IBMaths: AA HLDPTopic Questions1. Number & Algebra1.6 Binomial Theorem

Binomial Theorem (DP IB Maths: AA HL)

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1.6 Binomial Theorem

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1
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Find the coefficient of the term in x cubed in the expansion of left parenthesis 2 minus x right parenthesis to the power of 8.

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2
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Find the first three terms, in ascending powers of x comma in the expansion of open parentheses 3 plus x close parentheses to the power of 4.

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3
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In the expansion of open parentheses a minus x close parentheses to the power of 4 comma the coefficient of the x squared term is 96.

Given that a space greater than space 0 comma find the value of a.

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4
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Find the first three terms, in ascending powers of x comma in the expansion of open parentheses 9 minus 2 x close parentheses to the power of 5.

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5
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In the expansion of left parenthesis a minus 2 x right parenthesis to the power of 5 comma the coefficient of the x squared term is equal to the coefficient of the x cubed term. Find the value of a.

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6
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In the expansion of open parentheses 3 plus p x close parentheses to the power of 6, the coefficient of the x to the power of 4 term is four times the coefficient of the x squared term. Find the possible values ofspace p.

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7a
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Consider the expansion of open parentheses 4 a x minus 3 close parentheses to the power of 5.

Write down the number of terms in this expansion.

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7b
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The coefficient of the term in x to the power of 4 is negative 61440.

Find the value of a where a is a positive constant.

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8a
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Consider the expansion of open parentheses x cubed plus 4 over x close parentheses to the power of 4.

Write the first three terms in descending powers of x.

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8b
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Find the value of the constant term.

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9
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The coefficient of x to the power of 7 in the expansion of x cubed open parentheses a x plus 3 close parentheses to the power of 5 is 1215.

Find the possible values of a.     

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10a
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Consider the binomial expansion of fraction numerator 1 over denominator 1 plus x end fraction.

Write down the first four terms.

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10b
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Find the values of x such that the complete expansion converges.

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10c
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Use the terms found in part (a) to estimate fraction numerator 1 over denominator 1.1 end fraction.

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11a
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Consider the binomial expansion of  cube root of 4 open parentheses 2 plus x close parentheses. end root

Write down the first three terms.

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11b
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State the interval of convergence for the complete expansion.

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11c
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Use the terms found in part (a) to estimate cube root of 12 . Give your answer as a fraction.

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12a
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Consider the binomial expansion of  fraction numerator 1 over denominator square root of 4 plus x end root end fraction.

Write down the first four terms.

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12b
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State the interval of convergence for the complete expansion.

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1
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Find the coefficient of the x to the power of 16 term in the expansion left parenthesis 2 x squared minus x cubed right parenthesis to the power of 7.

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2a
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Consider the expansion of left parenthesis 5 x cubed minus x right parenthesis to the power of 6.

Write down the number of terms in this expansion.

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2b
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Find the first three terms, in descending powers of x, of the expansion.

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3a
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Consider the expansion of open parentheses fraction numerator a x over denominator 2 end fraction plus 3 over x squared close parentheses to the power of 5.

Find an expression, in terms of a, for the coefficient of the x to the power of negative 1 end exponent term.

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3b
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The coefficient of the x to the power of negative 1 end exponent term is 90.

Find the value of a.

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4a
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Consider the quadratic expression 5 x squared minus 15 x plus 10..

Write down the quadratic expression in the form left parenthesis p x minus q right parenthesis left parenthesis x minus r right parenthesis.

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4b
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Find the coefficient of the x to the power of 8 term in the expansion of left parenthesis 5 x squared minus 15 x plus 10 right parenthesis to the power of 5. Give your answer in the form a cross times 10 to the power of k comma where 1 less or equal than a less than 10 comma k element of straight integer numbers.

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5a
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The coefficient of x to the power of 7 in the expansion of open parentheses x over 3 close parentheses to the power of 5 space open parentheses a x plus 5 close parentheses squared is 1 third.  

Find the possible values of a.

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5b
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The sum of the coefficients of the expansion is 196 over 243.

Determine which value of a spacefound in part (a) is correct.

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6
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Consider the expansion left parenthesis 1 minus 3 x right parenthesis to the power of 4 left parenthesis 1 minus 2 k x right parenthesis squared.

The coefficient of the x to the power of 6 term is 36. Find the possible values of k.

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7a
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Consider the expansion of open parentheses x cubed over 3 plus k over x close parentheses to the power of 4. The constant term is negative 500 over 3. 

Find the value of k.

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7b
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Find the coefficient of the x to the power of 4 term.

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8
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In the expansion of open parentheses 1 half x plus 1 close parentheses to the power of n comma the coefficient of the x squared term is 8 n, where n element of straight integer numbers to the power of plus.

Find n.

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9a
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Consider the expansion of open parentheses 4 x minus 2 close parentheses to the power of 4.

Find the term in x to the power of 4 in the expansion.

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9b
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Hence find the term in x to the power of 6 in the expansion of open parentheses 3 x minus 5 close parentheses squared open parentheses 4 x minus 2 close parentheses to the power of 4.

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10a
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Consider the expansion of open parentheses 3 over 2 x minus 5 close parentheses to the power of 6.

Find the term in x cubed in the expansion.

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10b
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Hence find the term in x to the power of 4 in the expansion of open parentheses x minus 2 close parentheses open parentheses 3 over 2 x minus 5 close parentheses to the power of 6.

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11a
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Consider the binomial expansion of  fraction numerator 1 plus x squared over denominator 1 minus x squared end fraction.​

Find the first four terms, in ascending powers of x, of the expansion.

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11b
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State the interval of convergence for the complete expansion.

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11c
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By substituting an appropriate value into the expression found in part (a), find an approximation for the value of ​101 over 99.

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12a
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Consider the binomial expansion of  ​​square root of fraction numerator x over denominator 2 open parentheses x plus 3 close parentheses end fraction end root.

Write down the first three terms, in ascending powers of x, of the expansion.

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12b
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State the interval of convergence for the complete expansion.

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12c
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Using the expansion found in part (a), find an approximation for the value of ​fraction numerator 1 over denominator square root of 8 end fraction ,  giving your answer as an exact value in as simple a form as possible.

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13
Sme Calculator8 marks

Consider the binomial expansion of  open parentheses 1 plus 2 x close parentheses open parentheses 1 minus a x close parentheses to the power of 2 over 3 end exponent space comma space a element of straight natural numbers.

Given that the coefficient of the term in x squared is  negative 5,  find

i)
the value of a

ii)
the coefficient of the term in x

iii)
the first three terms of the expansion, in ascending powers of x.

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14a
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Consider the binomial expansion of  ​1 over open parentheses 2 x plus 3 close parentheses to the power of n ,  where n space element of space straight integer numbers to the power of plus.

For the case where the coefficient in x is negative 2 over 27,  show that n equals 3 to the power of n minus 2 end exponent.

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14b
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For the value of n found in part (a), find the coefficient of x squared. 

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15a
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Consider the binomial expansion of 1 over open parentheses 2 x squared plus x minus 3 close parentheses squared.

Show that  1 over open parentheses 2 x squared plus x minus 3 close parentheses squared  can be written in the form open parentheses 2 x plus a close parentheses to the power of negative 2 end exponent open parentheses x plus b close parentheses to the power of negative 2 end exponent, and find the values of a and b.

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15b
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Hence, or otherwise, find the first three terms of the expansion, in ascending powers of x.

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1
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Given that open parentheses 2 plus n x close parentheses squared open parentheses 1 minus 2 x close parentheses to the power of n equals 4 minus 24 x plus space horizontal ellipsis space


Find the value of n.

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2
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Given that open parentheses 1 plus n x close parentheses squared open parentheses 1 plus fraction numerator 2 x over denominator 3 end fraction close parentheses to the power of n equals 1 plus 40 x horizontal ellipsis

Find the value of n.

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3a
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Consider the expansion open parentheses 5 plus x close parentheses to the power of 5.

Write down and simplify the expansion in descending powers ofspace x.

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3b
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Hence, find the exact value of open parentheses 5.1 close parentheses to the power of 5.

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4a
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Consider the expansion open parentheses 2 minus x close parentheses cubed.

Write down and simplify the expansion in descending powers of x.

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4b
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Hence find the exact value of open parentheses 1.8 close parentheses cubed.

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5a
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Given that open parentheses 1 minus 2 x close parentheses squared open parentheses 1 plus y x close parentheses cubed equals 1 plus z x plus 40 x squared plus horizontal ellipsis plus k y cubed x to the power of 5.

Determine the value of k.

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5b
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Find the possible values of y and z.

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6a
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Given that open parentheses 1 minus 2 a x close parentheses cubed open parentheses 1 plus 3 x close parentheses cubed equals 1 plus b x minus 27 x squared plus horizontal ellipsis plus k a cubed x to the power of 6.

Determine the value of k.

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6b
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Find the possible values of a and b.

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7
Sme Calculator6 marks

In the expansion of 2 x squared open parentheses 3 plus k x close parentheses to the power of 7, the coefficient of the term in x to the power of 5 is 210. 

Find the value of k.

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8
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Consider the expansion of open parentheses x cubed over a plus 3 x to the power of 5 close parentheses to the power of 9 comma space a greater than 0. The coefficient of the x to the power of 39 term is five times the coefficient of the x to the power of 31 term. 

Find a, giving your answer to 3 significant figures.

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9
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Consider the expansion of open parentheses 2 x cubed minus k over x squared close parentheses to the power of 12, where k greater than 0. The coefficient of the term in x to the power of 6 is equal to the coefficient of the term in x to the power of 16 . 

Find k.

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10
Sme Calculator8 marks

The coefficient of the x to the power of 5 term in the expansion of left parenthesis 1 plus 2 x right parenthesis to the power of 4 left parenthesis 1 minus p x right parenthesis cubed is negative 120.

Find the value of p.

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11
Sme Calculator8 marks

Consider the binomial expansion of fraction numerator 1 plus x squared over denominator 1 minus x squared end fraction.

Find the first four terms, in ascending powers of x, of the expansion.

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12
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Find the coefficient of the term in x cubed in the expansion of open parentheses negative 2 x squared plus 7 x minus 3 close parentheses to the power of negative 1 end exponent.

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13a
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Consider the identity  ​fraction numerator 1 space minus space 7 x over denominator open parentheses x space plus space 2 close parentheses left parenthesis 3 space minus space x right parenthesis end fraction equals fraction numerator A over denominator x space plus space 2 end fraction plus fraction numerator B over denominator 3 space minus space x end fraction ,  where A and B are constants to be determined.

Find the values of Aand B.

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13b
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Hence, or otherwise, find the binomial expansion of ​ fraction numerator 1 space minus space 7 x over denominator open parentheses x space plus space 2 close parentheses left parenthesis 3 space minus space x right parenthesis end fraction ,  in ascending powers of x, up to and including the term in x squared.

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13c
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State the interval of convergence for the expansion found in part (b).

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14a
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Consider the binomial expansion of square root of 2 to the power of n open parentheses 1 minus x close parentheses to the power of n plus 1 end exponent end root , where n element of straight integer numbers.

Given that the coefficient in x squared is ​3 over 16  , show that 

2 to the power of fraction numerator n plus 2 over denominator 2 end fraction end exponent open parentheses n squared minus 1 close parentheses equals 3

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14b
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Given also that the constant term is ​1 half  , find

i)
the value of n

ii)
the first three terms of the expansion square root of 2 to the power of n open parentheses 1 minus x close parentheses to the power of n plus 1 end exponent end root, in ascending powers of x.

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