Find the expansion of (3x2 - 2ax + 3a2)3 using binomial theore
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    Binomial Theorem

    Multiple Choice QuestionsShort 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

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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

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    17. Find the expansion of (3x2 - 2ax + 3a2)3 using binomial theorem. 

    left parenthesis 3 straight x squared minus 2 ax plus 3 straight a squared right parenthesis cubed space equals space left square bracket left parenthesis 3 straight a squared minus 2 ax right parenthesis plus 3 straight x squared right square bracket cubed

                                                      = left square bracket straight p space plus space 3 straight x squared right square bracket cubed  where straight p equals 3 straight a squared minus 2 ax
                                                   space space equals space straight C presuperscript 3 subscript 0 space end subscript straight p cubed left parenthesis 3 straight x squared right parenthesis to the power of 0 plus space straight C presuperscript 3 subscript 1 straight p squared left parenthesis 3 straight x squared right parenthesis to the power of 1 plus straight C presuperscript 3 subscript 2 straight p to the power of 1 left parenthesis 3 straight x squared right parenthesis squared plus straight C presuperscript 3 subscript 3 space end subscript straight p to the power of 0 left parenthesis 3 straight x squared right parenthesis cubed
                                                   space space equals straight p cubed plus 9 straight p squared straight x squared plus 27px to the power of 4 plus 27 straight x to the power of 6
                                                     equals left square bracket 3 straight a squared minus 2 ax right square bracket cubed plus 9 straight x squared left square bracket 3 straight a squared minus 2 ax right square bracket squared plus 27 straight x to the power of 4 left parenthesis 3 straight a squared minus 2 ax right square bracket plus 27 straight x to the power of 6
    equals left square bracket straight C presuperscript 3 subscript 0 space left parenthesis 3 straight a squared right parenthesis cubed space left parenthesis negative 2 ax right parenthesis to the power of 0 plus straight C presuperscript 3 subscript 1 space left parenthesis 3 straight a squared right parenthesis squared space left parenthesis negative 2 ax right parenthesis space plus space straight C presuperscript 3 subscript 2 space left parenthesis 3 straight a squared right parenthesis to the power of 1 space left parenthesis negative 2 ax right parenthesis squared
                        plus straight C presuperscript 3 subscript 3 left parenthesis 3 straight a squared right parenthesis to the power of 0 space left parenthesis negative 2 ax right parenthesis cubed right square bracket space plus space 9 straight x squared space left square bracket 9 straight a to the power of 4 plus 4 straight a squared straight x squared minus 12 straight a cubed straight x right square bracket
                                                             plus 27 straight x to the power of 4 left parenthesis 3 straight a squared minus 2 ax right parenthesis plus 27 straight x to the power of 6
    equals space 27 straight a to the power of 6 minus 54 straight a to the power of 5 straight x plus 36 straight a to the power of 4 straight x squared minus 8 straight a cubed straight x cubed plus 81 straight a to the power of 4 straight x squared plus 36 straight a squared straight x to the power of 4 minus 108 straight a cubed straight x cubed
                                                                              plus 81 straight a squared straight x to the power of 4 minus 54 ax to the power of 5 plus 27 straight x to the power of 6
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    18. If a and b are distinct integers, prove that a-b is a factor of an - bn, whenever n is positive integer.
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    19. Find co-efficient of x5 in the expansion of (1 + 3x)6 (1 - x)5
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    20. In the expansion of (x + a)n, sums of odd and even terms are P and Q respectively. 
    Prove that:

    (a) 2 left parenthesis straight P squared plus straight Q squared right parenthesis space equals space left parenthesis straight x plus straight a right parenthesis to the power of 2 straight n end exponent plus left parenthesis straight x minus straight a right parenthesis to the power of 2 straight n end exponent  (b) left parenthesis straight P squared minus straight Q squared right parenthesis space equals space left parenthesis straight x squared minus straight a squared right parenthesis to the power of straight n
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