Hits: 47

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/23) | Q#7

Hits: 47   Question i.       Express  in the form , where  and , stating the exact  value of R and and the value of  correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval . 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 2013 | Oct-Nov | (P2-9709/23) | Q#5

Hits: 87 Question The parametric equations of a curve are  ,  , for . i. Show that .    ii. Find the coordinates of the point on the curve at which the gradient is -4. Solution      i.   We are required to show that  for the parametric equations given below; If a curve is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/22) | Q#6

Hits: 74   Question A.         Find B.   i.       Use the trapezium rule with 2 intervals to estimate the value of giving your answer correct to 3 decimal places. ii.       Using a sketch of the graph of  for , explain whether the trapezium rule  gives an under-estimate or an over-estimate of the true value of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/22) | Q#3

Hits: 31 Question Solve the equation , giving all solutions in the interval . Solution We are required to solve the equation; We know that;   provided that   provided that Hence; We have the trigonometric identity; From this we can get; Therefore; Let ; Now we have two options. Since Using calculator we can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/21) | Q#7

Hits: 92 Question i.       Express  in the form , where  and , stating the exact value of R and and the value of  correct to 2 decimal places.  ii.    Hence solve the equation Giving all solutions in the interval . Solution      i.  We are given the expression; We are required to write it […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | Oct-Nov | (P2-9709/21) | Q#5

Hits: 53 Question The parametric equations of a curve are  ,  , for . i.       Show that . ii.       Find the coordinates of the point on the curve at which the gradient is -4. Solution      i.  We are required to show that  for the parametric equations given below; If a curve is given parametrically by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/22) | Q#8

Hits: 111   Question      i.       Prove the identity    ii.       Hence solve the equation For . Solution      i.   We are required to prove the identity; We know that; Therefore;    provided that    ii.   We are required to solve the equation; From (i) we know that; Therefore; Since , therefore Let […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/22) | Q#7

Hits: 83     Question A.  Find the exact area of the region bounded by the curve , the x-axis and the lines   and . The diagram shows the curve , for and its minimum point M. Find the exact x- coordinate of M. Solution A.    We are required to find area under the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/23) | Q#7

Hits: 618   Question      i.       Express in the form , where  and , giving the  exact value of R and the value of a correct to 2 decimal places.    ii.      Hence solve the equation Giving all solutions in the interval  correct to 1 decimal place.   iii.       Determine the least value […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/23) | Q#6

Hits: 292     Question      i.       By sketching a suitable pair of graphs, show that the equation where x is in radians, has only one root for .    ii.       Verify by calculation that this root lies between x= 0.7 and x=0.9.   iii.       Show that this root also satisfies the equation   iv. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/23) | Q#3

Hits: 200   Question i.       Show that  .  ii.       Hence show that Solution      i.   We are given that; From this we can write; From this we can write; It can be formulated as; Hence;    ii.   We are required to show that; We have found in (i) that; Therefore; Rule for integration […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/21) | Q#7

Hits: 129   Question      i.       Express in the form , where  and , giving the exact value of R and the value of a correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval  correct to 1 decimal place.   iii.       Determine the least value of  as […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/21) | Q#6

Hits: 137     Question      i.       By sketching a suitable pair of graphs, show that the equation where x is in radians, has only one root for .    ii.       Verify by calculation that this root lies between x= 0.7 and x=0.9.   iii.       Show that this root also satisfies the equation   iv. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2013 | May-Jun | (P2-9709/21) | Q#3

Hits: 118 Question i.       Show that  . ii.       Hence show that Solution      i.   We are given that; From this we can write; From this we can write; It can be formulated as; Hence;      ii.   We are required to show that; We have found in (i) that; Therefore; Rule for integration […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 531   Question The function  is defined by  for .      i.       Solve the equation .    ii.       Find the range of .   iii.       Sketch the graph of . The function  is defined by  for .   iv.       State the maximum value of for which  has an inverse.    v.       Obtain an expression for . Solution i.   We have the function; We can write it as; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#5

Hits: 834   Question i.   Sketch, on the same diagram, the curves  and  for  . ii.     Hence state the number of solutions, in the interval , of the equations                      a.                          b.   Solution i.   We are required to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#3

Hits: 410 Question   i.      Express the equation  as a quadratic equation in .  ii.       Solve the equation  for , giving solutions in terms of . Solution i.   We are given that; We know that; Therefore; We have the trigonometric identity; It can be rearranged as follows; Substituting this expression of  in the above equation; It can be rewritten as; This […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/12) | Q#5

Hits: 498 Question It is given that  and , where .      i.       Show that  has a constant value for all values of .    ii.       Find the values of  for which . Solution      i.   We are given that; Taking squares of both sides of each equation. We have the algebraic formulae; We can simplify these to; Adding both sides […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/13) | Q#7

Hits: 491   Question a)   Find the possible values of  for which , giving your answers correct to 3 decimal places. b)  Solve the equation  for  giving  in terms of  in your answers. Solution a)     We have; We can rearrange the equation as; We know that; Therefore we can write above equation as; b)    To solve the equation  for , let […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#1

Hits: 447   Question Given that , where  is an acute angle in degrees, find, in terms of ,      i.           ii.          iii.        Solution      i.   We have; Since we are required to write it in terms of  where  , so we first write the given  expression in terms of  . We have the trigonometric identity; […]