Hits: 125

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#7

Hits: 125 Question The equation of a curve is      i.       Show that;    ii.       Find the x-coordinate of each stationary point of the curve in the interval . Give each answer correct to 3 significant figures. Solution      i.   We are required to show that; We are given; We utilize Quotient Rule to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/23) | Q#6

Hits: 53 Question     i.       Express  in the form , where  and . Give the  value of  correct to 2 decimal places.    ii.       Hence solve the equation for .   iii.       State the greatest and least values of as  varies. Solution      i.   We are given the expression; We are required to write […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#7

Hits: 53   Question      i.       Show that the exact value of    ii.   The diagram shows the curve  for . The shaded region is bounded by the  curve and the lines ,  and . Find the exact volume of the solid obtained when  the shaded region is rotated completely about the x-axis. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#6

Hits: 74 Question The diagram shows the curve with parametric equations for .     i.       Show that .    ii.       Find the equation of the tangent to the curve at the point where the curve crosses the positive y-axis. Give the answer in the form y = mx +c. Solution      i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/22) | Q#4

Hits: 48 Question The polynomial  is defined by where  is a constant. It is given that  is a factor of     i.       Use the factor theorem to show that .    ii.       When ;           a.  Factorise p(x) completely,           b.  Solve the equation  for . Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/21) | Q#7

Hits: 126   Question The parametric equations of a curve are for .  The curve crosses the x-axis at points B and D and the stationary points are A and C, as shown in the diagram. i.       Show that .    ii.       Find the values of t at A and C, giving each answer correct […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/21) | Q#5

Hits: 32   Question      i.       Find    ii.       Find the exact value of Solution      i.   We are required find; We know that;  Therefore; Hence; Rule for integration of  is: Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:        ii.   We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | Oct-Nov | (P2-9709/21) | Q#3

Hits: 39 Question      i.       Express in the form , where  and , Give the  value of correct to 2 decimal places.    ii.       Hence solve the equation for .   iii.       Determine the least value of  as  varies. Solution      i.   We are given the expression; We are required to write it in […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#6

Hits: 65 Question The diagram shows part of the curve and its point of intersection P with the x-axis.      i.       Find the exact x-coordinate of P.    ii.       Show that the equation of the curve can be written and use integration to find the exact area of the shaded region enclosed by the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/23) | Q#3

Hits: 50 Question It is given that  is an acute angle measured in degrees such that     i.       Find the value of .    ii.       Use an appropriate formula to find the exact value of . Solution      i.  We are given that; We know that; Therefore; Let ; then, Now we have two options. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#6

Hits: 106 Question The diagram shows part of the curve and its point of intersection P with the x-axis.      i.       Find the exact x-coordinate of P.    ii.       Show that the equation of the curve can be written and use integration to find the exact area of the shaded region enclosed by the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/22) | Q#3

Hits: 86 Question It is given that  is an acute angle measured in degrees such that     i.       Find the value of .    ii.       Use an appropriate formula to find the exact value of . Solution      i.  We are given that; We know that; Therefore; Let ; then, Now we have two options. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#6

Hits: 120 Question     i.       Prove that  .    ii.       Hence A.  Solve the equation  for . B.  Find the exact value of Solution      i.  We are given that;   provided that   provided that    ii.  a.  We are required to solve the equation; As demonstrated in (i); Hence;   provided that Now […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2015 | May-Jun | (P2-9709/21) | Q#3

Hits: 165 Question The equation of a curve is Find the equation of the tangent to the curve at the point  . Give the answer in the form y = mx + c, where the values of m and c are correct to 3 significant figures. Solution We are required to find the equation of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/13) | Q#7

Hits: 913 Question a.   Show that the equation  can be expressed as and hence solve the equation  for . b.     The diagram shows part of the graph of , where a and b are constants. The graph  crosses the x-axis at the point  and the y-axis at the point ,. Find c and d in  terms of a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/12) | Q#4

Hits: 940 Question      i.       Prove the identity .    ii.       Hence solve the equation  for . Solution i.   We are given the identity; We have the relation; Therefore; We have the trigonometric identity; It can be rearranged as; Hence; We have the algebraic identity; Therefore, we can write; ii.   We are required to solve following equation. From (i), we know that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#7

Hits: 4913 Question The diagram shows a circle with centre A and radius r. Diameters CAD and BAE are perpendicular  to each other. A larger circle has centre B and passes through C and D.      i.       Show that the radius of the larger circle is .    ii.       Find the area of the shaded region in terms of . Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#4

Hits: 677 Question      i.       Show that the equation  can be expressed as     ii.       Hence solve the equation  for . Solution i.   We are given that; Since ; We have the trigonometric identity; From this we can write; Therefore; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Let ; Since; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#3

Hits: 581 Question Solve the equation . Solution i.   We are given that; Using calculator; Let ; Now we have two options. Since ;  is not possible

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | May-Jun | (P1-9709/13) | Q#11

Hits: 6657 Question In the diagram, OAB is a sector of a circle with centre O and radius r. The point C on OB is such  that angle ACO is a right angle. Angle AOB is  radians and is such that AC divides the sector into  two regions of equal area.     i.       Show that   It […]