Let a,b,c,k be rational numbers such that k is not perfect cube if ` a + b k^(1/3) + c k^(2/3) = 0` prove that ` a = b = c =0`

Let a and b be positive real numbers with a^(3) +b^(3) = a -b and k = a^(2) + 4b^(2) , then (1) k lt 1 (2) k gt 1 (3) k = 1 (4) k gt 2

Let a, b, c be three vectors such that each of them are non-collinear, a+b and b+c are collinear with c and a respectively and a+b+c=k. Then (|k|, |k|) lies on

Let vec a = a_1 hat i + a_2 hat j+ a_3 hat k;vec b = b_1 hat i+ b_2 hat j+ b_3 hat k ; vec c= c_1hat i + c_2 hat j+ c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec a & vec b . If the angle between vec a and vec b is pi/6 , then |(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)|^2=

Let a, b be the roots of the equation x^(2) - 4 x +k_(1) = 0 and c , d the roots of the equation x^(2) - 36 x + k_(2) = 0 If a lt b lt c lt d and a, b,c,d are in G.P. then the product k_(1) k_(2) equals

If a, b, c are three distinct positive real numbers such that (b+c)/(a)+(c+a)/(b)+(a+b)/( c )gt k, then the grealtest value of k, is

Let a ,b ,c ,d be four distinct real numbers in A.P. Then the smallest positive valueof k satisfying 2(a-b)+k(b-c)^2+(c-a)^3=2(a-d)+(b-d)^2+(c-d)^3 is __________.

Let a ,b ,c ,d be four distinct real numbers in A.P. Then half of the smallest positive valueof k satisfying a(a-b)+k(b-c)^2=(c-a)^3=2(a-x)+(b-d)^2+(c-d)^3 is __________.

Let a,b,c are in A.P and k!=0 be a real number which of the following is/ are correct? (i) ka,kb,kc are in A.P (ii) k-a,k-b,k-c are in A.P. (iii) a/k,b/k,c/k are in A.P (a) i and ii only (b) ii and iii only (c) i and iii only (d) i, ii and iii

If a+b+c=0 and a,b,c are rational. Prove that the roots of the equation (b+c-a)x^(2)+(c+a-b)x+(a+b-c)=0 are rational.

If |a a^2 1+ahat3bb^2 1+b^3cc^2 1+c^3|=0 and the vectors vec A= hat i+a hat j+a^2 hat k , vec B= hat i+b hat j+b^2 hat k , vec C= hat i+c hat j+c^2 hat k are non-coplanar, then prove that a b c=-1 .