DP IB Maths: AA HL

Topic Questions

1.6 Binomial Theorem

1
Sme Calculator
4 marks

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.

Did this page help you?

2a
Sme Calculator
1 mark

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.

2b
Sme Calculator
4 marks

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

Did this page help you?

3a
Sme Calculator
3 marks

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.

3b
Sme Calculator
4 marks

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

Find the value of a.

Did this page help you?

4a
Sme Calculator
2 marks

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.

4b
Sme Calculator
5 marks

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.

Did this page help you?

5a
Sme Calculator
4 marks

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.

5b
Sme Calculator
4 marks

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

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

Did this page help you?

6
Sme Calculator
6 marks

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.

Did this page help you?

7a
Sme Calculator
4 marks

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.

7b
Sme Calculator
4 marks

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

Did this page help you?

8
Sme Calculator
5 marks

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.

Did this page help you?

9a
Sme Calculator
2 marks

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.

9b
Sme Calculator
3 marks

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.

Did this page help you?

10a
Sme Calculator
3 marks

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.

10b
Sme Calculator
5 marks

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.

Did this page help you?

11a
Sme Calculator
4 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.

11b
Sme Calculator
1 mark

State the interval of convergence for the complete expansion.

11c
Sme Calculator
3 marks

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

Did this page help you?

12a
Sme Calculator
5 marks

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.

12b
Sme Calculator
1 mark

State the interval of convergence for the complete expansion.

12c
Sme Calculator
3 marks

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.

Did this page help you?

13
Sme Calculator
8 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.

Did this page help you?

14a
Sme Calculator
4 marks

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.

14b
Sme Calculator
4 marks

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

Did this page help you?

15a
Sme Calculator
2 marks

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.

15b
Sme Calculator
6 marks

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

Did this page help you?

1
Sme Calculator
3 marks

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.

Did this page help you?

2
Sme Calculator
3 marks

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.

Did this page help you?

3
Sme Calculator
4 marks

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.

Did this page help you?

4
Sme Calculator
3 marks

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.

Did this page help you?

5
Sme Calculator
4 marks

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.

Did this page help you?

6
Sme Calculator
3 marks

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.

Did this page help you?

7a
Sme Calculator
1 mark

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.

7b
Sme Calculator
4 marks

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.

Did this page help you?

8a
Sme Calculator
3 marks

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.

8b
Sme Calculator
3 marks

Find the value of the constant term.

Did this page help you?

9
Sme Calculator
4 marks

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.     

Did this page help you?

10a
Sme Calculator
2 marks

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

Write down the first four terms.

10b
Sme Calculator
2 marks

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

10c
Sme Calculator
2 marks

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

Did this page help you?

11a
Sme Calculator
4 marks

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

Write down the first three terms.

11b
Sme Calculator
2 marks

State the interval of convergence for the complete expansion.

11c
Sme Calculator
2 marks

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

Did this page help you?

12a
Sme Calculator
4 marks

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.

12b
Sme Calculator
2 marks

State the interval of convergence for the complete expansion.

Did this page help you?

1
Sme Calculator
7 marks

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.

Did this page help you?

2
Sme Calculator
7 marks

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.

Did this page help you?

3a
Sme Calculator
3 marks

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.

3b
Sme Calculator
3 marks

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

Did this page help you?

4a
Sme Calculator
3 marks

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

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

4b
Sme Calculator
3 marks

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

Did this page help you?

5a
Sme Calculator
2 marks

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.

5b
Sme Calculator
7 marks

Find the possible values of y and z.

Did this page help you?

6a
Sme Calculator
2 marks

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.

6b
Sme Calculator
7 marks

Find the possible values of a and b.

Did this page help you?

7
Sme Calculator
6 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.

Did this page help you?

8
Sme Calculator
7 marks

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.

Did this page help you?

9
Sme Calculator
6 marks

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.

Did this page help you?

10
Sme Calculator
8 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.

Did this page help you?

11
Sme Calculator
8 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.

Did this page help you?

12
Sme Calculator
8 marks

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.

Did this page help you?

13a
Sme Calculator
3 marks

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.

13b
Sme Calculator
3 marks

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.

13c
Sme Calculator
2 marks

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

Did this page help you?

14a
Sme Calculator
4 marks

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

14b
Sme Calculator
4 marks

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.

Did this page help you?