Simplify: `-1/2a^2b^2c+1/3a b^2c-1/4a b c^2-1/5c b^2a^2+1/6c b^2a-1/7c^2a b+1/8c a^2bdot`

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a ,b ,c and a^prime ,b^(prime),c ' from the origin, then a. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0 b. 1/(a^2)-1/(b^2)-1/(c^2)+1/(a^('2))-1/(b^('2))-1/(c^('2))=0 c. 1/(a^2)+1/(b^2)+1/(c^2)-1/(a^('2))-1/(b^('2))-1/(c^('2))=0 d. 1/(a^2)+1/(b^2)+1/(c^2)+1/(a^('2))+1/(b^('2))+1/(c^('2))=0

Prove that [[1+a^2+a^4, 1+a b+a^2b^2 ,1+a c+a^2c^2],[ 1+a b+a^2b^2, 1+b^2+b^4, 1+b c+b^2c^2],[ 1+a c+a^2c^2, 1+b c+b^2c^2, 1+c^2+c^2]]=(a-b)^2(b-c)^2(c-a)^2

Delta =|(1+a^2+a^4,1+ab+a^2b^2, 1+ac+a^2c^2), (1+ab+a^2b^2, 1+b^2+b^4, 1+bc+b^2c^2),(1+ac+a^2c^2, 1+bc+b^2c^2, 1+c^2c^4)| is equal to

The factors of 8a^3+b^3-6a b+1 are (a) (2a+b-1)(4a^2+b^2+1-3a b-2a) (b) (2a-b+1)(4a^2+b^2-4a b+1-2a+b) (c) (2a+b+1)(4a^2+b^2+1-2a b-b-2a) (d) (2a-1+b)(4a^2+1-4a-b-2a b)

Which of the following has/have value equal to zero? |8 2 7 12 3 5 16 4 3| b. |1//a a^2b c1//bb^2a c1//cc^2a b| c. |a+b2a+b3a+b2a+b3a+b4a+b4a+b5a+b6a+b| d. |2 43 3 7 35 4 3 17 2|

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove: |(a^2+1,a b ,a c), (a b,b^2+1,b c), (c a,cb,c^2+1)|=1+a^2+b^2+c^2

A plane makes intercepts O A ,O Ba n dO C whose measurements are a,b and c on the O X ,O Ya n dO Z axes. The area of triangle A B C is a. 1/2(a b+b c+c a) b. 1/2a b c(a+b+c) c. 1/2(a^2b^2+b^2c^2+c^2a^2)^(1//2) d. 1/2(a+b+c)^2

If a ,b ,a n dc are in H.P., then th value of ((a c+a b-b c)(a b+b c-a c))/((a b c)^2) is ((a+c)(3a-c))/(4a^2c^2) b. 2/(b c)-1/(b^2) c. 2/(b c)-1/(a^2) d. ((a-c)(3a+c))/(4a^2c^2)

If a+b+c=0 and a^2+b^2+c^2=1, then (a) a b+b c+c a=1/2 (b) a b+b c+c a=-1/2 (c) a^4+b^4+c^4=1/2 (d) a^4+b^4+c^4=3/2