NDA Mathematics Question (2026) — Solution

Step-by-step solution

Since , , are the cube roots of -8, they are the roots of the equation z^3 + 8 = 0. Thus, we have ^3 = ^3 = ^3 = -8. The product of the roots of the cubic equation z^3 + 8 = 0 is given by = -8. Using these properties, we can find relations between the roots: ^3 = ^2 = ^3 = ^2 = ^3 = ^2 = Let the given expression be E = ^2 p^2+ ^2 q^2+ ^2 r^2 ^2 p^2+ ^2 q^2+ ^2 r^2 . Consider multiplying the denominator by the fraction : ( ^2 p^2 + ^2 q^2 + ^2 r^2) = p^2 + ^3 q^2 + ^2 r^2 Now, substitute the derived relations into each term of this expansion: For the first term, substitute = ^2 to get ^2 p^2. For the second term, since ^3 = ^3 = -8, we have ^3 = ^3 = ^2, which gives ^2 q^2. For the third term, substitute ^2 = to get ( ) r^2 = ^2 r^2. Adding these terms together, we get: ( ^2 p^2 + ^2 q^2 + ^2 r^2) = ^2 p^2 + ^2 q^2 + ^2 r^2 This shows that the numerator is exactly times the denominator. Therefore, the value of the given expression is . Answer: