Core 1 Worked solutions January 2008 paper

The link below is to a set of worked solutions for the January 2008 C1 paper.

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B2J1T-insOmpZDQyMmFiOWUtMGY0OS00YTAxLTllMmYtMzg4NDY1MDU2ZjVk&authkey=CKbR97wM&hl=en

There are then some notes below offering further explanations for each question.

**There is one small error in the woked solutions - see notes for 5(b).

1. Remember the general rule for integrating - increase the power and divide by that number - and don't forget the + C for the final mark

2. (a) power of 1/4 is fourth root : 2 x 2 x 2 x 2 = 16
(b) the power is applied to both the 16 and the x separately - for the x part the 12 and 3/4 are multiplied together and then simplified

3. First remember the general principle of multiplying top and bottom by the expression in the denominator but with the opposite sign. Then be careful multiplying out two pairs of double brackets. And then take care simplifying.

4. (a) First find the gradient from the two given points - be careful with negatives.
Then use one of two methods substituting the gradient and one of the points into either y-y1 = m(x-x1) or y = mx + c. Finally leave equations in correct form - the questions asks for integer coefficients.
(b) Pythagoras to find the distance between two given points. And simplify the surd.

5. (a) First split into separate terms. Then the first term gives x to the power of 1/2 divided by x to the power of 1 - so 1/2 subtract 1 gives minus 1/2. The second term is just power of -1.
(b) Remember the basic rule for differentiation - bring down the power and reduce the power.
**** ERROR in solutions - the second term should be minus (because it was power of -1/2 being brought down)

6. (a) stretch in y direction (vertically) scale factor 2 - so points on x axis don't change and maximum has y value doubled
(b) reflection in y-axis (horizontal direction - note minus is inside the bracket)
(c) needs translation two units left so a = 2 (remember f(x+2) is two left and f(x-2) is two right)

7. (a) subsitute in value of x1
(b) substitue in value of x2 and show all steps
(c) solve the quadratic - question gives p not equal to 0
(d) show pattern of alternating odd terms and even terms

8. (a) First make rearrange to make equations = 0
Then find a,b,c. And know that discriminate will be negative as no real roots. Then substitute values of a,b,c. Show all steps of rearranging.
(b) Factorise. Find roots. Sketch graph. Highlight "inside" part. Show inequality.

9. (a) Restate f ' (x) with powers of x. Remember basic integation rules. Remember dividing by 3/2 is same as multiplying by 2/3. Remember + C. Then use point given to find C. And finally restate whole equation including the C.
(b) From the original given equations for f ' (x) subst. x=4 to find gradient of tangent at P. Then find gradient of normal (minus one over gradient for tangent). Then use one of the two methods for finding equation with gradient found and point given.

10.(a) Basic shape from knowing it is cubic. Double root at x=1 so sits on axis there. Another root at x=-3 and subst. x=0 to get y=3 for crossing y-axis
(b) Just expand brackets
(c) Differentiate. Make equal to 3. Solve.

11. (a) Two possible approaches - informal or formal (equation for formal method given in formula book) - Remember 25th term is 24 steps after 1st term.
(b) Again formal or informal. To get from the 25th to the 21st is 4 steps. Also could have worked forwards from the 1st term - 20 steps to get to the 21st term.
(c) Lots of different methods - important point is identifying that largest total will be before terms become negative. Also realise that the total for 20 terms and 21 terms will be the same because the 21st term was zero.