Further maths Pure

Question 1

Express $r(\cos	heta +i\sin 	heta)$ in the exponential form

Accepted Answer

$re^{i	heta}$

Question 2

Express $\cos	heta$ in the exponential form

Accepted Answer

$\frac{e^{i	heta}+e^{-i	heta}}{2}$

Question 3

Express $i\sin	heta$ in the exponential form

Accepted Answer

$\frac{e^{i	heta}-e^{-i	heta}}{2}$

Question 4

State De Moivre's theorem

Accepted Answer

$z^n=r^n(\cos n	heta+i\sin n	heta)$

Question 5

What is the method for evaluating an indefinite integral?

Accepted Answer

For the value for which the integrand is undefined, substitute the value for a variable and consider the value of the integral as that variable tends to the original value

Question 6

What is the equation for determining the mean value of a function $f(x)$

Accepted Answer

$\mathcal{\overline{f(x)}=\frac{1}{b-a}\int_{a}^{b} f(x).dx}$

Question 7

Express $\mathcal{\frac{px^2+qx+r}{(x+a)(x^2+bx+c)}}$ in partial fractions

Accepted Answer

${\frac{A}{\mathcal{x-a}}+\frac{B\mathcal{x}+C}{\mathcal{x^2+bx+c}}}$

Question 8

What is the solution to $\mathcal{\int{(ax+b)^n}.dx}$

Accepted Answer

$\mathcal{\frac{(ax+b)^{n+1}}{a(n+1)}+c}$

Question 9

What is the solution to $\mathcal{\int{e^{ax+b}.dx}}$

Accepted Answer

$\mathcal{\frac{e^{ax+b}}{a}+c}$

Question 10

What is the solution to $\mathcal{\int{\frac{1}{ax+b}.dx}}$

Accepted Answer

$\mathcal{\frac{1}{a}\ln{|ax+b|}+c}$

Question 11

Give the exponential expansion to $\mathcal{\cosh{x}}$

Accepted Answer

$\mathcal{\frac{e^x+e^{-x}}{2}}$

Question 12

Give the exponential expansion to $\mathcal{\sinh{x}}$

Accepted Answer

$\mathcal{\frac{e^x-e^{-x}}{2}}$

Question 13

Give the exponential expansion to $\mathcal{	anh{x}}$

Accepted Answer

$\mathcal{\frac{e^x-e^{-x}}{e^x+e^{-x}}}$

Further maths Pure

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