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Binomial Theorem

Short Answer Type

11.

Find $left parenthesis straight a plus straight b right parenthesis to the power of 4 minus left parenthesis straight a minus straight b right parenthesis to the power of 4.$ then evaluate $space space left parenthesis square root of 3 plus square root of 2 right parenthesis to the power of 4 minus left parenthesis square root of 3 minus square root of 2 right parenthesis to the power of 4$

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12.

Find $left parenthesis straight x plus 1 right parenthesis to the power of 5 plus left parenthesis straight x minus 1 right parenthesis to the power of 5$ and hence evaluate $left parenthesis square root of 2 plus 1 right parenthesis to the power of 5 plus space left parenthesis square root of 2 minus 1 right parenthesis to the power of 5$

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13.
Simplify: $left parenthesis straight x plus 2 straight y right parenthesis to the power of 10 plus left parenthesis straight x minus 2 straight y right parenthesis to the power of 10$

left parenthesis straight x plus 2 straight y right parenthesis to the power of 10 plus left parenthesis straight x minus 2 straight y right parenthesis to the power of 10 equals 2 left square bracket straight C presuperscript 10 subscript 0 space end subscript straight x to the power of 10 left parenthesis 2 straight y right parenthesis to the power of 0 plus straight C presuperscript 10 subscript 2 space end subscript straight x to the power of 8 left parenthesis 2 straight y right parenthesis squared plus straight C presuperscript 10 subscript 4 space straight x to the power of 6 left parenthesis 2 straight y right parenthesis to the power of 4 plus straight C presuperscript 10 subscript 6 space straight x to the power of 4 left parenthesis 2 straight y right parenthesis to the power of 6

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space equals 2 open square brackets straight x to the power of 10 plus 45 cross times 4 cross times straight x to the power of 8 straight y squared plus fraction numerator 10 times 9 times 8 times 7 over denominator 4 times 3 times 2 times 1 end fraction. space 2 to the power of 4 space straight x to the power of 6 space straight y to the power of 4 plus open parentheses fraction numerator 10 times 9 times 8 times 7 over denominator 4 times 3 times 2 times 1 end fraction close parentheses 2 to the power of 6 space straight x to the power of 4 space straight y to the power of 6 space plus 45 cross times 2 to the power of 8 cross times straight x squared straight y to the power of 8 plus 1024 space straight y to the power of 10 space close square brackets

space space equals space 2 left square bracket straight x to the power of 10 plus 180 space straight x to the power of 8 straight y squared space plus space 3360 space straight x to the power of 6 straight y to the power of 4 space plus space 13440 space straight x to the power of 4 straight y to the power of 6 divided by space 11520 space straight x squared straight y to the power of 8 plus 1024 space straight y to the power of 10 right square bracket

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14. Show that 9ⁿ ^{+ 1} - 8n - 9 is divisible by 64, whenever n is a positive integer.

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15. Using Binomial Theorem, prove that 6ⁿ - 5n always leaves remainder 1 when divided by 25.

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16.

Find the value of $space space open parentheses straight a squared plus square root of straight a squared minus 1 end root close parentheses to the power of 4 plus open parentheses straight a squared minus square root of straight a squared minus 1 end root close parentheses to the power of 4$