# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#5

Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#3

Question The variables x and y satisfy the equation y = Kxa, where K and a are constants. The  graph of ln y against ln x is a straight line passing through the points (0.22,  3.96) and (1.32, 2.43), as shown in the diagram. Find the values of K and a, correct to 3 significant […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/22) | Q#2

Question                     i.       Solve the equation                   ii.       Hence, using logarithms, solve the equation giving the answer correct to 3 significant figures. Solution SOLVING EQUALITION: PIECEWISE      i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#6

Question The diagram shows the curve with equation The maximum point is denoted by M.      i.       Find an expression for  and determine the gradient of the curve at the point  where the   curve crosses the x-axis.    ii.       Show that the x-coordinate of the point M satisfies the equation   iii.       Use an […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#3

Question Find the equation of the normal to the curve at the point (3, 1). Solution  We are given equation of the curve as; We are required to find the equation of normal to the curve at the point (3, 1). To find the equation of the line either we need coordinates of the two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/21) | Q#2

Question      i.       Solve the inequality .    ii.       Hence find the greatest integer satisfying the inequality  Solution SOLVING INEQUALITY: PIECEWISE      i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#7

Question The equation of a curve is      i.       Find an expression for  and show that the gradient of the curve at the point  (−1, 2)  is .   ii.       Show that the curve has no stationary points.   iii.       Find the x-coordinate of each of the points on the curve at which the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#3

Question A curve has equation Find the exact gradient of the curve at the point for which y = 4. Solution We are required to find the gradient of the curve at a point where y=4. Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/23) | Q#1

Question      i.       Solve the inequality .    ii.       Hence find the largest integer n satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE      i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#7

Question The equation of a curve is      i.       Find an expression for  and show that the gradient of the curve at the point  (−1, 2)  is .    ii.       Show that the curve has no stationary points.   iii.       Find the x-coordinate of each of the points on the curve at which the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#3

Question A curve has equation Find the exact gradient of the curve at the point for which y = 4. Solution We are required to find the gradient of the curve at a point where y=4. Gradient (slope) of the curve at the particular point is the derivative of equation of the  curve at that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Oct-Nov | (P2-9709/21) | Q#1

Question      i.       Solve the inequality .    ii.       Hence find the largest integer n satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE      i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#3

Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/23) | Q#2

Question      i.       Solve the equation .    ii.       Hence solve the equation , giving the answer correct to 3  significant figures. Solution i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/22) | Q#3

Question Find the exact coordinates of the stationary point of the curve with equation Solution We are required to find the exact coordinates of the stationary point of the curve. A stationary point on the curve is the point where gradient of the curve is  equal to zero; Therefore, we find the expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | May-Jun | (P2-9709/22) | Q#2

Question      i.       Solve the equation .    ii.       Hence solve the equation , giving the answer correct to 3  significant figures. Solution i.   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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#7

Question The parametric equations of a curve are for . At the point P on the curve, the gradient of the curve is 2.      i.       Show that the value of the parameter at P satisfies the equation .    ii.       By first expressing in the form , where R > 0 and   , find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#5

Question The curve with equation and the point P on the curve has y-coordinate 10. i.               Show that the x-coordinate of P satisfies the equation ii.               Use the iterative formula to find the x-coordinate of P correct to 4 significant figures. Give the  result  of each iteration to 6 significant figures. iii.               Find the gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#3

Question The variables x and y satisfy the equation y = Aepx+p, where A and p are constants.  The graph of ln y against x is a straight line passing through the points (1, 2.835) and  (6, 6.585), as shown in the diagram. Find the values of A and p. Solution We are given; Taking […]