Answer: If a2, b2, c2 are in AP then, b2 – a2 = c2 – b2 If a/(b+c), b/(c+a), c/(a+b) are in AP then, b/(c+a) – a/(b+c) = c/(a+b) – b/(c+a) Let us take LCM on both the sides, b2 – a2 = c2 – b2...
Answers: (i) If b2(c + a) – a2(b + c) = c2(a + b) – b2(c + a) b2c + b2a – a2b – a2c = c2a + c2b – b2a – b2c Given, b – a = c – b a, b, c are in AP, c(b2 – a2 ) + ab(b – a) = a(c2 – b2 ) + bc(c – b)...
Answer: If (ca – b2) – (bc – a2) = (ab – c2) – (ca – b2) bc – a2, ca – b2, ab – c2 are in A.P. Consider LHS and RHS, (ca – b2) – (bc – a2) = (ab – c2) – (ca – b2) (a – b2 – bc + a2) = (ab – c2 – ca...
Answer: (i) If 1/a, 1/b, 1/c are in AP 1/b – 1/a = 1/c – 1/b Consider LHS, 1/b – 1/a = (a-b)/ab = c(a-b)/abc Consider RHS, 1/c – 1/b = (b-c)/bc = a(b-c)/bc [by multiplying with ‘a’ on both the...
Answer: (i) Expanding, a2 + c2 – 2ac = 4(ab – ac – b2 + bc) a2 + 4c2b2 + 2ac – 4ab – 4bc = 0 (a + c – 2b)2 = 0 a + c – 2b = 0 a, b, c are in AP b – a = c – b a + c – 2b = 0 a + c = 2b (a – c)2 = 4...
Answer: Expanding, a3 + c3 + 6abc = 8b3 a3 + c3 – (2b)3 + 6abc = 0 a3 + (-2b)3 + c3 + 3a(-2b)c = 0 if a + b + c = 0, a3 + b3 + c3 = 3abc (a – 2b + c)3 = 0 a – 2b + c = 0 a + c = 2b b = (a+c)/2 a, b,...
Answer: Given, a(1/b + 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP Also, a(1/b + 1/c) + 1, b(1/c + 1/a) + 1, c(1/a + 1/b) + 1 are in AP Let us take LCM for each expression, (ac+ab+bc)/bc ,...
Answer: Given, x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are in AP (z2 + zx + x2) – (x2 + xy + y2) = (y2 + yz + z2) – (z2 + zx + x2) d = common difference, Y = x + d and x = x + 2d Consider the...
Answer: (i) If a, b, c are in AP b – a = c – b 1/a, 1/b, 1/c are in AP 1/b – 1/a = 1/c – 1/b If (b+c)/a, (c+a)/b, (a+b)/c are in AP (c+a)/b – (b+c)/a = (a+b)/c – (c+a)/b Let us take LCM, 1/a, 1/b,...