Cambridge International AS and A Level Mathematics Pure Mathematics 1

Exercise 3C

P1^

3

ExERCISE 3C 1 Write out the following binomial expansions.

(i) (x + 1)^4 (ii) (1 + x)^7 (iii) (x + 2)^5

(iv) (2x + 1)^6 (v) (2x − 3)^4 (vi) (2x + 3 y)^3

(vii) (^) x
()−x
(^23) (viii) (^) x
x

()+^22

4

(ix) 32 2

5

()x −x

2 Use a non-calculator method to calculate the following binomial coefficients.

Check your answers using your calculator’s shortest method.

(i) 4

2









(ii) 6

2









(iii) 6

3









(iv) 6

4









(v) 6

0









(vi) 12

9









3 In these expansions, find the coefficient of these terms.

(i) x^5 in (1 + x)^8 (ii) x^4 in (1 − x)^10 (iii) x^6 in (1 + 3 x)^12

(iv) x^7 in (1 − 2 x)^15 (v) x^2 in x

x

2

210

()+^

4 (i) Simplify (1 + x)^3 − (1 − x)^3.

(ii) Show that a^3 − b^3 = (a − b)(a^2 + ab + b^2 ).

(iii) Substitute a = 1 + x and b = 1 − x in the result in part (ii) and show that

your answer is the same as that for part (i).

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

of 2 2

4

x

x

()−.

6 Find the first three terms, in ascending powers of x, in the expansion (2 + kx)^6.

7 (i) Find the first three terms, in ascending powers of x, in the expansion

(1 − 2 x)^6.

(ii) Hence find the coefficients of x and x^2 in the expansion of (4 − x)(2 − 4 x)^6.

8 (i) Find the first three terms, in descending powers of x, in the expansion

(^42)
6
x k
x

()−.

(ii) Given that the value of the term in the expansion which is independent of

x is 240, find possible values of k.

9 (i) Find the first three terms, in descending powers of x, in the expansion of

(^) x
x
2

16

()−.

(ii) Find the coefficient of x^3 in the expansion of (^) x
x
2

16

()−.