Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#10

Question Functions  and  are defined by  , , , ,      i.       Solve the equation .    ii.       Express  and  in terms of .   iii.       Show that the equation   has no solutions.   iv.       Sketch in a single diagram the graphs of  and , making clear the relationship between the graphs. Solution i.   We have the functions; We can write these functions […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#2

Question The equation of a curve is . The equation of a line is . On the same diagram, sketch the curve and the line for . Solution First we sketch  for . We can find the points of the graph as follows. Now we sketch the line  for . We can write it as; Slope-Intercept form of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#9

Question Relative to an origin , the position vectors of the points ,  and  are given by and i.       Find angle AOB.    ii.       Find the vector which is in the same direction as  and has magnitude 30.   iii.       Find the value of the constant  for which  is perpendicular to . Solution i.   We recognize that  is angle between  and  . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#7

Question The equation of a curve is . i.       Obtain an expression for .    ii.       Find the equation of the normal to the curve at the point P(1, 3).   iii.       A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#6

Question A curve is such that , where k is a constant.     i.       Given that the tangents to the curve at the points where  and  are perpendicular, find the value of .    ii.       Given also that the curve passes through the point (4, 9), find the equation of the curve. Solution      i.   If two lines are perpendicular […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#4

Question The equation of a curve is .     i.       Find the coordinates of the stationary point on the curve and determine its nature.    ii.       Find the area of the region enclosed by the curve, the x-axis and the lines    and . Solution      i.   Coordinates of stationary point on the curve  can be found from the derivative of equation […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#5

Question The diagram shows a semicircle ABC with centre O and radius 6 cm. The point B is such that angle BOA is 90o and BD is an arc of a circle with centre A. Find i.       the length of the arc BD,    ii.       the area of the shaded region. Solution      i.   Expression for length of a […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#1

Question Solve the equation  for . Solution To solve this equation for , we can substitute . Hence, Since given interval is  , for  interval can be found as follows; Multiplying the entire inequality with 2; Adding  to entire inequality; Since ; Hence the given interval for  is . To solve  equation for interval , Using calculator we can find the value of . To find all […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#3

Question i.     Find the first three terms in the expansion of   in ascending powers of . ii.       Given that the coefficient of in the expansion of  is  , find the value           of  the constant . Solution i.   First rewrite the given expression in standard form. Expression for the Binomial expansion of  is: In the given […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#8

Question The first term of an arithmetic progression is 8 and the common difference is d, where d ≠ 0. The  first term, the fifth term and the eighth term of this arithmetic progression are the first term, the  second term and the third term, respectively, of a geometric progression whose common ratio is r.      i.       Write down two equations connecting […]