Question

Find the first three terms,in ascending powers of x,in the expansion of

(i)$$\left ( 1-x \right )^{6}$$

(ii)$$\left ( 1+2x \right )^{6}$$

(ii)Hence find the coefficient of $$x^{2}$$  in the expansion of $$\left [ \left ( 1-x \right )\left ( 1+2x \right ) \right ]^{6}$$

(i)(a)$$(1-x)^{6}=1-6x+15x^{2}$$

(b)$$\left ( 1+2x \right )^{6}=1+12x+60x^{2}$$

(ii) Product of (a) and (b)

$$\rightarrow 60-72+15=3$$

Question

Find the term independent of x in the expansion of $$\left ( 4x^{3} +\frac{1}{2x}\right )^{8}$$

$$\left [ ^{8}C_{6} \right ]\times \left [ 16 \right ]\times \left ( x^{6} \right )\times \left [ \frac{1}{\left ( 2^{6} \right )\left ( x^{6} \right )} \right ]$$

7

Question

Find the coefficient of $$x^{6}$$ in the expansion of $$\left ( 2x^{3} -\frac{1}{x^{2}}\right )^{7}$$.

$$\left [ ^{7} C_{3}\right ]\times \left [ \left ( 2x^{3} \right ) ^{4}\right ]\times \left [ \left ( -\frac{1}{x^{2}} \right ) ^{3}\right ]$$

$$35\times 2^{4}\times (-1)^{3}$$ leading to their answer

$$-560\left ( x^{6} \right )$$ as answer

Question

Find the term independent of x in the expansion of $$\left ( 2x+\frac{1}{2x^{3}} \right )^{8}$$.

$$^{8}C_{6}\left ( 2x \right )^{6}\left ( \frac{1}{2x^{3}} \right )^{2}$$

$$28\times 64\times \frac{1}{4}$$ (powers and factorials evaluated)

448

Question

(i) Find the first three terms when $$\left ( 2+3x \right )^{6}$$ is expanded in ascending powers of x.

(ii) In the expansion of $$\left ( 1+ax \right )\left ( 2+3x \right )^{6}$$ , the coefficient of x is zero. Find the value of a.

(i)$$64+576x+2160x^{2}$$

(ii)$$576a(x^{2})+216(x^{2})=0$$

$$a=-\frac{2160}{576}$$

$$\left ( -\frac{15}{4} \right )$$ or -3.75

Question

In the expansion of $$\left ( 1-\frac{2x}{a} \right )\left ( a+x \right )^{5}$$, where a is a non-zero constant, show that the coefficient of $$x^{2}$$  is zero.

$$(a+x)^{5}=a^{5}+^{5}C_{1}a^{4}x+^{5}C_{2}a^{3}x^{2}+….$$

$$\left ( -\frac{2}{a}\times (their 5a^{4})+(their 10a^{3}) \right )(x^{2})$$

0

Question

In the expansion of $$\left ( 2+ax \right )^{7}$$, the coefficient of x is equal to the coefficient of $$x^{2}$$. Find the value of the non-zero constant a.

$$^{7}C_{1}\times 2^{6}\times a=^{7}C^{2}\times 2^{5}\times a^{2}$$

a=\left ( \frac{7\times 2^{6}}{21\times 2^{5}} \right )=\frac{2}{3}

#### Question.

The coefficients of $$x^2$$ and $$x^3$$ in the expansion of $$(3 − 2x)^6$$ are a and b respectively. Find the value of $$\frac{a}{b}$$.

$$\left ( 3-2x \right )^{6}$$

Coefficient of $$x^{2}=3^{4}+(-2)^{2}\times _{6}C_{2}=a$$

Coefficient of x^{3}=3^{3}+(-2)^{3}\times _{6}C_{3}=b

#### Question.

Find the term independent of x in the expansion of $$(x –\frac{3}{2x})^2$$.

$$\left ( x-\frac{3}{2x} \right )^{6}$$

Term is $$^{6}C_{3}\times x^{3}\times \left ( \frac{-3}{2x} \right )3$$

$$\rightarrow -67.5$$ oe

#### Question.

The function f is defined for x≥ 0 by $$f(x) =(4x + 1)^\frac{3}{2}$$.

(i) Find f′(x) and f′′(x).

The first, second and third terms of a geometric progression are respectively f(2), f′(2) and kf′′(2).
(ii) Find the value of the constant k.

(i) $$f{}’\left ( x \right )=\left [ \frac{3}{2}\left ( 4x+1 \right )^{\frac{1}{2}} \right ]\left [ 4 \right ]$$

$$f{}”\left ( x \right )=6\times \frac{1}{2}\times \left ( 4x+1 \right )^{-\frac{1}{2}}\times 4$$

(ii) f(2),$$f{}’\left ( 2 \right )$$,$$kf{}”\left ( 2 \right )=27$$,4k OR 12

$$\frac{27}{18}=\frac{18}{4k}$$ oe OR $$kf{}”\left ( 2 \right )=12\Rightarrow k=3$$

#### Question.

In the expansion of $$\left ( \frac{1}{ax}+2ax^{2} \right )^{5}$$ the coefficient of x is 5. Find the value of the constant a.

$$^{5}C_{2}\left ( \frac{1}{ax} \right )^{3}\left ( 2ax^{2} \right )^{2}$$

$$10\times \frac{1}{a^{3}}\times 4a^{2}=5$$ soi

a=8 cao

#### Question.

(i) Find the coefficients of $$x^{4}$$ and  $$x^{5}$$ in the expansion of $$(1 – 2x)^{5}$$.

(ii) It is given that, when $$(1 + px)(1 – 2x)^{5}$$ is expanded, there is no term in $$x^{5}$$. Find the value of the constant p.

Ans:(i) $$80(x^4)$$, $$-32(x^5)$$

(ii) $$(-32 +80p)(x^{5})=0$$

$$p=\frac{2}{5}$$ or  $$\frac{32}{80}$$

Question

The term independent of x in the expansion of $$\left ( 2x+\frac{k}{x} \right )^{6}$$ ,, where k is a constant, is 540.

(i) Find the value of k.

(ii) For this value of k, find the coefficient of $$x^{2}$$ in the expansion.15\times $$16\times k^{2}=(or540x^{2})$$

(i)  Ind term =$$(2x) ^{3}\times (\frac{k}{x})^{3}\times _{6}C^{3}\textrm{}=540\rightarrow K=1^{\frac{1}{2}}$$

(ii) Term, in $$15\times 16\times k^{2}$$=$$(or540x^{2})$$

Question

The coefficient of $$x^{3}$$ in the expansion of $$(1-px)^{5}$$ is −2160. Find the value of the constant p.

$$_{3}^{5}\textrm{C}\left [ (-) (px)^{3}\right ](-1)10p^{3}=-2160$$

then divide and take cube root,

$$p=6$$

### Question

In the expansion of $$(2x^{2}+\frac{1}{x})^{6}$$, the coefficients of x6 and x3 are equal.
(a) Find the value of the non-zero constant a.                                                                                 [4]

(b) Find the coefficient of x6 in the expansion of $$(1-x^{3})(2x^{2}+\frac{a}{x})^{6}$$    [1]

Ans

5 (a) $$6C2\times \left [ 2(x^{2}) \right ]^{4}\times \left [ \frac{a}{(x)} \right ]^{2}, \ 6C3\times \left [ 2(x^{2}) \right ]^{3}\times \left [ \frac{a}{(x)}^{3} \right ]$$

$$15\times 2^{4}\times a^{2}=20\times 2^{3}\times a^{3}$$

$$a=\frac{15\times 2^{4}}{20\times 2^{3}}=\frac{3}{2}$$

5 (b) 0

### Question.

A curve has equation $$y = kx2 + 2x − k$$ and a line has equation $$y = kx − 2$$, where k is a constant. Find the set of values of k for which the curve and line do not intersect.

kx2 + 2x – k = kx – 2  leading to kx2 + (-k + 2) x – k + 2 [=0]

(-k + 2)2 – 4k( – k + 2)

5k2 – 12k + 4 or (-k + 2) (-k + 2 -4k)

(- k + 2) (-5k + 2)

$$\frac{2}{5} < k < 2$$

### Question

The coefficient of $$\frac{1}{x}$$ in the expansion of $$(2x+\frac{a}{x^2})^5$$ is 720.
(a) Find the possible values of the constant a.
(b) Hence find the coefficient of $$\frac{1}{x^7}$$ in the expansion.

Ans:

(a) $$5C2[2(x)]^3[\frac{a}{(x^2)}]^2$$
$$10 \times 8 \times a^2 (\frac{x^3}{x^4})=720(\frac{1}{x})$$
a = ±3

(b) $$5C4[2(x)][\frac{their a}{(x^2)}]^4$$
810 identified

### Question

(a) Find the first three terms in the expansion of (3 − 2x)5 in ascending powers of x.     [3]

(b) Hence find the coefficient of x2 in the expansion of (4 + x)2 (3 − 2x)5

Ans

3 (a) 243
−810x
+1080x2
3 (b) (4 +x)2 = 16 + 8x +x2
Coefficient of x2 is 16 × 1080 + 8 × (−810) + 243
11043

### Question

a) Find the first three terms in the expansion, in ascending powers of x, of (1+x)5.
b) Find the first terms in the expansion, in ascending powers of x, (1-2x)6.
c) Hence find the coefficient of x2 in the expansion of (1+x)5(1-2x)6.

Ans:
a) 1-5x+10x2
b) 1-12x+60x2
10

### Question

Find the term independent of x in each of the following expansions.

(a)  $$\left ( 3x + \frac{2}{x^{2}} \right )^{6}$$

(b) $$\left ( 3x + \frac{2}{x^{2}} \right )^{6}$$ $$\left ( 1 – x^{3} \right )$$

(a) $$6_{C_{2}}\times \left ( 3x \right )^{4} \left ( \frac{2}{x^{2}} \right )^{2}$$

$$15 \times 3^{4}\times 2^{2}$$

4860

(b) Their 4860 and one other relevant term

Other term = $$6 C3 \left ( 3x \right )^{3}\left ( \frac{2}{x^{2}}\right )^{3} or 6C3\times 3 ^{3} \times 2^{3} or 4320$$

[4860‒ 4320 =] 540

### Question.

The coefficient of $$\frac{1}{x}$$ in the expansion of $$(kx+\frac{1}{x})^{5}+(1-\frac{2}{x})^{8}$$ is 74. find the value of positive constant k.

$$\left ( kx+\frac{1}{x} \right )^{5}+\left ( 1-\frac{2}{x} \right )^{8}$$

Coefficient in $$\left ( kx+\frac{1}{x} \right )^{5}=10\times k^{2}$$

Coefficient in $$\left ( 1-\frac{2}{x} \right )^{8}=8\times -2$$

10k² − 16 = 74→k = 3