Mathematics for Chemistry/Tests and Exams

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A possible final test with explanatory notes[edit]

This test was once used to monitor the broad learning of university chemists at the end of the 1st year and is intended to check, somewhat lightly, a range of skills in only 50 minutes. It contains a mixture of what are perceived to be both easy and difficult questions so as to give the marker a good idea of the student's algebra skills and even whether they can do the infamous integration by parts.


(1) Solve the following equation for x

x^2+2x-15 = 0

It factorises with 3 and 5 so : (x+5)(x-3) = 0 therefore the roots are -5 and +3, not 5 and -3!


(2) Solve the following equation for x

2x^2-6x -20 = 0

Divide by 2 and get x^2-3x -10 = 0.

This factorises with 2 and 5 so : (x-5)(x+2) = 0 therefore the roots are 5 and -2.


(3) Simplify

\ln w^6 - 4\ln w

Firstly 6\ln w - 4\ln w so it becomes 2\ln w.


(4) What is

\log_{2} \frac 1 { 64}

64 = 8 x 8 so it also equals 2^3 x 2^3 i.e. \frac 1 { 64} is 2^{-6}, therefore the answer is -6.


(5) Multiply the two complex numbers

3+5i  ~~~~ {\rm and} ~~~~ 3-5i

These are complex conjugates so they are 3^2 minus i^2 x 5^2 i.e. plus 25 so the total is 34.


(6) Multiply the two complex numbers

(5,-2)  ~~~~ {\mathrm and}  ~~~~ (-5,-2)

The real part is -25 plus the 4i^2. The cross terms make -10i and +10i so the imaginary part disappears.


(7) Differentiate with respect to x:

\frac 1 {3x^2}-3x^2

Answer:  ~~~~~~~- \frac 2 {3x^3}-6x


(8)  \frac  6 {x^4} +3  x^3

Answer:  ~~~~~~~ 9  {x^2} - \frac  {24} {x^5}


(9)  \frac{2}{\sqrt x} + 2 \sqrt x

Answer:  ~~~~~~\frac 1 {\sqrt x} - \frac 1 {\sqrt {x^3}}


(10) x^3 (  x - ( 2x + 3 ) (2x - 3) )

Expand out the difference of 2 squares first.....collect and multiply....then just differentiate term by term giving:  ~~~~ 20x^4 -4x^3 + 27x^2


(11) 3x^3  \cos 3x

This needs the product rule.... Factor out the 9x^2 ....  9x^2( \cos 3x - x \sin 3x)


(12) \ln ( 1 - x)^2

This could be a chain rule problem....... \frac 1 { ( 1 - x)^2}  . 2 . (-1) . ( 1 - x )

or you could take the power 2 out of the log and go straight to the same answer with a shorter version of the chain rule to:- \frac 2 {( 1 - x)}.


(13) Perform the following integrations:

\int  \left( 2 \cos^2 \theta + 2 \theta \right) {\rm d}\theta

\cos^2 must be converted to a double angle form as shown many times.... then all 3 bits are integrated giving .......

\cos \theta \sin \theta +  \theta +  \theta^2


(14) \int \left( 8 x^{-3} - \frac 4 x + \frac 8 {x^3} \right) {\rm d}x

Apart from  - \frac 4 x , which goes to \ln, this is straightforward polynomial integration. Also there is a nasty trap in that two terms can be telescoped to \frac {16} { x^3}.

-(\frac 8 {x^2} + 4 \ln x)


(15) What is the equation corresponding to the determinant:

\begin{vmatrix}
b & \frac 1 {\sqrt 2}  & 0\\
\frac 1 {\sqrt 2}& b & 1\\
0 & 1 & b\\
\end{vmatrix}
= 0

The first term is b(b^2-1) the second -\frac 1 {\sqrt 2} (\frac b {\sqrt 2} - 0 ) and the 3rd term zero. This adds up to b^3 -3 /2 b.


(16) What is the general solution of the following differential equation:

\frac {{\rm d} \phi} { {\rm d} r} = \frac  {\rm A} r

where A is a constant..

\theta = A \ln r + k.


(17) Integrate by parts: \int x \sin x  {\rm d} x

Make x the factor to be differentiated and apply the formula, taking care with the signs... \sin x - x \cos x.


(18)The Maclaurin series for which function begins with these terms?

1 + x +  x^2/  2! + x^3 / 3! + x^4/  4! + \dots

It is e^x....


(19)Express

\frac {x-2} {(x-3)(x+4)} as partial fractions.

It is ..... ~~~~~\frac 1 {7(x-3)} + \frac 6 {7(x+4)}


(20) What is 2e^{i4\phi} - \cos 4 \phi in terms of sin and cos

This is just Euler's equation..... 2e^{i4\phi} = 2\cos 4 \phi - 2 i \sin 4 \phi

so one \cos 4 \phi disappears to give ... \cos 4 \phi - 2 i \sin 4 \phi.

50 Minute Test II[edit]

(1) Simplify 2\ln (1 / x^3) + 5\ln x


(2)What is \log_{10} \frac 1 { 10~000}


(3) Solve the following equation for t

t^2-3t-4 = 0


(4) Solve the following equation for w

w^2+4w-12 = 0


(5) Multiply the two complex numbers (-4,3) ~~~~ {\rm and } ~~~~~ (-5,2)


(6) Multiply the two complex numbers 3+2i ~~~~ {\rm and }  ~~~~~~ 3-2i


(7) The Maclaurin series for which function begins with these terms?

x - x^3/  6 + x^5 / 120 + \dots


(8) Differentiate with respect to x:

x^3 ( 2 - 3x )^2


(9) \frac {\sqrt x} 2 - \frac {\sqrt 3} {2 \sqrt x}


(10)x^4-3x^2+{\rm k}

where k is a constant.


(11) \frac 2 {3x^4}-{\rm A}x^4

where A is a constant.


(12) 3x^3 e^{3x}


(13) \ln ( 2 - x)^3


(14) Perform the following integrations:

\int \left( 3 w^4 - 2 w^2 + \frac 6 {5 w^2} \right) {\rm d}w


(15) \int  \left( 3 \cos \theta  + \theta \right) {\rm d}\theta


(16) What is the equation belonging to the determinant \begin{vmatrix} x & 0 & 0\\ 0 & x & i \\ 0 & i & x \\ \end{vmatrix} = 0</math>


(17) What is the general solution of the following differential equation:

\frac {{\rm d} y} { {\rm d} x} = k y


(18) Integrate by any appropriate method:

\int \left(  \ln x  + \frac 4 x \right)  {\rm d} x


(19) Express \frac {x+1} {(x-2)(x+2)}

as partial fractions.


(20) What is 2e^{i2\phi} + 2i \sin 2 \phi in terms of sin and cos.

50 Minute Test III[edit]

(1) Solve the following equation for t

t^2-4t-12 = 0


(2) What is \log_{4} \frac 1 { 16}


(3) The Maclaurin series for which function begins with these terms?

1 - x^2/ 2 + x^4 / 24 + \dots
---- (4) Differentiate with respect to x:

\frac  5  {x^2} - 8 {x^4}


(5) \frac 4 {\sqrt x} - {\sqrt 2 x}


(6) 5 {\sqrt x} + \frac  6 {x^3  }


(7) \frac 5 {x^3}-5x^3


(8) x^2 (2x^2 - ( 5 + 2x ) (5 - 2x) )


(9) 2x^2  \sin x


(10) Multiply the two complex numbers (2,3) ~~~~~~ {\rm  and } ~~~~~~ (2,-3)


(11) Multiply the two complex numbers 3 -i ~~~~~ {\rm  and } ~~~~ -3 +i


(12) Perform the following integrations:

\int  \left(  \frac 1 {3x} + \frac 1 {3x^2} -  5 x^{-6}  \right) {\rm d}x


(13)

\int \left( 6 x^{-2} + \frac 2 x - \frac 8 {x^2} \right) {\rm d}x


(14) \int  \left( \cos^2 \theta + \theta \right) {\rm d}\theta


(15) \int  \left( \sin^3 \theta \cos \theta + 2 \theta \right) {\rm d}\theta


(16) Integrate by parts: \int 2 x \cos x  {\rm d} x


(17) What is the equation corresponding to the determinant:

\begin{vmatrix}
x & -1  & 0\\
-1 & x & 0 \\
0 & 0 & x   \\
\end{vmatrix}
= 0


(18) Express \frac {x-1} {(x+3)(x-4)} as partial fractions.


(19)What is the general solution of the following differential equation:

\frac {{\rm d} \theta} { {\rm d} r} = \frac { r} A


(20) What is e^{i2\phi} - 2i \sin 2 \phi in terms of sin and cos.