EXAMSHARE

Question 1

1. Solve the integral $\int e^{\mathrm{5x}}\sin \mathrm{3x}\mathrm{dx}$ (7.5 marks)
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2. Integrate by partial function (10 marks)
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Question 2

1. Solve the integral $\int x^3{e}^{\mathrm{2x}}\mathrm{dx}$ (7.5 marks)
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2. Prove that cos x = 1 $-\frac{x^2}{\mathrm{2!}}+\frac{x^4}{\mathrm{4!}}-\frac{x^6}{\mathrm{6!}}+\mathrm{...}$ and that the series is valid for all values of x (10 marks)
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Question 3

Solve the following equations using D operator

1. ($D^2+4D-3$)($e^{\mathrm{2x}}$)
2. $\frac{1}{{\mathrm{(D}}^2+\mathrm{4)}}$($e^{\mathrm{-3x}}$)
3. ($D^2-\mathrm{7D}+2$)($e^{\mathrm{x/2}}$)
4. $\frac{1}{D^2-\mathrm{3D}-2}$($e^{\mathrm{5x}}$)
5. ($D+4$)($e^{\mathrm{3x}}$)

Question 4

Find $\frac{\mathrm{\partial z}}{\mathrm{\partial x}}$ and $\frac{\mathrm{\partial z}}{\mathrm{\partial x}}$:

1. z = 4x² + 3xy + 5y²
2. z = (3x + 2xy)(4x - 5y)
3. z = tan(3x + 4y)
4. z =$\frac{\sin \mathrm{(3x}+\mathrm{2y)}}{\mathrm{xy}}$

Question 5

Solve the following equations by operator D method

1. ($D^2-5D-4$)($x^2+\mathrm{4x}+1$)
2. ($D^2-7D+3$)($\sin \mathrm{2x}+3\cos \mathrm{2x}$)
3. ($D^2-3D+6$)(${\mathrm{4e}}^{\mathrm{2x}}$)
4. $\frac{1}{D}$($2x^2+8+\frac{3}{x}$)
5. $\frac{1}{D^2}$($3x^2+\cos \mathrm{2x}$)

Question 6

Find all the first and second partial differential coefficients for each of the following function:

1. z = 3x² + 2xy + 4y² (3.5 marks)
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2. z =$\frac{x+y}{x-y}$(7 marks)
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3. If z = 5x² + 3x²y4y³, find $\frac{\mathrm{\partial z}}{\mathrm{\partial x}}$, $\frac{\mathrm{\partial z}}{\mathrm{\partial y}}$, $\frac{{\mathrm{\partial }}^2\mathrm{z}}{\mathrm{\partial }{\mathrm{x}}^2}$, $\frac{{\mathrm{\partial }}^2\mathrm{z}}{\mathrm{\partial }{\mathrm{y}}^2}$, $\frac{{\mathrm{\partial }}^2\mathrm{z}}{\mathrm{\partial x}\mathrm{\partial y}}$ and $\frac{{\mathrm{\partial }}^2\mathrm{z}}{\mathrm{\partial y}\mathrm{\partial x}}$
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