Saved Bookmarks
| 1. |
Simplify \(\rm\left ( x+\sqrt{x} \right )^{6}+\left ( x-\sqrt{x} \right )^{6}\)1. 2x6 + 30x5 + 30x4 +4x32. 2x6 + 15x5 + 15x4 +4x33. 2x6 + 30x5 + 30x4 +2x34. x6 + 15x5 + 15x4 +x35. None of these |
|
Answer» Correct Answer - Option 3 : 2x6 + 30x5 + 30x4 +2x3 Concept: Binomial expansion for \(\rm\left ( x+a \right )^{n}+\left ( x-a \right )^{n}\) is given by \(\rm\left ( x+a \right )^{n}+\left ( x-a \right )^{n}\) = 2{nC0 xn-0 a0 + nC2 xn-2 a2 + nC4 xn-4 a4 +....} \(\rm\left ( x+a \right )^{n}+\left ( x-a \right )^{n}\) = 2{sum of terms at odd places} Calculation: For this \(\rm\left ( x+\sqrt{x} \right )^{6}+\left ( x-\sqrt{x} \right )^{6}\) assume \(\rm a=\sqrt{x}\) \(\rm\left ( x+a \right )^{6}+\left ( x-a \right )^{6}\) = 2{6C0 x6-0 a0 + 6C2 x6-2 a2 + 6C4 x6-4 a4 + 6C6 x6-6 a6} \(\rm\left ( x+a \right )^{6}+\left ( x-a \right )^{6}\) = 2{x6 + 15x4a2 + 15x2a4 + a6} Put \(\rm a=\sqrt{x}\) \(\rm\left ( x+\sqrt{x} \right )^{6}+\left ( x-\sqrt{x} \right )^{6}\) = 2{x6 + 15x4.x + 15x2.x2+ x3} \(\rm\left ( x+\sqrt{x} \right )^{6}+\left ( x-\sqrt{x} \right )^{6}\) = 2{x6 + 15x5 + 15x4+ x3} \(\rm\left ( x+\sqrt{x} \right )^{6}+\left ( x-\sqrt{x} \right )^{6}\) = 2x6 + 30x5 + 30x4 +2x3 |
|