Binomial extended [59 marks] 1. [Maximum mark: 8] EXN.2.AHL.TZ0.7 Consider the identity 2+7x ≡ ( 1+2x ) ( 1−x ) A 1+2x + (a) Find the value of A and the value of B. (b) Hence, expand 2+7x ( 1+2x ) ( 1−x ) B 1−x , where A, B ∈ Z [3] in ascending powers of x, up to [4] and including the term in x . 2 (c) Give a reason why the series expansion found in part (b) is not valid for x 2. = 3 4 . [1] [Maximum mark: 6] (a) Write down and simplify the first three terms, in ascending EXM.2.AHL.TZ0.1 1 powers of x, in the Extended Binomial expansion of (1 − x) . [3] By substituting x [3] 3 (b) . = 1 9 find a rational approximation to √9. 3 3. [Maximum mark: 27] EXM.3.AHL.TZ0.1 This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices. A power series in x is defined as a function of the form f (x) = a0 + a1 x + a2 x 2 3 + a3 x +. . . where the a i ∈ R . It can be considered as an infinite polynomial. (a) Expand (1 + x) using the Binomial Theorem. [2] 5 This is an example of a power series, but is only a finite power series, since only a finite number of the a are non-zero. i (b) Consider the power series 1 − x + x 2 − x 3 4 + x −. . . By considering the ratio of consecutive terms, explain why this series is equal to (1 + x) −1 and state the values of x for which this equality is true. (c) [4] Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for (1 + x) (d) . [2] Repeat this process to find the first four terms in a power series for (1 + x) (e) −2 −3 [2] . Hence, by recognising the pattern, deduce the first four terms in a power series for (1 + x) −n ,n ∈ Z + . [3] We will now attempt to generalise further. Suppose (1 + x) a0 + a1 x + a2 x (f ) q 2 , q ∈ Q can be written as the power series 3 + a3 x +. . . By substituting x = 0 . , find the value of a . 0 [1] (g) By differentiating both sides of the expression and then substituting x = 0, find the value of a . [2] (h) Repeat this procedure to find a and a . [4] (i) Hence, write down the first four terms in what is called the Extended Binomial Theorem for (1 + x) , q ∈ Q. [1] (j) Write down the power series for [2] (k) Hence, using integration, find the power series for arctan x, giving the first four non-zero terms. 1 2 3 q 4. 1 2 1+x . [Maximum mark: 5] [4] 22M.1.AHL.TZ1.6 Consider the expansion of (8x 3 − n 1 2x ) where n ∈ Z + . Determine all possible values of n for which the expansion has a non-zero constant term. 5. [Maximum mark: 7] [5] 21N.1.AHL.TZ0.9 Consider the expression 1 − √1 − x √1+ax where a ∈ Q, a ≠ 0 . The binomial expansion of this expression, in ascending powers of x, as far as the term in x is 4bx + bx , where b ∈ 2 2 Q . (a) Find the value of a and the value of b. [6] (b) State the restriction which must be placed on x for this expansion to be valid. [1] 6. [Maximum mark: 6] 18M.2.AHL.TZ2.H_5 (a) 3n + 1 ) as a polynomial in Express the binomial coefficient ( 3n − 2 . [3] n (b) Hence find the least value of n for which ( 3n + 1 ) > 10 3n − 2 © International Baccalaureate Organization, 2023 6 . [3]