University Maths Solution

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Maths Question
Question 1

$\text{Verify that}{{\text{ }}^{18}}{{C}_{7}}{{+}^{18}}{{C}_{6}}{{=}^{19}}{{C}_{7}}$ 

Question 2

$\text{Show that}{{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{n+1}}{{C}_{r}}$

Question 3

$\text{Prove that}{{\text{ }}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}{{+}^{n}}{{C}_{2}}+--{{-}^{n}}{{C}_{n}}={{2}^{n}}$

Question 4

$\text{Write down the expansion of }{{(1-2x)}^{7}}$

Question 5

$\text{Write down the expansion of }{{(2x+3y)}^{5}}$

Question 6

$\text{Write down the expansion of }{{(3+x)}^{8}}\left| \text{Hint:}(3+x)=3(1+\tfrac{x}{3}) \right.$

Question 7

$\begin{align}  & \text{Expand }{{(1+3x)}^{-\tfrac{1}{2}}} \\ & \text{Giving the first four terms and stating the series is valid} \\\end{align}$

Question 8

$\begin{align}  & \text{Expand }\sqrt{9+{{x}^{2}}} \\ & \text{Giving the first four terms and stating the series is valid} \\\end{align}$

Question 9

$\begin{align}  & \text{Expand }{{(5x-1)}^{-2}} \\ & \text{Giving the first four terms and stating the series is valid} \\\end{align}$

Question 10

$\text{Calculate the coefficient of }{{x}^{3}}{{y}^{4}}\text{ in }{{(2x-3y)}^{7}}$

Question 11

$\text{Find the term independent of }x\text{ in the expansion }{{\left( 2x-\frac{3}{{{x}^{2}}} \right)}^{6}}$

Question 12

$\text{Find the term independent of }x\text{ in the expansion of }{{\left( {{x}^{2}}-\frac{2}{3x} \right)}^{9}}$

Question 13

$\begin{align}  & \text{If }x\text{ is so small that we can neglect terms in }{{x}^{4}}\text{ and higher power of }x,\text{ } \\ & \text{show that }\sqrt[3]{\frac{1-2x}{1+2x}}=1-\frac{4}{5}x+\frac{8}{9}{{x}^{2}}-\frac{170}{81}{{x}^{3}} \\\end{align}$

Question 14

$\begin{align} & \text{Express }\frac{1}{(x-1)(x+2)}\text{ as sum of two partial fractions}\text{. Hence, write down the } \\ & \text{expansion of fraction as a series when }x\text{ is small, up to the term in }{{x}^{2}}.\text{ } \\ & \text{For values of }x\text{ is the expansion valid?} \\ \end{align}$

Question 15

$\begin{align}
& \text{Express }f(x)=\frac{7{{x}^{2}}+12x+4}{(x+2){{(x+1)}^{2}}}\text{ in the form }\frac{A}{x+2}+\frac{B}{x+1}+\frac{C}{{{(x+1)}^{2}}} \\
& \text{Where }A,B,C\text{ are independent of }x.\text{ Hence or otherwise show that }{{x}^{2}}<1,\text{ } \\
& \text{the first terms in the expansion of }f(x)\text{ as a series of ascending powers of }x\text{ } \\
& \text{are }2+x-3{{x}^{2}} \\
\end{align}$

Question 16

$\begin{align}
& \text{Show that if }x\text{ is small that }{{x}^{3}}\text{ and higher powers of }x\text{ can be neglected,} \\
& \sqrt{\frac{1+x}{1-x}}=1+x+\frac{1}{2}{{x}^{2}} \\
& \text{By putting }x=\frac{1}{7},\text{ show that }\sqrt{3}\cong \frac{196}{113} \\
\end{align}$

Question 17

$\begin{align}  & \text{Express }\frac{1}{(x+2)(2x+1)}\text{ in partial fraction and hence expand in } \\ & \text{ascending powers of }x,\text{ giving the four terms and the coefficient of }{{x}^{n}} \\\end{align}$

Question 18

$\begin{align}  & \text{Find the series expansion }f(x)={{\left( \frac{2-x}{1+3x} \right)}^{\tfrac{1}{2}}}\text{ in ascending powers of }x \\ & \text{ as far as the terms in }{{x}^{2}}.\text{ For what values of }x\text{ may }f(x)\text{ be used to } \\ & \text{obtain an approximation to }\sqrt{3} \\\end{align}$

Question 19

$\begin{align}  & \text{Express the function }f(x)=\frac{11{{x}^{2}}-12x+4}{{{(x-1)}^{2}}(4x-1)}\text{ in partial fractions}\text{. } \\ & \text{Hence or otherwise expand }f(x)\text{ in the form} \\ & f(x)={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdot \cdot \cdot +{{a}_{n}}{{x}^{n}}\cdot \cdot \cdot  \\ & \text{State the value of }x\text{ for which the expansion is valid} \\\end{align}$