If `f(x)={:abs(((1+)^(a),(1+2x)^(b),1),(1,(1+x)^(a),(1+2x)^(b)),((1+2x)^(b),1,(1+x)^(0))):}` then find (i) constant term (ii) coefficient of x.

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

Consider the polynomial fucntion f(x) = |{:((1+x)^(a),,(1+2x)^(b),,1),(1,,(1+x)^(a),,(1+2x)^(b)),((1+2x)^(b),,1,,(1+x)^(a)):}| a,b being positive integers. The constant term in f(x) is

If (1)/((1+2x)(1-x^(2)))=(A)/(1+2x)+(B)/(1+x)+(C)/(1-x) then assending order of A, B, C.

If f(x)=|(1, x, x+1), (2x, x(x-1), (x+1)x), (3x(x-1), x(x-1)(x-2), (x+1)x(x-1))| then f(5000) is equal to 0 b. 1 c. 500 d. -500

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))|

If (x^(2)+x+2)/(x^(2)+2x+1)=A+(B)/(x+1)+(C)/((x+1)^(2)) , then find the value of A+B+C .

If sin^(-1)((2a)/(1+a^2))-cos^(-1)((1-b^2)/(1+b^2))=tan^(-1)((2x)/(1-x^2)) , then prove that x=(a-b)/(1+a b)

If f(x)=|(1,x,x+1),(2x,x(x-1),(x+1)x),(3x(x-1),x(x-1)(x-2),(x+1)x(x-1))|, then f(100) is equal to - (i) 0 (ii) 1 (iii) 100 (iv) -100

Find the coefficients of x^7 in (a x^2+1/(b x))^(11)and x^(-7)i n(a x-1/(b x^2))^(11) and find the relation between a and b so that coefficients are equal.

If (2x^(3)+1)/((x-1)(x+1)(x+2))=A+(B)/(x-1)+(C)/(x+1)+(D)/(x+2) , then find the value of A+B+C+D .