Mathematics for Chemistry/Tests and Exams

A possible final test with explanatory notes
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
(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

(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
(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.