A LevelMaths: PureOCRExam Questions4. Sequences and Series4.1 Binomial Expansion

Binomial Expansion (OCR A Level Maths: Pure)

Exam Questions

2 hours39 questions
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1
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Evaluate

(i)
4 factorial
(ii)
 C presuperscript 5 subscript 2
(iii)
 C presuperscript 6 subscript 3

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2
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Show that, for all values of k,

      C presuperscript k subscript 1 space equals k

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3
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Expand left parenthesis x plus 2 right parenthesis to the power of 4.

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4
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Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 3 plus 2 x right parenthesis to the power of 8.

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5
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Find the coefficient of the x squared term in the expansion of left parenthesis 2 minus x right parenthesis to the power of 5.

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6
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Expand .left parenthesis 2 x minus 3 right parenthesis to the power of 6.

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7
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In the expansion of left parenthesis p plus x right parenthesis to the power of 12, the coefficient of the x to the power of 5 term is 12 976 128.
Find the value of p.

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8a
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Find the first three terms in the expansion of left parenthesis 5 plus 2 x right parenthesis to the power of 5.

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8b
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 Use your answer to part (a) to estimate the value of left parenthesis 5.04 right parenthesis to the power of 5.

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9
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In the expansion of left parenthesis p plus x right parenthesis to the power of 4, where p is a non-zero constant, the coefficient of the x squared term is twice the coefficient of the x term.  Find the value of p.

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1
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Expand left parenthesis 2 plus x right parenthesis to the power of 4.

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2
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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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3a
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Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 3 plus x right parenthesis to the power of 4.

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3b
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Use your answer to part (a) to estimate left parenthesis 3.1 right parenthesis to the power of 4.

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4
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In the expansion of left parenthesis a minus x right parenthesis to the power of 4, the coefficient of the x squared term is 96.
Given that a greater than 0, find the value of a.

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5a
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Find the first three terms in the expansion of left parenthesis 9 minus 2 x right parenthesis to the power of 5.

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5b
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Use your answer to part (a) to estimate left parenthesis 8.9 right parenthesis to the power of 5.

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6
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In the expansion of left parenthesis a minus 2 x right parenthesis to the power of 5, 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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7
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In the expansion of left parenthesis 3 plus p x right parenthesis 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 of p.

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8a
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Find the first three terms in the expansion of left parenthesis 3 plus 2 x right parenthesis to the power of 8.

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8b
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Given that x is small such that x cubedand higher powers of x can be ignored show that

         left parenthesis 1 plus x right parenthesis left parenthesis 3 plus 2 x right parenthesis to the power of 8 almost equal to 6561 plus 41553 x plus 116640 x squared 
                 

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9
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In the expansion of left parenthesis p plus q x right parenthesis to the power of 5, the coefficients of the x squared term and the x cubed term are equal.

Find p in terms of q.

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10a
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In the expansion of left parenthesis a plus b x right parenthesis to the power of 4, the coefficient of the x squared term is equal to the coefficient of the x cubed term.

Show that a over b equals 2 over 3.

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10b
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Given that a and b are integers, and that 10 less than b less than 15, find the values of a and b.

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1
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Fully expand left parenthesis 4 minus x right parenthesis to the power of 4.

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2
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Fully expand  left parenthesis 2 minus begin inline style 1 third end style x right parenthesis to the power of 4.

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3
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Find the coefficient of the term in x to the power of 4 in the expansion of left parenthesis 3 plus 2 x right parenthesis to the power of 9.

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4a
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Find the first three terms, in ascending powers of x, in the expansion of left parenthesis 5 minus 2 x right parenthesis to the power of 4.

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4b
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Use your answer to part (a) to estimate left parenthesis 4.5 right parenthesis to the power of 4.

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5
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In the expansion of left parenthesis 4 minus p x right parenthesis to the power of 6, the coefficient of the x to the power of 4 term is 19 440.
Given that p is a positive integer find the value of p.

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6
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In the expansion of left parenthesis 3 a minus 2 x right parenthesis to the power of 6 comma the coefficient of the x cubed term is equal to the coefficient of the x to the power of 4 term.  Find the value of a.

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7a
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Find the first three terms in the expansion of left parenthesis 2 minus 3 x right parenthesis to the power of 7.

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7b
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Given that x is small such that x cubed and higher powers of x can be ignored show that

         left parenthesis 1 minus 2 x right parenthesis left parenthesis 2 minus 3 x right parenthesis to the power of 7 almost equal to 128 minus 1600 x plus 8736 x squared 
                 

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8
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In the expansion of left parenthesis p plus q x right parenthesis to the power of 8, the coefficients of the x squared term and the x to the power of 6 term are equal.

Find p in terms of q.

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9
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In the expansion of left parenthesis 1 plus x right parenthesis to the power of n comma the coefficient of the x cubed term is 84.

Find the value of n.

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10
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In the expansion of left parenthesis a plus b x right parenthesis to the power of 4 comma the coefficient of the x cubed term is 216.

In the expansion of left parenthesis a plus b x right parenthesis to the power of 6 comma the coefficient of the x to the power of 4 term is 4860.

Find the possible values of a and b.

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1
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Expand .left parenthesis 3 minus 2 x right parenthesis to the power of 5

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2
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Find the coefficient of the term in x to the power of 4 in the expansion of left parenthesis 4 minus 3 x right parenthesis to the power of 7..

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3
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Given that C presuperscript n subscript 3 equals 35 find the value of n.

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4a
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Use the first three terms, in ascending powers of x, in the expansion of left parenthesis 3 minus 5 x right parenthesis to the power of 4 to find an approximation for left parenthesis 2.6 right parenthesis to the power of 4.

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4b
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Using your calculator, find the percentage error in the approximation from part (a) to the exact value of left parenthesis 2.6 right parenthesis to the power of 4.

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5
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In the expansion of left parenthesis m minus 1 fourth x right parenthesis to the power of 5, the coefficient of the x cubed term is -10.
Find the possible values of m.

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6
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In the expansion of  left parenthesis 3 a plus begin inline style 1 half end style x right parenthesis to the power of 6, the coefficient of the x cubed term is equal to the coefficient of the x to the power of 5 term.  Find the values of a, giving your answers in the form fraction numerator square root of m over denominator n end fraction, where m and n are integers to be found.

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7a
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Find the first three terms in the expansion of left parenthesis 4 minus 3 x right parenthesis to the power of 9.

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7b
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Given that x is small such that x cubed and higher powers of x can be ignored show that

            left parenthesis 3 minus 2 x squared right parenthesis left parenthesis 4 minus 3 x right parenthesis to the power of 9 almost equal to 786432 minus 5308416 x plus 15400960 x squared
                 

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8
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In the expansion of left parenthesis p plus q x right parenthesis to the power of 9, the coefficient of the x cubed term is double that of the x to the power of 5 term.  Find p in terms of q.

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9
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In the expansion of left parenthesis 1 minus 3 x right parenthesis to the power of n comma the coefficient of the x cubed term is -3240.

Find the value of n.

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10
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In the expansion of left parenthesis a plus b x right parenthesis to the power of 8 comma the coefficient of the x to the power of 5 term is -870 912.

In the expansion of left parenthesis a plus b x right parenthesis to the power of 12 comma the coefficient of the x cubed term is -1 557 135 360.

Find the possible values of a and b.

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