# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#8

Question i.       Express  in the form , where  and , Give the exact value of R and the value of  correct to 2 decimal places.    ii.       Hence solve the equation for .   iii.       Find the greatest and least possible values of  as  varies. Solution      i.   We are given the expression; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#7

Question      i.       By sketching a suitable pair of graphs, show that the equation has exactly two real roots.    ii.       Show by calculation that this root lies between 1.2 and 1.3.   iii.       Show that this root also satisfies the equation   iv.       Use an iteration process based on the equation in […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#6

Question a.   Find b.   Show that Solution a.     We are required to find; We can expand the integrand as; Rule for integration of  is: Rule for integration of , or ; b.     We are required to show that; We have trigonometric identity; Therefore; Hence; Rule for integration of  is: Rule for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#5

Question Find the value of  when  for each of the following cases: i.       . ii.        . Solution      i.   We are given that; We are required to find the value of when , therefore, first we need to find . If  and  are functions of , and if , then; If , then; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#2

Question A curve has parametric equations Find the exact gradient of the curve at the point for which . Solution      i.   Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point. Gradient (slope) of the curve at a particular point can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#4

Question The polynomial  is defined by where  is a constant. i.       Given that  is a factor of , find the value of . ii.       When  has the value found in part (i), find the quotient when  is divided by . Solution      i.   We are given that; We are also given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#3

Question The variables x and y satisfy the equation y = Kxm, where K and m are constants. The graph of ln y  against ln x is a straight line passing through the points (0, 2.0) and (6, 10.2), as shown in the  diagram. Find the values of K and m, correct to 2 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#1

Question Solve the inequality . Solution SOLVING EQUATION: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have to consider both moduli separately and it leads to following cases; Hence, the only solution for the given equation is; SOLVING EQUATION: […]