# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#8

Question The equation of a curve is 2×2 − 3x − 3y + y2 = 6.      i.       Show that    ii.       Find the coordinates of the two points on the curve at which the gradient is −1. Solution      i.   We are given equation of the curve as; We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve . The curve has a gradient of 3 at the point P.      i.       Show that the x-coordinate of P satisfies the equation    ii.       Verify that the equation in part (i) has a root between x = 3.1 and x = 3.3.   iii.       Use the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#6

Question i.       The polynomial , where  and  are constants, is denoted by . It is  given that   and  are factors of . Find the values of  and .    ii.       When  and  have these values, find the quotient when p(x) is divided by x2 + x − 2. Solution      i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#5

Question Solve the equation , giving all solutions in the interval . Solution We are required to solve; We have trigonometric identity; 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 2; Since ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#4

Question i.       Express  in terms of .    ii.       Hence show that Solution      i.   We are given that; From this we can write; We have the trigonometric identity; From this we can write; Hence;      ii.   We are required to show that; We have found in (i) that; Therefore; Rule for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#1

Question Find the gradient of the curve y = ln(5x + 1) at the point where x = 4. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is: Therefore; Rule for differentiation natural logarithmic function , for  is; Gradient (slope) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#3

Question Solve the equation 2 ln(x + 3) − ln x = ln(2x − 2). Solution We are given; Power Rule; Division Rule; Taking anti-logarithm of both sides; For any ; We have algebraic formula; Now we have two options. Since we have in the given equation and logarithm of a negative number is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#2

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following with […]