Advertisements
Advertisements
प्रश्न
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Advertisements
उत्तर
Firstly, the expression (x + y)4 + (x – y)4 is simplified by using Binomial Theorem.
(x + y)4 = 4C0x4 + 4C1x3y + 4C2x2y2 + 4C3xy3 + 4C4y4
= x4 + 4x3y + 6x2y2 + 4xy3 + y4
(x - y)4 = 4C0x4 - 4C1x3y + 4C2 x2y2 - 4C3xy3 + 4C4y4
= x4 - 4x3y + 6x2y2 - 4xy3 + y4
∴ (x + y)4 + (x - y)4 = 2(x4 + 6x2 y2 + y4)
Putting x = a2 and y = `sqrt(a^2 - 1)`, we obtain
`a^2 + sqrt((a^2 - 1)^4) + a^2 - sqrt((a^2 - 1)^4) = 2[(a^2)^4 + 6 (a^2)^2 sqrt((a^2 - 1)^2) + sqrt(a^2 - 1)^4]`
= `2[a^8 + 6a^4 (a^2 - 1) + (a^2 - 1)^2]`
= `2[a^8 + 6a^6 - 6a^4 + a^4 - 2a^2 + 1]`
= `2[a^8 + 6a^6 - 5a^4 - 2a^2 + 1]`
= `2a^8 + 12a^6 - 10a^4 - 4a^2 + 2`
APPEARS IN
संबंधित प्रश्न
Expand the expression: (1– 2x)5
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: (2x – 3)6
Expand the expression: `(x/3 + 1/x)^5`
Using Binomial Theorem, evaluate the following:
(96)3
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.
If a1, a2, a3 and a4 are the coefficient of any four consecutive terms in the expansion of (1 + x)n, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that 4OE = (x + a)2n – (x – a)2n
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The coefficient of a–6b4 in the expansion of `(1/a - (2b)/3)^10` is ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
