Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#10

Question Functions and  are defined by  for  for  ,      i. Obtain expressions, in terms of , for  and , stating the value of  for which  is not defined.    ii. Sketch the graphs of  and  on the same diagram, making clear the relationship between the two graphs.   iii. Given that the equation , where  is a constant, has no solutions, find the set of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#9

Question The diagram shows part of the curve  which passes through the points A and B. The curve has a maximum point at A and the gradient of the line BA is 2.     i.      Find the coordinates of A and B.    ii.    Find a  and hence evaluate the area 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 2012 | May-Jun | (P1-9709/12) | Q#8

Question      i.       Find the angle between the vectors  and . The vector  has a magnitude of 15 units and is in the same direction as the vector . The vector  has a magnitude of 14 units and is in the same direction as the vector .    ii.       Express  and  in terms of ,  and .   iii.       Find the unit […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#6

Question The diagram shows a metal plate made by removing a segment from a circle with centre O and radius 8 cm. The line AB is a chord of the circle and angle AOB = 2.4 radians. Find i.        the length of AB, ii.      the perimeter of the plate,  iii.   the area of the plate. Solution i. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#5

Question i.       Prove the identity ii.       Solve the equation      for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; Therefore; ii.   Solve the equation   for . We can rewrite the given equation as; As demonstrated in (i), we can rewrite the given equation as; To solve this equation […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#4

Question The point A has coordinates (−1, −5) and the point B has coordinates (7, 1). The perpendicular bisector of AB meets the x-axis at C and the y-axis at D. Calculate the length of CD. Solution i.   Expression to find distance between two given points  and is: Therefore, to find the length of CD we need […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#2

Question The equation of a curve is . i.     Obtain an expression for . ii.   A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the y-coordinate when x = 4. 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 | (P1-9709/12) | Q#1

Question The diagram shows the region enclosed by the curve  , the x-axis and the lines  and . Find, in terms of , the volume obtained when this region is rotated through  about the x-axis. Solution i.   Expression for the volume of the solid formed when the shaded region under the curve  is rotated completely about the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#3

Question The coefficient of  in the expansion of  is 90. , Find the value of positive constant . Solution Expression for the Binomial expansion of  is: We need to expand both terms one-by-one. First we expand In the given case: Hence; Similarly we also need to expand First rewrite the given expression in standard form. In the given case: Hence; Now […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/12) | Q#7

Question a)   In an arithmetic progression, the sum of the first  terms, denoted by , is given by . Find the first term and the common difference. b)  In a geometric progression, the second term is 9 less than the first term. The sum of the second  and third terms is 30. Given that all the terms of […]