# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#4

Question i.       Sketch the curve  for . ii.       By adding a suitable straight line to your sketch, determine the number of real roots of the equation State the equation of the straight line. Solution i.   We are required to sketch  for ; We can sketch the graph of  for  as follows. We can find the points of the graph as follows. ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#3

Question Solve the equation  for   . Solution We can rearrange the given equation as; We have the trigonometric identity; This identity can be rearranged as; Substituting  expression in the given equation; Let ; We can solve this quadratic equation through factorization. Now we have two options. Since ; Using calculator we can find the values of . We utilize the symmetry property of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#7

Question The diagram shows the function  defined for  by  for  for      i.       State the range of .    ii.       Copy the diagram and on your copy sketch the graph of .   iii.       Obtain expressions to define , giving the set of values of  for which each expression is valid. Solution i.   We have the function;  for  for It is evident from […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows parts of the curves  and  and their points of intersection  and . The x-coordinates of  and  are and  respectively.     i.       Show that  and  are roots of the equation . Solve this equation and hence state the value of  and the value of .    ii.       Find the area of the shaded region between the two curves. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#10

Question The diagram shows triangle , in which the position vectors of  and  with respect to  are given by &  is a point on  such that , where  is a constant.     i.       Find angle .    ii.     Find  in terms of  and vectors ,  and .   iii.   Find the value of  given that  is perpendicular to . Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#8

Question The diagram shows a rhombus ABCD. Points P and Q lie on the diagonal AC such that BPD is an arc of a circle with centre C and BQD is an arc of a circle with centre A. Each side of the rhombus has length 5 cm and angle BAD = 1.2 radians. i.       Find the area of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#6

Question A curve has equation . It is given that .     i.       Find the set of values of  for which  is an increasing function.    ii.       Given that the curve passes through (1, 3), find . Solution i.   To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#5

Question A curve has equation .     i.       Find  and .    ii.       Find the coordinates of the maximum point A and the minimum point B on the curve. Solution i.   Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation of  is: Second derivative is the derivative of the derivative. If we have derivative of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#2

Question Points A, B and C have coordinates (2, 5), (5, −1) and (8, 6) respectively.     i.       Find the coordinates of the mid-point of AB.    ii.       Find the equation of the line through C perpendicular to AB. Give your answer in the form ax + by + c = 0. Solution i.   To find the mid-point of a line we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#3

Question Solve the equation  for . Solution We have the equation; We have the trigonometric identity; We can write it as; Therefore; To solve this equation for , we can substitute . Hence, Now we have two options; Since; Using calculator we can find the values of . We utilize the symmetry property of   to find other solutions (roots) of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#1

Question Find the term, independent of , in the expansion of . Solution First rewrite the given expression in standard form. Expression for the Binomial expansion of  is: In the given case: Hence; The term independent of  i.e coefficient of   ;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#9

Question a)   A geometric progression has first term 100 and sum to infinity 2000. Find the second term.  b)  An arithmetic progression has third term 90 and fifth term 80. i.       Find the first term and the common difference. ii.       Find the value of m given that the sum of the first m terms is equal to the sum of […]