Simplify:  from Mathematics Binomial Theorem
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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

    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. 
    283 Views
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    15. Using Binomial Theorem, prove that 6n - 5n always leaves remainder 1 when divided by 25.
    394 Views
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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. 
    199 Views
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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
    124 Views
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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
    142 Views
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