# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#9

Question Relative to an origin , the position vectors of the points , ,  and  are given by and where  and  are constants. Find  i.        the unit vector in the direction of , ii.      the value of p for which angle , iii.       the values of  for which the length of   is 7 units. Solution i.   A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#10

Question Functions  and  are defined by    ,    ,      i.       Find the set of values of  for which .    ii.       Find the range of  and state, with a reason, whether  has an inverse.   iii.       Show that the equation  has no real solutions.   iv.       Sketch, on a single diagram, the graphs of  and , making clear the relationship between the graphs. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#3

Question      i.       Show that the equation  can be written as a quadratic equation in .    ii.       Hence, or otherwise, solve the equation in part (i) for . Solution i.   First we need to manipulate the given equation to write it in a single trigonometric ratio i.e. ; Dividing the entire equation by ; Using the relation ; This is the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#8

Question The diagram shows a glass window consisting of a rectangle of height  m and width  m and a semicircle of radius  m. The perimeter of the window is 8 m.      i.       Express  in terms of .    ii.       Show that the area of the window, A m2, is given by Given that  can vary,   iii.       find the value […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#7

Question The diagram shows part of the graph of   and the normal to the curve at . This normal meets the -axis at R. The point Q on the -axis and the point S on the curve are such that PQ and SR are parallel to the -axis.      i.       Find the equation of the normal at P […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#6

Question The curve   and the line  intersect at two points. Find i.       the coordinates of the two points,    ii.       the equation of the perpendicular bisector of the line joining the two points. Solution      i.   To find the coordinates of intersection points; If two lines (or a line and a curve) intersect each other at a point then […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#5

Question In the diagram, OCD is an isosceles triangle with  cm and angle COD =0.8 radians. The points A and B, on OC and OD respectively, are joined by an arc of a circle with centre O and radius 6 cm. Find      i.       the area of the shaded region,    ii.       the perimeter of the shaded region. Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#2

Question Evaluate   Solution Writing the given expression in standard form; Rule for integration of  is: In the given case;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#4

Question Find the coefficient of  in the expansion of i.       ii.    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 with , we can  equate Finally substituting  in: Therefore the coefficient of  is . ii.   To find the coefficient of  in the expansion […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2004 | May-Jun | (P1-9709/01) | Q#1

Question A geometric progression has first term 64 and sum to infinity 256. Find      i.       the common ratio    ii.       the sum of the first ten terms. Solution i.   From the given information, we can compile following data for Arithmetic Progression (A.P); Expression for the sum to infinity of the Geometric Progression (G.P) when  or ; ii.   Sum of […]