# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/22) | Q#7

Question i.       Express  in the form , where  and , giving the  value of  correct to 2 decimal places.    ii.       Hence solve the equation for .   iii.       Find the greatest possible value of as  varies, and determine the smallest positive value of  for which this greatest value occurs. Solution      i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/23) | Q#7

Question The angle lies between  and and is such that       i.       Show that and hence find the exact value of .    ii.       It is given that the angleis such that . Without using a calculator, find the exact  value of . Solution      i.   We are given; We know that; Hence; Let […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | Oct-Nov | (P2-9709/21) | Q#7

Question The angle lies between  and and is such that      i.       Show that and hence find the exact value of .    ii.       It is given that the angleis such that . Without using a calculator, find the exact  value of . Solution      i.   We are given; We know that; Hence; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#8

Question The diagram shows the curve , for and its maximum point M.     i.       Show that    ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are functions of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#3

Question A.         Find B.   i.       Use the trapezium rule with three intervals to find an approximation to giving your answer correct to 3 significant figures. Solution A.   We are required to find; Rule for integration of  is: B.   i.   We are required to apply Trapezium Rule to evaluate; The trapezium rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/23) | Q#2

Question Solve the equation , giving all solutions in the interval . Solution We are required to solve the equation; We know that; Therefore; Hence; Now we have two options. Using calculator we can find that; We utilize the symmetry property of   to find another solution (root) of : Properties of Domain Range Odd/Even […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#8

Question The diagram shows the curve , for and its maximum point M. i.       Show that      ii.       Hence find the x-coordinate of M, giving your answer correct to 2 decimal places.. Solution      i.   We are given that; Therefore; We apply product rule to find the derivative. If  and  are functions of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#3

Question A.         Find B.   i.       Use the trapezium rule with three intervals to find an approximation to giving your answer correct to 3 significant figures. Solution A.   We are required to find; Rule for integration of  is:   B.   i.   We are required to apply Trapezium Rule to evaluate; The […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/22) | Q#2

Question Solve the equation , giving all solutions in the interval . Solution We are required to solve the equation; We know that; Therefore; Hence; Now we have two options. Using calculator we can find that; We utilize the symmetry property of   to find another solution (root) of : Properties of Domain Range Odd/Even […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#5

Question i.       Prove that  .    ii.       Hence a.  Find the exact value of b.  Evaluate Solution      i.   We are given that; except where  or undefined   provided that We have the trigonometric identity; Therefore; Hence;      ii.   a.   We are required to find the exact value of; As […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2014 | May-Jun | (P2-9709/21) | Q#2

Question Find the gradient of each of the following curves at the point for which x = 0.     i.           ii.         Solution      i.   We are required to find the gradient of the curve at the point for which x = 0. Therefore first we need to find the expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#5

Question i.       Show that .    ii.  Hence solve the equation  for . Solution      i.   We are given; We choose the left hand side; It can be rewritten as; Utilizing the algebraic formula; Utilizing the trigonometric identity; We can write now; We can rewrite the trigonometric identity as; Therefore; Hence;    ii.   We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/13) | Q#2

Question In the diagram, OADC is a sector of a circle with centre O and radius 3 cm. AB and CB are tangents  to the circle and angle  radians. Find, giving your answer in terms of  and ,       i.       the perimeter of the shaded region,    ii.       the area of the shaded region. Solution     i.   It is evident […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#11

Question The function  is defined for .      i.       Find the exact value of  for which .    ii.       State the range of .   iii.       Sketch the graph of .   iv.       Find an expression for . Solution i.   We are given that; We can write it as; We are given that , therefore; Using calculator; ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#5

Question      i.       Show that the equation  can be expressed as    ii.       Hence solve the equation  for . Solution i.   We are given; We know that; We can write the given equation as; We have the trigonometric identity; We can rearrange this identity as; Substituting in above equation; ii.   From (i) we know that we can write the given equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#3

Question Solve the equation  for . Solution We are given the equation; We have the trigonometric identity; We can rewrite it as; Substituting this in above equation; Now we have two options. Using calculator we can find that; Since we are required to solve the equation  for , we consider angles only in this range. Therefore; We utilize  the symmetry property […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/11) | Q#2

Question Find the value of  satisfying the equation . Solution We are given the equation; Using calculator we can find; Therefore; Using calculator we can find; Therefore;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/13) | Q#4

Question      i.       Prove the identity    ii.       Hence solve the equation  for . Solution i.   We utilize the relation; We have the trigonometric identity; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Using calculator; We utilize the periodic/symmetry property of   to find other solutions (roots) of :  Periodic/Symmetry […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#5

Question      i.       Prove the identity    ii.       Solve the equation  for . Solution i.   We have the trigonometric identity; From this we can substitute  in the above equation. Since ; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Using calculator; We utilize the periodic/symmetry property of   to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | May-Jun | (P1-9709/12) | Q#3

Question The reflex angle  is such that , where .      i.       Find an expression, in terms of , for a.   b.      ii.       Explain why  is negative for . Solution i.   A Reflex Angle is one which is more than 180° but less than 360°. a.     We are given that; We have the trigonometric identity; From this we can […]