Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#6

Question The equation of a curve is        i.       Show, by differentiation, that the gradient of the curve is always negative.    ii.       Use the trapezium rule with 2 intervals to estimate the value of giving your answer correct to 2 significant figures.     iii.   The diagram shows a […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#7

Question      i.       By differentiating  , show that if y = cot x then    ii.       Hence, show that   By using appropriate trigonometrical identities, find the exact value of     iii.     iv.   Solution      i.   We are given; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#4

Question i.       Express  in the form , where  and , giving exact  value of . ii.       Hence show that one solution of the equation , and find the other solution in the interval 0 < θ < 2π. Solution      i.   We are given that; We are required to write it in the form; If  and  are positive, then; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#7

Question The parametric equations of a curve are      i. Show that    ii. Find the equation of the tangent to the curve at the point where .   iii. For the part of the curve where , find the coordinates of the points where the tangent  is parallel to the x-axis. Solution      i.   We are required to show that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#4

Question i. Show that the equation Can be written in the form ii. Hence solve the equation to For . Solution i. We are given; We apply following two addition formulae on both sides of given equation. Therefore; Since; ii. We are required to solve following equation doe . We have found in (i) that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2002 | Oct-Nov | (P2-9709/02) | Q#4

Question      i.       By sketching a suitable pair of graphs, show that there is only one value of x in  the interval    that is a root of the equation    ii.       Verify by calculation that this root lies between 1 and 1.5.   iii.       Show that this value of x is also a root of the equation   iv.       Use the iterative […]

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

Question      i. Sketch the graph of the curve , for . The straight line , where  is a constant, passes through the maximum point of this curve for .    ii. Find the value of k in terms of .   iii. State the coordinates of the other point, apart from the origin, where the line and the curve intersect. Solution i. […]

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 | May-Jun | (P1-9709/01) | Q#2

Question Find all the values of  in the interval  which satisfy the equation; Solution First we need to manipulate the given equation to write it in a single trigonometric ratio; Dividing the entire equation by ; Using the relation  we can rewrite the equation as, To solve this equation for , we can substitute . Hence, Since given interval is  , for  interval can […]

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

Question The diagram shows a semicircle ABC with centre O and radius 8 cm. Angle  radians. i.       In the case where , calculate the area of the sector BOC.    ii.       Find the value of  for which the perimeter of sector AOB is one half of the perimeter of sector BOC.   iii.       In the case where , show that the exact length […]