# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#10

Question Functions  and  are defined by    ,    ,      i.       Find the value of  for which .    ii.       Express each of  and  in terms of .   iii.     Show that the equation  has no real roots.   iv.     Sketch, on a single diagram, the graphs of  and , making clear the relationship between these two graphs. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#7

Question The diagram shows a triangular prism with a horizontal rectangular base , where  units and  units. The vertical ends  and  are isosceles triangles with  units. The mid-points of  and  are  and  respectively. The origin  is at the mid-point of . Unit vectors ,  and  are parallel to ,  and  respectively. i.       Find the length of OB.    ii.       Express […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#2

Question   i.  Show that the equation  may be written in the form  where  .    ii.       Hence solve the equation , for . Solution i.   We have; We need to express the equation in terms of  only because we are given that . We have the trigonometric identity; We can write it as; Hence the equation becomes; We can also write it as; Let; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows points  and  on the curve . The tangent to the curve at B crosses the x-axis at C. The point D has coordinates (2, 0). i.       Find the equation of the tangent to the curve at B and hence show that the area of triangle BDC is    ii.       Show that the volume of the solid formed when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#8

Question A solid rectangular block has a base which measures  cm by  cm. The height of the block is  cm and the volume of the block is 72 cm3.      i.       Express  in terms of  and show that the total surface area, A cm2, of the block is given by  Given that x can vary,    ii.       find the value of  for which A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#6

Question The diagram shows the sector OPQ of a circle with centre O and radius  cm. The angle POQ is  radians and the perimeter of the sector is 20 cm.      i.       Show that .    ii.       Hence express the area of the sector in terms of .   iii.       In the case where , find the length of the chord PQ. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#5

Question The diagram shows a trapezium ABCD in which BC is parallel to AD and angle . The coordinates of A, B and D are (2, 0), (4, 6) and (12, 5) respectively.      i.       Find the equations of BC and CD.    ii.       Calculate the coordinates of C. Solution i.   First we find the equation of . To […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#4

Question A curve is such that  . The curve passes through the point (1, 5).      i.       Find the equation of the curve.     ii.       Find the set of values of  for which the gradient of the curve is positive. Solution i.   To find the equation of the curve; We can find equation of the curve from its […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#1

Question Find the coordinates of the points of intersection of the line  and the curve . Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the curve). Therefore, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#3

Question a) A debt of $3726 is repaid by weekly payments which are in arithmetic progression. The first payment is$60 and the debt is fully repaid after 48 weeks. Find the third payment. b) Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4. […]