Find co-efficient of x5 in the expansion of (1 + 3x)6 (1 - x)5

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 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. 
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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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                                                                      space space space space space space cross times space left square bracket 1 minus 5 straight x space plus space 10 straight x squared minus 10 straight x cubed plus 5 straight x to the power of 4 minus straight x to the power of 5 right square bracket

Co-efficient of space space straight x to the power of 5 space equals space 1 cross times left parenthesis negative 1 right parenthesis space plus 18 cross times 5 plus 135 cross times left parenthesis negative 10 right parenthesis space plus 540 cross times 10 plus 1215 cross times left parenthesis negative 5 right parenthesis space plus space 1458 space cross times space 1
                           equals negative 1 plus 90 minus 1350 plus 5400 minus 6075 plus 1458
                          space space equals 6948 minus 7426
                           space equals negative 478
                         

   


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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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