If f(x) `=|{:(x+c_(1),x+a,x+a),(x+b,x+c_(2),x+a),(x+b,x+b,x+c_(3)):}|` show that f(x) is linear in X. Hence deduce that f(0) = `(bg(a)-ag(b))/((b-a)),`
where g(x)=`(c_(1)-x)(c_(2)-x)(c_(3)-x)`.

If f(x)=|(x+c_1,x+a,x+a),(x+b,x+c_2,x+a),(x+b,x+b,x+c_3)| then show that f(x) is linear in x. Hence deduce f(0)= (bg(a)-ag(b))/((b-a)) where g(x)=(c_1-x)(c_2-x)(c_3-x)

f(x) = |{:(x+c_(1),,x+a,,x+a),(x+b,,x+c_(2),,x+a),(x+b,,x+b,,x+c_(3)):}| " and " g(x)= (C_(1) -x)(c_(3)-x) Coefficient of x in f(x) is

If f(x)=|{:(0,x-a,x-b),(x+a,0,x-c),(x+b,x+c,0):}| , then

f(x) = |{:(x+c_(1),,x+a,,x+a),(x+b,,x+c_(2),,x+a),(x+b,,x+b,,x+c_(3)):}| " and " g(x)= (C_(1) -x)(c_(3)-x) Which of the following is not true ?

If f(x)=|(0, x-a, x-b), (x+a, 0, x-c),(x+b, x+c, 0)| , then

If a,b,c are in A.P.and f(x)= |(x+a,x^2+1,1),(x+b, 2x^2-1,1),(x+c,3x^2-2 , 1)| , then f'(x) is :

If f(x)=|[x+a^2 , a b , a c],[a b, x+b^2,b c],[ ac,b c, x+c^2]| . then f^(prime)(x)

Let f(x)=(x-a)(x-b)(x-c), a lt b lt c. Show that f'(x)=0 has two roots one belonging to (a, b) and other belonging to (b, c).

If f(x) = |(a_1+x, b_1+x, c_1+x), (a_2+x, b_2+x, c_2+x),(a_3+x, b_3+x, c_3+x)| , show that f''(x)=0 and that f(x)=f(0)+S ,where S denotes the sum of all the cofactors of all the element in f(0)dot

"Let "g(x)=|{:(f(x+c),f(x+2c),f(x+3c)),(f(c),f(2c),f(3c)),(f'(c),f'(2c),f'(3c)):}|, where c is constant, then find lim_(xrarr0) (g(x))/(x).