IBMaths: AA SLDPTopic Questions1. Number & Algebra1.5 Binomial Theorem

Binomial Theorem (DP IB Maths: AA SL)

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1.5 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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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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1
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Given that left parenthesis 2 plus n x right parenthesis squared left parenthesis 1 minus 2 x right parenthesis to the power of n equals 4 minus 24 x plus....

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 space 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...

Find the value of n.

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3a
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Consider the expansion left parenthesis 5 plus x right parenthesis to the power of 5.

Write down and simplify the expansion in descending powers of 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 left parenthesis 2 minus x right parenthesis 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 left parenthesis 1 minus 2 x right parenthesis squared left parenthesis 1 plus y x right parenthesis cubed equals 1 plus z x plus 40 x squared plus midline 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 left parenthesis 1 minus 2 a x right parenthesis cubed left parenthesis 1 plus 3 x right parenthesis cubed equals 1 plus b x minus 27 x squared plus midline 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
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In the expansion of 2 x squared left parenthesis 3 plus k x right parenthesis to the power of 7 comma 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 39term is five times the coefficient of the x to the power of 31term.

Find a comma 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 comma 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
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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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