Link to the worked solutions forthe May 2007 C1 paper:
https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B2J1T-insOmpYWE2ZmE5MWQtMzkwMC00ODMzLTk2OGMtM2VjYzhhYTM0NTYx&hl=enFurther supporting notes for worked solutions:
1. Just expand and simplify
2. (a) cube root of 8 is 2 then 2 the power of 4
(b) dividing means subtract the indices (4/3 - 1 = 1/3)
3. first change root x to power of 1/2
(a) differentiate - bring power down and reduce the power
(b) differentiate again teh answer to (a) to get the second differential
(c) integrate the original equation - increase the power and divide by that - don't forget the +C
4. identify a & d from given information
(a) 200th term is 199 steps from the first term so add 199 lots of the difference
(b) part (a) found the last term so formula used in method 1 can be used or alternatively that used in method 2 (both formulae are in the formulae book)
5. (a) as shown the graph is translated 2 units left - don't forget the show the asymptote with a dashed line
(b) state equations for both asymptotes (state y=0 rather than y-axis as the question specifically asks for equations)
6. (a) substitute equation 1 into 2 and simplify
(b) first solve the quadratic formed in (a) to find solutions for x and then substitute these values into equation 1 to find solutions for y (two possible methods for solving the quadratic - completing the square or using the quadratic formula - which must be recalled
7. different real roots means discriminant must be greater than zero
(a) identify a,b,c from original equation, substitute into discriminant, simplify to reach the required inequality
(b) to solve the quadratic inequality, first factorise, then identify roots, then sketch, then identify the correct part of the graph (in this case outside of the roots) and finally state solutions (in this case two separate inequalities)
8. (a) substitue a1 into the equation to find a2
(b) substitute a2 into the equation to find a3
(c) (i) question means find the sum of the first four terms, so first find a4 by substituting a3 into the equation, then add the terms
(ii) show that the expression is divisible by 10 by factorising the common factor of 10
9. (a) integrate - increase the power and divide by that - don't forget the +C - then use the given point to substitue x and y to find C - then restate the equation for f(x) with the value for C
(b) first factorise the common factor x and then factorise the remaining quadratic
(c) basic cubic shape with roots as found by solving the answer to (b) equal to zero
10. (a) find the y coordinates of P and Q by substituting the x value into the given equation then use pythagoras on the two pairs of coordinates to find the length PQ
(b) to find the gradients of the tangents at P and Q need to first differentiate and then substitue the values of the x coordinates of P and Q - these gradients being equal shows the tangents are parallel
(c) the normal at P is perpendiculat to the tangent so as gradient of tangent is -13 the gradient of the normal is 1/13 - use this gradient and the coordinates of P in either straight line equation method to find the equation of the normal - leave in the correct form with all coefficients as integers (key step here is to multiply through by 13)
11. (a) simplest way to find the gradient of l2 is to rearrange into the form y=mx+c
(b) x coordinate of the point of intersection is found by equating the equations of the two lines and then substitute the value into either equation to find the y coordinate
(c) find x coordinates of A and B by making each equation equal to 1 (as y=1) then the sketch shows the points ABP to show the calculation required to find the area of the triangle
