# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#3

Question The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the  curve for 0 ≤ x ≤ 2π, and determine the nature of each of these stationary points. Solution We are required to find the x-coordinates of stationary points of the curve. A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#8

Question The diagram shows the roof of a house. The base of the roof, OABC, is rectangular and horizontal with  and  . The top of the roof  is 5m above the base and . The sloping edges , ,  and  are all equal in length. Unit vectors  and  are parallel to ,  respectively and  is vertically upwards. i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#2

Question Solve the equation  for . Solution First we need to manipulate the given equation to write it in a single trigonometric ratio i.e. ; Dividing the entire equation with ; Using the relation ; 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#11

Question Functions and  are defined by   for   where  is constant   for   ,      i. Find the values of  for which the equation  has two equal roots and solve the equation  in these cases.    ii. Solve the equation  when .   iii. Express  in terms of . Solution i.   We have the functions; We can write these functions as; We can equate both functions; Standard form […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#10

Question The diagram shows the curve , where  is a constant. The curve has a minimum point on the x-axis.      i.       Find the value of .    ii.       Find the coordinates of the maximum point of the curve.   iii.       State the set of values of  for which  is a decreasing function of .   iv.       Find the area of the shaded region. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#7

Question The diagram shows a circle with centre O and radius 8 cm. Points A and B lie on the circle. The tangents at A and B meet at the point T, and  cm. i.       Show that angle AOB is 2.16 radians, correct to 3 significant figures.    ii.       Find the perimeter of the shaded region.   iii.       Find the area of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#6

Question In the diagram, ABC is a triangle in which  cm,  cm and angle . The line CX is perpendicular to the line ABX. i.       Find the exact length of BX and show that .    ii.       Show that the exact length of AC is Solution From the given information we can compile following data; i.   To find  we consider the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#9

Question A curve is such that  and P (1, 8) is a point on the curve. i.       The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the coordinates of the mid-point of QR.    ii.   Find the equation of the curve. Solution      i.   To find the mid-point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#5

Question The curve  intersects the line  at two points. Find the distance between the two points. Solution Expression to find distance between two given points  and is: So first we need to find the coordinates of points of intersection of the curve and the line. If two lines (or a line and a curve) intersect each other […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#1

Question A curve has equation . . Given that the gradient of the curve is −3 when x = 2, find the value of the constant k. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: In this case; Therefore; We can rewrite the expression; Rule for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#4

Question The first three terms in the expansion of , in ascending powers of , are . Find the values of the constants . Solution Expression for the Binomial expansion of  is: In the given case: Hence; We are given the first three terms of the expansion: Comparing 1st terms of both above equations: Comparing 2nd terms of both above […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#3

Question Each year a company gives a grant to a charity. The amount given each year increases by 5% of its  value in the preceding year. The grant in 2001 was \$5000. Find i.         the grant given in 2011, ii.       the total amount of money given to the charity during the years 2001 to 2011 inclusive. […]