# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows the curve . i.       Find the equation of the tangent to the curve at the point . ii.    Show that the x-coordinates of the points of intersection of the line   and the curve are given by the equation . Hence find these x- coordinates. iii.     The region shaded in the diagram is rotated through […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#9

Question A curve has equation , where  is a positive constant. Find, in terms of ,  the values of  for which the curve has stationary points and determine the nature of  each stationary point. Solution A stationary point  on the curve  is the point where gradient of the curve is  equal to zero; We are given that; Therefore, we need […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#6

Question The diagram shows sector OAB with centre O and radius 11 cm. Angle  radians. Points C and D lie on OA and OB respectively. Arc CD has centre O and  radius 5 cm. i.       The area of the shaded region ABDC is equal to k times the area of the unshaded  region OCD. Find k.    ii.    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#3

Question The point A has coordinates  and the point B has coordinates . The  point C is the mid-point of AB.   i.       Find the equation of the line through A that is perpendicular to .  ii.     Find the distance AC. Solution i.   We are required to find equation of a line that passes through  and is  perpendicular to the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#2

Question A curve has equation . It is given that  and that . Find . Solution We are given that curve  and we are required  to find the equation of the curve  for which . We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule for integration of  is: Rule for integration […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#1

Question Solve the inequality . Solution We have; We solve the following equation to find critical values of ; Now we have two options; Hence the critical points on the curve for the given condition are  & 2. Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If  (‘a’ is positive) then parabola  opens upwards […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#7

Question a)   Find the possible values of  for which , giving your answers correct to 3 decimal places. b)  Solve the equation  for  giving  in terms of  in your answers. Solution a)     We have; We can rearrange the equation as; We know that; Therefore we can write above equation as; b)    To solve the equation  for , let Since given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#4

Question The diagram shows a pyramid OABC in which the edge OC is vertical. The horizontal base OAB is a triangle, right-angled at O, and D is the mid-point of AB. The edges OA, OB and OC have lengths of 8 units, 6 units and 10 units respectively. Unit vectors ,  and  are parallel to ,  and  respectively. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#10

Question The function  is defined by  for , where  is a constant. It is given that f is a one-one function.      i.       State the range of  in terms of  and find the smallest possible value of . The function  is defined by  for , where  and  are positive constants. It is given that, when ,  and .    ii.       Write down two equations in  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#8

Question      i.       Find the coefficient of  in the expansion of .    ii.       Find the coefficient of  in the expansion of .   iii.       Hence find the coefficient of  in the expansion of . Solution i.   Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the term of   : […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#5

Question a)   In a geometric progression, the sum to infinity is equal to eight times the first term. Find the  common ratio. b)  In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference. Solution a)     From the given information, we can compile following data […]