Prove that `|(1+a^(2)+a^(4),1+ab+a^(2)b^(2),1+ac+a^(2)c^(2)),(1+ab+a^(2)b^(2),1+b^(2)+b^(4),1+bc+b^(2)c^(2)),(1+ac+a^(2)c^(2),1+bc+b^(2)c^(2),1+c^(2)+c^(4))|=(a-b)^(2)(b-c)^(2)(c-a)^(2)`

Show that |(1+a^(2)-b^(2),2ab,-2b),(2ab,1-a^(2)+b^(2),2a),(2b,-2a,1-a^(2)-b^(2))|=(1+a^(2)+b^(2))^(3)

If |(a^(2),b^(2),c^(2)),((a+1)^(2),(b+1)^(2),(c+1)^(2)),((a-1)^(2),(b-1)^(2),(c-1)^(2))|=k(a-b)(b-c)(c-a) then find the value of k.

Area of triangle whose vertices are (a,a^(2)), (b,b^(2)) and (c,c^(2)) is (1)/(2) . And area of another triangle whose vertices are (p,p^(2)), (q,q^(2)) and (r,r^(2)) is 4, then the value of |((1+ap)^(2),(1+bp)^(2),(1+cp)^(2)),((1+aq)^(2),(1+bq)^(2),(1+cq)^(2)),((1+ar)^(2),(1+br)^(2),(1+cr)^(2))| is a)2 b)4 c)8 d)16

Prove that |(a,b+c,a^(2)),(b,c+a,b^(2)),(c,a+b,c^(2))|=-(a+b+c)xx(a-b)(b-c)(c-a).

Prove that Prove that Delta=|(1,1,1),(a,b,c),(bc+a^(2),ac+b^(2),ab+c^(2))|=2(a-b)(b-c)(c-a)

The value of the determinant |(ka,k^(2)+a^(2),1),(kb,k^(2)+b^(2),1),(kc,k^(2)+c^(2),1)| is a)k(a+b)(b+c)(c+a) b)kabc (a^(2)+b^(2)+c^(2)) c)k(a-b) (b -c) (c - a) d)k(a + b - c) (b + c - a) (c + a - b)

Prove that |[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]|=4a^2b^2c^2

If the line segment joining the points P (a,b) and Q(c,d) subtends an angle theta at the origin , then the value of costheta is a) (ab+cd)/(sqrt(a^(2)+b^(2))sqrt(c^(2)+d^(2))) b) (ab)/(sqrt(a^(2)+b^(2)))+(bd)/(sqrt(c^(2)+d^(2))) c) (ac+bd)/(sqrt(a^(2)+b^(2))sqrt(c^(2)+d^(2))) d) (ac-bd)/(sqrt(a^(2)+b^(2))sqrt(c^(2)+d^(2)))

If a^(2) + b^(2), ab+bc and b^(2)+c^(2) are in G.P then a,b,c are in