2) (x + 2/x)^6` is
6C3 · 23 – 6C4 · 24
= `(6!)/(3!3!) xx 8 - (6!)/(4!2!) xx 16`
= `(6 xx 5 xx 4)/(1 xx 2 xx 3) xx 8 - (6 xx 5)/(1 xx 2) xx 16`
= 160 – 240
= – 80
उत्तर २
`(1 - x^2) (x + 2/x)^6`
= `(1 - x^2)[""^6"C"_0 x^6 + ""^6"C"_1 x^5(2/x) + ""^6"C"_2 x^4(2/x)^2 + ""^6"C"_3 x^3 (2/x)^3 + ""^6"C"_4 x^2(2/x)^4 + ""^6"C"_5 x(2/x)^5 + ""^6"C"_6 (2/x)^6]`
= `(1 - x^2)[""^6"C"_0 x^6 + ""^6"C"_1 (2x^4) + ""^6"C"_2 (4x^2) + ""^6"C"_3 (8) + ""^6"C"_4 (16/x^2) + ""^6"C"_5 (32/x^4) + ""^6"C"_6(64/x^6)]`
∴ the term independent of x
= 6C3 (8) – 6C4 (16)
= `(6!)/(3!3!) xx 8 - (6!)/(4!2!) xx 16`
= `(6 xx 5 xx 4)/(1 xx 2 xx 3) xx 8 - (6 xx 5)/(1 xx 2) xx 16`
= 160 – 240
= – 80
APPEARS IN
संबंधित प्रश्न
In the following expansion, find the indicated term.
`(2x^2 + 3/(2x))^8`, 3rd term
In the following expansion, find the indicated term.
`((4x)/5 - 5/(2x))^9`, 7th term
In the following expansion, find the indicated term.
`(1/3 + "a"^2)^12`, 9th term
In the following expansion, find the indicated term.
`(3"a" + 4/"a")^13`, 10th term
Find the constant term (term independent of x) in the expansion of `(2x + 1/(3x^2))^9`
Find the constant term (term independent of x) in the expansion of `(x - 2/x^2)^15`
Find the constant term (term independent of x) in the expansion of `(x^2 - 1/x)^9`
Find the constant term (term independent of x) in the expansion of `(2x^2 - 5/x)^9`
Find the term containing x6 in the expansion of (2 − x) (3x + 1)9
The coefficient of x2 in the expansion of (1 + 2x)m is 112. Find m
Select the correct answer from the given alternatives.
The total number of terms in the expression of (x + y)100 + (x − y)100 after simplification is:
Select the correct answer from the given alternatives.
In the expansion of (x2 − 2x)10, the coefficient of x16 is
Select the correct answer from the given alternatives.
The term not containing x in expansion of `(1 - x)^2 (x + 1/x)^10` is
Select the correct answer from the given alternatives.
The number of terms in expansion of (4y + x)8 − (4y − x)8
Answer the following:
Find tenth term in the expansion of `(2x^2 + 1/x)^12`
Find the coefficients of x6 in the expansion of `(3x^2 - 1/(3x))^9`.
Find the coefficients of x60 in the expansion of `(1/x^2 + x^4)^18`
Answer the following:
Find the constant term in the expansion of `((4x^2)/3 + 3/(2x))^9`
Answer the following:
If the coefficient of x2 and x3 in the expansion of (3 + kx)9 are equal, find k
Answer the following:
If the constant term in the expansion of `(x^3 + "k"/x^8)^11` is 1320, find k
Answer the following:
Show that there is no constant term in the expansion of `(2x - x^2/4)^9`
Answer the following:
State, first four terms in the expansion of `(1 - x)^(-1/4)`
Answer the following:
State, first three terms in the expansion of `(5 + 4x) ^(-1/2)`
Answer the following:
(a + bx) (1 − x)6 = 3 − 20x + cx2 + ..... then find a, b, c
Answer the following:
Suppose (1 + kx)n = 1 − 12x + 60x2 − .... find k and n.
The greatest value of the term independent of x in the expansion of (x sin p + x–1 cos p)10, p ∈ R is ______.
