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O LevelAdditional MathsCambridgeExam QuestionsSequences & SeriesBinomial Theorem

Binomial Theorem (Cambridge O Level Additional Maths)

Exam Questions

59 mins11 questions
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1a
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Find the first 3 terms in the expansion of open parentheses 4 minus x over 16 close parentheses to the power of 6in ascending powers of x. Give each term in its simplest form.

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1b
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Hence find the term independent of x in the expansion ofĀ open parentheses 4 minus x over 16 close parentheses to the power of 6 open parentheses x minus 1 over x close parentheses squared

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2
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Find the term independent ofĀ x in the binomial expansion of open parentheses 3 x minus 1 over x close parentheses to the power of 6.

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3
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Find the coefficient of x squared in the expansion ofĀ open parentheses x minus 3 over x close parentheses open parentheses x plus 2 over x close parentheses to the power of 5.

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1a
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Expand left parenthesis 2 minus x right parenthesis to the power of 5, simplifying each coefficient.

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1b
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Hence solveĀ  fraction numerator straight e to the power of open parentheses 2 minus x close parentheses to the power of 5 end exponent cross times straight e to the power of 80 x end exponent over denominator straight e to the power of 10 x to the power of 4 plus 32 end exponent end fraction equals straight e to the power of negative x to the power of 5 end exponent.

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2
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In the expansion of open parentheses 2 k minus x over k close parentheses to the power of 5 , whereĀ k is a constant, the coefficient ofĀ x squared is 160.
Find the value of k.

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3
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(i)
Find the first 3 terms in the expansion of space left parenthesis 1 space plus space 3 x right parenthesis to the power of 6 space, in ascending powers of x.
Simplify the coefficient of each term.

(ii)
When the expansion ofĀ left parenthesis 1 space plus space 3 x right parenthesis to the power of 6 left parenthesis a space plus space x right parenthesis squared is written in ascending powers of x, the first three terms are space 4 space plus space 68 x space plus space b x squared space, whereĀ a andĀ b are constants.
Find the value ofĀ a and the value of b.

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4
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The first 3 terms in the expansion ofĀ left parenthesis 3 minus a x right parenthesis to the power of 5 , in ascending powers of x, can be written in the form b minus 81 x plus c x squared. Find the value of each of a comma space b space and space c.

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1
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The first 3 terms in the expansion of open parentheses a plus x close parentheses cubed space open parentheses 1 minus space x over 3 close parentheses to the power of 5, in ascending powers of x, can be written in the formĀ 27 plus space b x plus c x squared , where a,Ā b and c are integers. Find the values of a,Ā b and c.

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2
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The first three terms in the expansion ofĀ open parentheses a plus b x close parentheses to the power of 5 space left parenthesis 1 plus space x right parenthesis space are space 32 minus 208 x plus c x squared . Find the value of each of the integersĀ a comma space b space and space c.

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3
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Given that the coefficient ofĀ x squared in the expansion of open parentheses 1 plus x close parentheses open parentheses 1 minus x over 2 close parentheses to the power of n is 25 over 4, find the value of the positive integer n.Ā 

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4
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In the expansion of open parentheses 1 plus x over 2 close parentheses to the power of nthe coefficient ofĀ x to the power of 4 is half the coefficient of x to the power of 6. Find the value of the positive constant n.

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