Question 12 of 20intermediate🔧 ApplyLong Answer5 marks

Find (x+1)6+(x−1)6(x + 1)^6 + (x - 1)^6. Hence or otherwise evaluate (2+1)6+(2−1)6(\sqrt{2} + 1)^6 + (\sqrt{2} - 1)^6.

(i)
Answer

(x + 1)^6 + (x - 1)^6 = 2[x6x^{6} + 15x415x^{4} + 15x215x^{2} + 1]

(ii)
Answer

(2\sqrt{2} + 1)^6 + (2\sqrt{2} - 1)^6 = 198

Explanation

This question tests the application of Binomial Theorem. In part (i), when we expand both (x+1)6(x+1)^6 and (x−1)6(x-1)^6 using the theorem and add them, the terms with odd powers of xx cancel out due to opposite signs, leaving only even-powered terms. This simplification technique is commonly used in NCERT problems.

In part (ii), we substitute x=2x = \sqrt{2} into the simplified expression from part (i), making the calculation straightforward.

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