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Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#7

Hits: 3   Question (i)          Show that (2 sin x + cos x)2 can be written in the form (ii)        Hence find the exact value of  Solution      i.   We are given that; We have formula; Therefore; Therefore; Therefore;    ii.   We are required to find the exact value of; From (i) we […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#6

Hits: 2   Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#5

Hits: 4     Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M can be written in the form , where the value of a is to be  stated.    ii.       Find the exact value of the area of the region enclosed by the curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#4

Hits: 1     Question      i.       Given that , find the value of .    ii.       Hence, showing the use of an appropriate formula in each case, find the exact value of a.   b.   Solution      i.   We are given that; We have trigonometric identity; Let , then; Since , therefore;      […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#3

Hits: 2   Question The polynomial  is defined by where  is a constant. i.       Given that  is a factor of , find the value of .    ii.       When  has this value,                      a.  Factorise p(x) completely,                   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#7

Hits: 1   Question (i)          Show that and hence find the exact value of; (ii)   The region enclosed by the curve y = tan x + cos x and the lines x = 0,  and y = 0 is shown in  the diagram. Find the exact volume of the solid produced when this region […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#6

Hits: 3   Question The diagram shows the curve , for . The x-coordinate of the maximum point  M is denoted by . i.       Find  and show that  satisfies the equation tan 2x = 2x + 4.    ii.       Show by calculation that  lies between 0.6 and 0.7.   iii.       Use the iterative formula to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#5

Hits: 1   Question The parametric equations of a curve are      i.       Find an expression for in terms of t.      i.       Find the equation of the normal to the curve at the point for which t = 0. Give your answer in  the form ax + by + c = 0, where a, […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#4

Hits: 1   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 largest value of k for which the equation  has any solutions. Solution      i.   We are given the expression; We are […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#3

Hits: 1   Question i.       Find the quotient when the polynomial   is divided by  ,  and  show that the remainder is 4.    ii.       Hence, or otherwise, factorise the polynomial Solution      i.   Hence quotient is and remainder is .      ii.   We are required to factorise; When a polynomial, , is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#7

Hits: 26   Question (i)       Show that (2 sin x + cos x)2 can be written in the form (ii)        Hence find the exact value of  Solution      i.   We are given that; We have formula; Therefore; Therefore; Therefore;    ii.   We are required to find the exact value of; From […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#6

Hits: 19   Question The parametric equations of a curve are The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.      i.       Show that .    ii.       Use an iterative process based on the equation in part (i) to find the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#5

Hits: 13     Question The diagram shows the curve  and its minimum point M.      i.       Show that the x-coordinate of M can be written in the form , where the value of a is to be  stated.    ii.       Find the exact value of the area of the region enclosed by the curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#4

Hits: 7     Question      i.       Given that , find the value of .    ii.       Hence, showing the use of an appropriate formula in each case, find the exact value of a.   b.   Solution      i.   We are given that; We have trigonometric identity; Let , then; Since , therefore;      […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#3

Hits: 5   Question The polynomial  is defined by where  is a constant. i.       Given that  is a factor of , find the value of .    ii.       When  has this value,               a.  Factorise p(x) completely,              b.   Find the remainder when p(x) […]

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

Hits: 14     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 […]

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

Hits: 11     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. […]

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

Hits: 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.   […]

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

Hits: 15     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; […]

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

Hits: 8   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; […]