F.4 M1 questions

2012-12-30 6:01 am

1. If the coefficient of x^3 is triple that of x in the expansion of (1- x/2 )^n, where n is a positive integer, find the value of n.
answer:10
2. Given that the expansion of (a+x)(1-2x)^n in ascending powers of x is 3-41x+bx^2+..., find the values of the constants a, n and b.
answer: a=3, n=7, b=238
3. If n is a positive integer and the coefficient of x^2 in the expansion of (1+x)^n+(1+2x)^n is 75, find the value(s) of n.
answer: 6
4.Given e^x + e^-x =20, find the value of e^2x + e^-2x.
answer: 398
5. Let y=1-e^-3x. Prove that x=1/3ln1/1-y.
6. In the expansion of (x- 3/x)^2(1+2x)^n, the constant term is 210. Find the value of n.
7(a). Given that (1+ax)^n=1-12x+60x^2+bx^3+ terms involving the higher powers of x, find the values of a, b and n.
(b). Using (a) or otherwise, in the expansion of (1+ax)^n(1-x), where a and n are the values found in (a), find:
(i) the coefficient of x^2;
(ii) the coefficient of x^3.

回答 (1)

myisland8132

2012-12-30 7:52 pm

✔ 最佳答案

1 (1 - x/2)^n

= 1 - nx/2 + n(n - 1)x^2/4 - n(n - 1)(n - 2)x^3/48 - ...

3n/2 = n(n - 1)(n - 2)/48

72 = (n - 1)(n - 2)

n = 10

2 (a + x)(1 - 2x)^n = 3 - 41x + bx^2

(a + x)(1 - 2nx + 2n(n - 1)x^2 - ...) = 3 - 41x + bx^2

a = 3

1 - 2na = -41

n = 7

b = 2an(n - 1) - 2n = 238

3 (1 + x)^n + (1 + 2x)^n

The coefficient of x^2 is

n(n - 1)/2 + 2n(n - 1) = 75

5n(n - 1)/2 = 75

n(n - 1) = 30

n = 6

4 exp(2x) + exp(-2x)

= [exp(x) + exp(-x)]^2 - 2

= 400 - 2

= 398

5 y = 1 - exp(-3x)

exp(-3x) = 1 - y

-3x = ln(1 - y)

x = (-1/3)ln(1 - y)

6 (x - 3/x)^2 (1 + 2x)^n

= (x^2 - 6 + 9/x^2)(1 + 2nx + 2n(n - 1)x^2 + ..)

The constant term is

-6 + 18n(n - 1) = 210

n(n - 1) = 12

n = 4

7(a) (1 + ax)^n = 1 - 12x + 60x^2 + bx^3

1 + nax + n(n - 1)a^2 x^2/2 + n(n - 1)(n - 2)a^3 x^3/6 = 1 - 12x + 60x^2 + bx^3

na = -12

n(n - 1)a^2/2 = 60

(n - 1)a = -10 => a = -2, n = 6

b = -160

(b)(i) (1 + ax)^n (1 - x)

= (1 - 2x)^6 (1 - x)

= (1 - 12x + 60x^2 - 160x^3 + ...)(1 - x)

The coefficient of x^2 is 12 + 60 = 72

(ii) The coefficient of x^3 is -160 - 60 = -180

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