# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#2

Question Find the coefficient of  in the expansion of  .   Solution First rewrite the given expression in standard form. Expression for the Binomial expansion of  is: In the given case: Hence;     Hence the coefficient of   is .

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#1

Question  Solve the equation , for  Solution i.   We have the equation; Dividing both sides of the equation by ; We have the relation , therefore, To solve this equation, we can substitute . Hence,   Since given interval is  , for  interval can be found as follows; Multiplying both sides of the inequality with 2; Since ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#11

Question The diagram shows the line  and part of the curve  .     i.       Show that the equation  can be written in the form .    ii.       . Hence find the area of the shaded region.   iii.       The shaded region is rotated through  about the y-axis. Find the exact value of the volume of revolution obtained. Solution i.   We are given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#10

Question It is given that a curve has equation , where .     i.       Find the set of values of  for which the gradient of the curve is less than 5.    ii.       Find the values of  at the two stationary points on the curve and determine the nature of each stationary point. Solution i.   Gradient (slope) of the curve is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#9

Question The coordinates of A are (−3, 2) and the coordinates of C are (5, 6). The mid-point of AC is M and the perpendicular bisector of AC cuts the x-axis at B. i.       Find the equation of MB and the coordinates of B.    ii.       Show that AB is perpendicular to BC.   iii.       Given that ABCD is a square, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#6

Question Two vectors  and  are such that   and  , where  is a constant.      i.       Find the value of  for which  is perpendicular to .    ii.       For the case where , find the angle between the directions of  and  . Solution      i.   If  and  & , then  and  are perpendicular. Therefore, we need the scalar/dot product […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#5

Question The diagram shows the curve  and the line , where  is a constant. The curve and the line intersect at the points A and B.     i.       For the case where , find the x-coordinates of A and B.    ii.       Find the value of  for which  is a tangent to the curve . Solution i.   For the case where […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#4

Question A watermelon is assumed to be spherical in shape while it is growing. Its mass, Mkg, and radius, r cm, are related by the formula , where  is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#3

Question In the diagram, ABC is an equilateral triangle of side 2 cm. The mid-point of BC is Q. An arc of a circle with centre A touches BC at Q, and meets AB at P and AC at R. Find the total area of the shaded regions, giving your answer in terms of  and . Solution It […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#8

Question The function  is defined for the domain , where  and  are constants.      i.       Express  in the form , where  and  are constants.    ii.       State the range of  in terms of .   iii.       State the smallest value of  for which  is one-one.   iv.       For the value of  found in part (iii), find an expression for  and state the domain of  , giving your answers […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#7

Question a)   The first two terms of an arithmetic progression are 1 and  respectively. Show that the sum  of the first ten terms can be expressed in the form , where a and b are constants to be  found. b)  The first two terms of a geometric progression are 1 and  respectively, where . i.       Find the […]