If `a_i, b_i in N` for i 1,2,3, then coefficient of x in the determinant;`|((1+x)^(a_1b_1),(1+x)^(a_1b_2),(1+x)^(a_1b_3)),((1+x)^(a_2b_1),(1+x)^(a_2b_2),(1+x)^(a_2b_3)), ((1+x)^(a_3b_1),(1+x)^(a_3b_2),(1+x)^(a_3b_3))|`

Find the coefficient of x in the determinant |{:((1+x)^(a_(1)b_(1)),(1+x)^(a_(1)b_(2)),(1+x)^(a_(1)b_(3))),((1+x)^(a_(2)b_(1)),(1+x)^(a_(2)b_(2)),(1+x)^(a_(2)b_(3))),((1+x)^(a_(3)b_(1)),(1+x)^(a_(3)b_(2)),(1+x)^(a_(3)b_(3))):}|

If A and B are the coefficients of x^n in the expansion (1 + x)^(2n) and (1 + x)^(2n-1) respectively, then A/B is

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

Show that if x_1, x_2, x_3!=0 |x_1+a_1b1a_1b_2a_1b_3a_2b_1x_2+a_2b_2a_2b_3a_3b_1a_3b_2x_3+a_3b_3|=x_1x_2x_3(1+(a_1b_1)/(x_1)+(a_2b_2)/(x_2)+(a_3b_3)/(x_3)) .

If f(x)=|((1+x)^a,(1+2x)^b,1),(1,(1+x)^a,(1+2x)^b), ((1+2x)^b,1,(1+x)^a)| then find constant term and coefficient of x

The coefficient of the term independent of x in the exampansion of ((x+1)/(x^(2//3)-x^(1//3)+1)-(x-1)/(x-x^(1//2)))^(10) is 210 b. 105 c. 70 d. 112

Let a, b, c such that (1)/((1-x)(1-2x)(1-3x))=(a)/(1-x)+(b)/(1-2x)+(c)/(1-3x), a/1+b/3+c/5=

The value of |(a_(1) x_(1) + b_(1) y_(1),a_(1) x_(2) + b_(1) y_(2),a_(1) x_(3) + b_(1) y_(3)),(a_(2) x_(1) +b_(2) y_(1),a_(2) x_(2) + b_(2) y_(2),a_(2) x_(3) + b_(2) y_(3)),(a_(3) x_(1) + b_(3) y_(1),a_(3) x_(2) + b_(3) y_(2),a_(3) x_(3) + b_(3) y_(3))| , is

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|= (a -b) (b -c) (c -a) (a + b+c) where a, b, c are all different, then the determinant |(1,1,1),((x-a)^(2),(x -b)^(2),(x -c)^(2)),((x -b) (x -c),(x -c) (x -a),(x -a) (x -b))| vanishes when

If x is positive, the sum to infinity of the series 1/(1+x)-(1-x)/((1+x)^2)+(1-x) ^2/((1+x)^3)-(1-x)^3/((1+x)^4)+....... i s (a)1/2 (b) 3/4 (c) 1 (d) none of these