### Maths for Foundation Science Coursework

**Question 1 **

(a) Express in the form, where *n *is an integer.

(b) Expressin the form , where *n *is an integer.

**Question 2 **

The energy liberated when burning propane C3H8 is 2220 kJ mol-1 and when burning pentane C5H12 is 3540 kJ mol-1. Use the data to calculate the mean enthalpies of the C-C and C-H bonds.

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**Question 3 **

The points A and B have coordinates (6, -1) and (2, 5) respectively.

- a)

(i)Show that the gradient of AB is -3/2.

(ii) Hence find an equation of the line AB, giving your answer in the form

a*x *+ b*y *= c, where a, b and c are integers.

(b)

(i)Find an equation of the line which passes through B and which is

perpendicular to the line AB.

(ii) The point C has coordinates (k, 7) and angle ABC is a right angle.

Find the value of the constant k.

**Question 4 **

In a kinetics experiment, a rate constant ** k **varies with temperature

**according to the**

*T***Arrhenius equation**:

Where Ea= Activation energy, Jmol-1 and R = the gas constant which has a value of 8.314 JK-1mol-1.

Data for ** k **and

**were obtained during an investigation of a chemical reaction and recorded in the table below.**

*T*

Temperature T / K |
293 | 303 | 313 | 323 | 333 |

Rate constant k / s-1 |
2.20 | 2.89 | 3.72 | 4.72 | 5.91 |

(a) Show, using the laws of logarithms, how the **Arrhenius equation **can be linearized in order to provide an equation of the form y = mx + c.

(b) Provide a second table containing the data that should be plotted on the **y**– and **x**– axes to obtain the relevant straight line graph.

(c) Determine the value of the gradient, and its units, associated with the straight line graph produced from the data provided in the table, and therefore determine the value of the activation energy *E**a*.

(d) Determine the value of the intercept associated with the straight line graph produced from the data provided in the table, and therefore determine the value of the constant *A*.

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**Question 5 **

An analytical scientist wants to prepare a calibration graph relating the concentration of a compound in solution (controlled *x *variable) with its optical absorbance (observed *y *variable). When a solution of 0.23 moldm-3 was prepared an optical absorbance of 0.8 was observed. A second solution with a concentration of 0.21 moldm-3 was observed to have an absorbance of 0.2.

Determine the equation that describes the linear relationship between the concentration and the absorbance of the compound.

**Question 6 **

The viscosity h of a polymer in solution changes as a function of the polymer’s molar mass M and is described in the following relationship:

*h = K M**a *

where *K *and *a *are constants.

A scientist investigating a particular polymer, P1, found that when P1 had a molar mass of 500 gmol-1 the polymer displayed a viscosity of 2.56 x10-3, and with a molar mass of 3000 gmol-1 the viscosity increased to a value of 6.72 x10-3.

- Linearize
*h = K M**a*to obtain an equation of a straight line of the form y = mx + c and use the data provided to determine values of the constants,*K*and*a*, for this polymer, P1.

(b) Data plotted for a second polymer, P2, was found to have a trend parallel to polymer P1 where a molar mass of 1000 gmol-1 had a viscosity of 1.32.

(i) Determine the viscosity of polymer P2 with a molar mass of 750 gmol-1.

(ii) Calculate the constant K for polymer P2.

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**Question 7 **

The quadratic equation, (2*k *– 3)*x*2 + 2*x *+ (*k *– 1) = 0 where *k *is a constant, has real roots.

- Showthat2
*k*2 –5*k*+2≤0

(b) (i) Factorise 2*k*2 – 5*k *+ 2

(ii) Hence, or otherwise, solve the quadratic inequality 2*k*2 – 5*k *+ 2 ≤ 0

**Question 8 **

(a)

(i) Express *x*2 + 10*x *+ 19 in the form (*x *+ p)2 + q, where p and q are integers.

(ii) Write down the coordinates of the vertex (minimum point) of the curve with equation*y*=*x*2 +10*x*+19

(iii) Write down the equation of the line of symmetry of the curve *y*= *x*2 + 10*x *+ 19

(iv) Describe geometrically the transformation that maps the graph of *y *= *x*2 onto the graphof*y*=*x*2 +10*x*+19

(b) Determine the coordinates of the points of intersection of the line *y *= *x *+ 11 and thecurve*y*=*x*2 +10*x*+19

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