If` n in N`such that is not a multiple of 3 and`(1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r )`, then find the value of `sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r )`.
1 Answers
`underset(r=0)overset(n)sum(-1)^(r).a_(r )..^(n)C_(r )`
`= a_(0) xx .^(n)C_(0) - a_(1) xx .^(n)C_(1) + a_(2) xx .^(n)C_(2)-"….."+(-1)^(n)a_(n) xx .^(n)C_(0)`
`=` Coefficient of `x^(n)` in `(1-x+x^(2))^(n)(x-1)^(n)`
`=` Coefficient of `x^(n)` in `(x^(3)-1)^(n)`
`= 0` (as n is a multiple of 3)
0
like
0 dislike
Talk Doctor Online in Bissoy App
Evaluate the following limit: `lim_(nto oo)(sum_(r=1)^(n) sqrt(r)sum_(r=1)^(h)1/(sqrt(r)))/(sum_(r=1)^(n)r)`
Correct Answer - `8/3` `lim_(n to oo) (sum_(r=1)^(n)sqrt(r) sum_(r=1)^(n)1/(sqrt(4)))/(sum_(r=1)^(n)r)` `:.` Limit `=lim_(nto oo) (1/n sum_(r=1)^(n)sqrt(4/n)(1/nsum_(r=1)^(n)sqrt(n/r)))/(1/n sum_(r=1)^(n)r/n)` `=(int_(0)^(1)sqrt(x)dxint_(0)^(1)(dx)/(sqrt(x)))/(int_(0)^(1)x dx) =8/3`
2 Answers 1 views
[ The value of int_(-pi)^( pi)sum_(r=0)^(999)cos rx(1+sum_(r=1)^(999)sin rx)dx, is [ (1) 2 pi, (2) 999 pi, (3) 0]]
Correct Answer - A `I=int_(-pi)^(pi) sum_(r=0)^(999) cos rx (1+ sum_(r=1)^(999) sin rx)dx` `:. I=int_(-pi)^(pi) sum_(r=0)^(999) cosrx(1-sum_(r=1)^(999) sinrx)dx` (Replacing `x` by `-x`) Adding we get `2I=2int_(-pi)^(pi)sum_(r=0)^(999) cos rx dx` `implies I=int_(-pi)^(pi) 1...
2 Answers 1 views
Find the value of `sum_(p=1)^(n) (underset(m=p)overset(n)sum""^(n)C_(m)""^(m)C_(p))`. And hence, find the value of `lim_(nrarroo) (1)/(3^(n)) sum_(m=p
Correct Answer - `(3^(n) - 2^(n))` and `1` `S = underset(p=1)overset(n)sum(underset(m=p)overset(n)sum.^(n)C_(m).^(m)C_(p))` `= underset(p=1)overset(n)sum(.^(n)C_(p).^(p)C_(p)+.^(n)C_(p+1).^(p)+1)C_(p) + "....."+.^(n)C_(n).^(n)C_(p))` `= underset(p=1)overset(n)sum` [coefficient of `x^(p)` in `{.^(n)C_(p)(1+x)^(p)+.^(n)C_(p+1)(1+x)^(p+1)+"....."+.^(n)C_(n)(1+x)^(n)}]` `= underset(p=1)overset(n)sum` [Coefficeint of `x^(p)` in `{.^(n)C_(0)+.^(n)C_(1)(1+x)+.^(n)C_(2)(1+x)^(2)+"....."+.^(n)C_(p-1)(1+x)^(p-1)+.^(n)C_(p)(1+x)^(p)+"....."+.^(n)C_(n)(1+x)^(n)}]` `= underset(p=1)oversetg(n)sum`...
2 Answers 1 views
If `(1+x^(2))^(n) =sum_(r=0)^(n) a_(r)x^(r )= (1+x+x^(2)+x^(3))^(100)`. If `a = sum_(r=0)^(300)a_(r)`, then `sum_(r=0)^(300) ra_(r)` is
Correct Answer - D `(1-x)^(n)(1+x)^(n)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n)(1-x)^(n-r)` or `(1-x+2x)^(n) = underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)` or `underset(r=0)overset(n)sum.^(n)C_(r)(1-x)^(n-r)(2x)^(r)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)` Comparing general term, we get `a_(r) = .^(n)C_(r)2^(r)`.
2 Answers 1 views
Let `P = sum_(r=1)^(50) (""^(50+r)C_(r)(2r-1))/(""^(50)C_(r)(50+r)), Q = sum_(r=0)^(50)(""^(50)C_(r))^(2), R = sum_(r=0)^(100)(-1)^(r) (""^(100)C_(r))
Correct Answer - C In P, general term of the series is `T_(r)=(.^(50-r)C_(r)(2r-1))/(.^(50)C_(r)(50+r))` `=(.^(50+r)C_(r))/(.^(50)C_(r))(1-(50-r+1)/(50+r))` `= (.^(50+r)C_(r))/(.^(50)C_(r))-(.^(50+r)C_(r))/(.^(50)C_(r))((50-r+1)/(50+r))` Now, `(.^(50+r)C_(r))/(.^(50)C_(r))((50-r+1)/(50+r))` `= ((50-r+1)(50+r)!r!(50-r)!)/(r!50!(50+r)50!)` `=((50-r+1)!(50+r-1)!)/(50!50!)` `= (.^(50+r-1)C_(r-1))/(.^(50)C_(r-1))` `rArr T(r) = (.^(50+r)C_(r))/(.^(50)C_(r)) - (.^(50+r-1)C_(r-1))/(.^(50)C_(r-1))= V(r) -...
2 Answers 1 views
For `0ltphiltpi//2," if" x=sum_(n=0)^(oo) cos^(2n)phi,y=sum_(n=0)^(oo) sin^(2m)phi,z=sum_(n=0)^(oo)cos^(2n)phisin^(2n)phi`,then
Correct Answer - B::C All are infinte geometric progression with common ratio lt 1 `x=1/(1-cos^2phi)=1/sin^2phi,y=1/(1-sin^2phi)=1/cos^2phi`, `z=1/(1-cos^2phisin^2phi)` Now, `xy+z=1/(sin^2phicos^2phi)+1/(1-sin^2phicos^2phi)` `=1/(sin^2phicos^2phi(1-sin^2phicos^2phi))` `or xy+z=xyz ...(i)` Clearly, `x+y=(sin^2phi+cos^2phi)/(sin^2phicos^2phi)=xy` `:. x+y+z=xyz` [using Eq. (i)]
2 Answers 1 views
`sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k))` is equal to (where `|a| gt 1`)
Correct Answer - A `(a)` `S=sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k))`, `|a| gt 1` or `0 lt (1)/(|a|) lt 1` `=sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i)a^(j)a^(k))` `=(sum_(i=1)^(oo)(1)/(a^(i)))(sum_(j=1)^(oo)(1)/(a^(j)))(sum_(k=1)^(oo)(1)/(a^(k)))` `=((1)/(a))/(1-(1)/(a))*((1)/(a))/(1-(1)/(a))*((1)/(a))/(1-(1)/(a))=(1)/((a-1)^(3))`
2 Answers 1 views
The value of `sum_(i=1)^(n)sum_(j=1)^(i)sum_(k=1)^(j)=220` , then the value of `n` equals
Correct Answer - C `(c )` `sum_(i=1)^(n)sum_(j=1)^(i)sum_(k=1)^(j)1` `=sum_(i=1)^(n)sum_(j=1)^(i)j` `=sum_(i=1)^(n)(i(i+1))/(2)` `=(1)/(2)sum_(i=1)^(n)(i^(2)+i)` `=(n(n+1)(n+2))/(6)=220` `:.n=10`
2 Answers 1 views
If `sum_(i=1)^(7) i^(2)x_(i) = 1` and `sum_(i=1)^(7)(i+1)^(2) x_(i) = 12` and `sum_(i=1)^(7)(i+2)^(2)x_(i) = 123` then find the value of `sum_(i=1)^(7
Correct Answer - `334`
2 Answers 1 views
Let `P = sum_(r=1)^(50)(""^(50+r)C_(r)(2r-1))/(""^(50)C_(r)(50+r)), Q = sum_(r=0)^()(""^(50)C_(r))^(2), R = sum_(0)^(100)(-1)^(r)(""^(100)C_(r))^(2)`
Correct Answer - `ubrace(333"......"3)_("n times")`
2 Answers 1 views