Prove that `|{:(a, a, x), (m, m, m), (b, x, b):}|=m(x-a)(x-b)`

If | (a, a, x,),(m, m, m),(b, x, b) | =0 then x may be equal to- a. a b. b c. a+b d. m

If M is the mean of x1 , x2 , x3 , x4 , x5 and x6, prove that (x1 − M) + (x2 − M) + (x3 − M) + (x4 − M) + (x5 — M) + (x6 − M) = 0

If y=(x-a)^(m)(x-b)^(n) , prove that (dy)/(dx)=(x-a)^(m-1)(x-b)^(n-1)[(m+n)x-(an+bm) ].

Roots of the equations |{:(x,,m,,n,,1),(a,,x,,n,,1),(a,,b,,x,,1),(a,,b,,c,,1):}|=0 are

If |{:(a-b,b-c,c-a),(x-y,y-z,z-x),(p-q,q-r,r-p):}|=m|{:(c,a,b),(z,x,y),(r,p,q):}|," then m"=

If a, b, c are in G.P., x and y are the A.M.'s of a, b and b, c respectively, then prove that: (i)(a)/(x)+(c)/(y)=2" "(ii) (1)/(x)+(1)/(y)=(2)/(b)

If a = x^(m + n) . Y^(l), b = x^(n + l). Y^(m) and c = x^(l + m) . Y^(n) , Prove that : a^(m-n) . b^(n - 1) . c^(l-m) = 1

If y=|(f(x),g(x),h(x)),(l, m, n),( a, b, c)| , prove that (dy)/(dx)=|(fprime(x),gprime(x),hprime(x)),(l, m, n),( a, b, c)|

If a, x_1, x_2, …, x_k and b, y_1, y_2, …, y_k from two A. Ps with common differences m and n respectively, the the locus of point (x, y) , where x= (sum_(i=1)^k x_i)/k and y= (sum_(i=1)^k yi)/k is: (A) (x-a) m = (y-b)n (B) (x-m) a= (y-n) b (C) (x-n) a= (y-m) b (D) (x-a) n= (y-b)m