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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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space space minus left square bracket straight C presuperscript 4 subscript 0 straight a to the power of 4 left parenthesis negative straight b right parenthesis to the power of 0 plus straight C presuperscript 4 subscript 1 straight a cubed left parenthesis negative straight b right parenthesis to the power of 1 plus straight C presuperscript 4 subscript 2 straight a squared left parenthesis negative straight b right parenthesis squared

straight C presuperscript 4 subscript 3 straight a left parenthesis negative straight b right parenthesis cubed plus straight C presuperscript 4 subscript 4 straight a to the power of 0 left parenthesis negative straight b right parenthesis to the power of 4

]

space space equals space 2 left square bracket straight C presuperscript 4 subscript 1 straight a cubed straight b to the power of 1 plus straight C presuperscript 4 subscript 3 straight a to the power of 1 straight b cubed right square bracket space equals space 8 straight a cubed straight b plus 8 ab cubed equals 8 ab left square bracket straight a squared plus straight b squared right square bracket

Putting

straight a equals square root of 3 comma space straight b equals square root of 2 comma

we get

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 space equals space 8 left parenthesis square root of 3 right parenthesis space left parenthesis square root of 2 right parenthesis space left square bracket 3 plus 2 right square bracket space equals space 40 square root of 6

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

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