Hits: 82

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#8

Hits: 82     Question      i.       Given that   and find in terms of .    ii.       Solve the equation giving all solutions in the interval 0◦ ≤ x ≤ 360◦. Solution      i.   We are given; We are given that ; therefore;    ii.   We are required to solve  giving all solutions […]

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

Hits: 85     Question The diagram shows the part of the curve  for .      i.       Use the trapezium rule with 2 intervals to estimate the value of giving your answer correct to 2 decimal places.    ii.       The line y=x intersects the curve at point P. Use the iterative formula to determine the […]

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

Hits: 58     Question The curve with equation has one stationary point  in the interval . Find the exact  x-coordinate of this point. Solution We are required to find the x-coordinate of stationary point of the curve with equation; A stationary point on the curve is the point where gradient of the curve is […]

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

Hits: 60     Question Solve the equation , for . Solution We are required to solve; Therefore; We have the trigonometric identity; Let , then; Since , therefore; Using calculator we can find that; Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity/symmetry property of   to find other solutions (roots):  […]

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

Hits: 57     Question Solve the equation , for . Solution We are required to solve; Therefore; We have the trigonometric identity; Let , then; Since , therefore; Using calculator we can find that; Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity/symmetry property of   to find other solutions (roots):  […]

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: 75   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#4

Hits: 34     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 | (P2-9709/22) | Q#7

Hits: 102   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: 111   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#4

Hits: 50   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 | (P2-9709/21) | Q#7

Hits: 87   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#4

Hits: 47     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 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/11) | Q#1

Hits: 3844   Question  Solve the equation , for  Solution i.   We have the equation; Dividing both sides of the equation by ; We have the relation , therefore, To solve this equation, we can substitute . Hence,   Since given interval is  , for  interval can be found as follows; Multiplying both sides of the inequality with […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#4

Hits: 308 Question      i. Solve the equation  for . ii. How many solutions has the equation  for ? Solution    i.   We are given; 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; Since ; Hence the given interval for  is . To solve  equation […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 682   Question In the diagram, D lies on the side AB of triangle ABC and CD is an arc of a circle with centre A and radius 2 cm. The line BC is of length 2√3 cm and is perpendicular to AC. Find the area of the shaded region BDC, giving your answer in terms of  and […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#11

Hits: 3024   Question The diagram shows a sector of a circle with centre  and radius 20 cm. A circle with centre  and radius  cm lies within the sector and touches it at P, Q and R. Angle POR = 1.2 radians. i. Show that , correct to 3 decimal places. ii. Find the total area of the three parts […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#1

Hits: 416   Question i. Prove the identity ii. Use this result to explain why  for . Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can rewrite the identity as; Therefore; Again using the relation , we can rewrite; ii.   It is evident that in equation  for […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 184   Question Solve the equation , for . Solution We have the equation; We have the trigonometric identity; We can rewrite the identity as; Therefore; To solve this equation  for ,  we can substitute . Hence, Now we have two options; Since; NOT POSSIBLE Using calculator we can find the values of . We utilize the symmetry property of   to […]