# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Feb-Mar | (P2-9709/22) | Q#8

Question      i.       Show that .    ii.       Using the identity in part (i),                a.   find the least possible value of as x varies.                             b.  find the exact value of Solution      i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#7

Question      i.       Express in the form , where a and b are integers.    ii.       Hence express in the form where R > 0 and  .   iii.       Using the result of part (ii), solve the equation  for . Solution      i.   We are given the expression;   provided that   provided that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/23) | Q#3

Question      i.       Find    ii.       Without using a calculator, find the exact value of Solution      i.   We are required to find; We know that ; Therefore; Hence; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:    ii.   We are required to find the exact […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#7

Question The diagram shows the curve with parametric equations for .     i.       Show that can be expressed in the form    ii.       Find the equation of the normal to the curve at the point where the curve crosses the positive y-axis. Give your answer in the form y = mx +c, where the constants […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/22) | Q#6

Question      i.       Show that    ii.       Solve the equation for .   iii.       Find the exact value of Solution      i.   We are required to show that; Since ; Therefore; provided that ii.   We are required to solve the equation for . We are given that; As demonstrated in (i); Since ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#7

Question The polynomial  is defined by where  and  are constants. It is given that  is a factor of . It is also given that the remainder is 18 when  is divided by .     i.       Find the values of a and b.    ii.       When a and b have these values         […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | Oct-Nov | (P2-9709/21) | Q#5

Question      i.       Show that    ii.       Hence find the exact value of; Solution      i.   We are required to show that; Since ; Therefore; ii.   We are required to find the exact value of; As demonstrated in (i); Therefore; Rule for integration of  is: 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 2016 | Oct-Nov | (P2-9709/21) | Q#3

Question A curve has equation y = 2 sin 2x − 5 cos 2x +6 and is defined for 0 ≤ x ≤ π. Find the x-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. Solution We are required to find the x-coordinates of stationary points of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#7

Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q.     i.       Show that .    ii.       Find the coordinates of M.   iii.       Find the gradient of the curve at P and at Q. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#4

Question Show that i.       Hence a)   find the exact value of , b)  Solve the equation , for . Solution      i.   We are given that; We know that; Therefore;    ii.   a)     We are required to find the exact value of From (i) we have found that; Therefore; b)    We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/23) | Q#3

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

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#7

Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q.     i.       Show that .    ii.       Find the coordinates of M.   iii.       Find the gradient of the curve at P and at Q. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#4

Question Show that i.Hence a)   find the exact value of , b)  Solve the equation , for . Solution      i.   We are given that; We know that; Therefore;      ii.   a)     We are required to find the exact value of From (i) we have found that; Therefore; b)    We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/22) | Q#3

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

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#7

Question      i.       Find      ii.       Without using a calculator, find the exact value of giving your answer in the form , where a and b are integers.   Solution      i.   We are required to find;   provided that   Rule for integration of  is: We integrate both parts  and  one by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#5

Question A curve is defined by the parametric equations for .       i.       Show that .    ii.       Find the coordinates of the stationary point.   iii.       Find the gradientof the curve at point . Solution      i.   We are required to show that  for the parametric equations given below; If a curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2016 | May-Jun | (P2-9709/21) | Q#2

Question Solve the equation for . Solution We are given;  except where  or undefined Now we have two options. Using calculator; Now we find all solutions in the interval . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity/symmetry property of   to find other solutions (roots) of :  Therefore; For ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#9

Question a.   Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and  second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r  respectively. Find the value of the common sum to infinity. b.   The first term of an arithmetic progression […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#3

Question Showing all necessary working, solve the equation  for . Solution We are given equation; We know that ; therefore, Now we have two options. Using calculator we can find that; We have following properties of . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity property of   to find other solutions (roots) of . Therefore; For; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#10

Question A function f is defined by  for .     i.       Find the range of f.    ii.       Sketch the graph of   iii.       Solve the equation , giving answers in terms of . The function  is defined by  for , where k is a constant.   iv.       State the largest value of k for which g has an inverse.    v.       For this value of k, find […]