Hits: 63

# 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: 63     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: 59   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; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#8

Hits: 100   Question i.       Express  in the form , where  and , stating the  exact value of R and and giving the value of  correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval .   iii.       Write down the least value of  as  varies. Solution      […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#6

Hits: 135     Question The parametric equations of a curve are x = 1 + 2 sin2θ , y = 4 tanθ , i. Show that    ii. Find the equation of the tangent to the curve at the point where , giving your answer in  the form y = mx + c. Solution […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#5

Hits: 105     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=1.1 and x=1.2.   iii.       Use the iterative formula to determine this root correct to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#3

Hits: 74     Question The diagram shows the part of the curve  for . Find the x-coordinates of the  points on this part of the curve at which the gradient is 4. Solution We are required to find the x-coordinate of the points on the curve where gradient is 4. Therefore first we need […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | Q#8

Hits: 61   Question i.       Prove that    ii.       Hence a.   Solve for  the equation b.   find the exact value of . Solution      i.   We are given that; We utilize following relations;   provided that   provided that Therefore; We utilize following relation; Therefore;    ii.   We are required to solve; As […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/22) | Q#8

Hits: 123   Question i.       Prove that    ii.       Hence a.   Solve for  the equation b.   find the exact value of . Solution      i.   We are given that; We utilize following relations;   provided that   provided that Therefore; We utilize following relation; Therefore;    ii.   We are required to solve; As […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#8

Hits: 146   Question i.       Express  in the form , where  and , Give the exact value of R and the value of  correct to 2 decimal places.    ii.       Hence solve the equation for .   iii.       Find the greatest and least possible values of  as  varies. Solution      i.   We are given […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#2

Hits: 148   Question A curve has parametric equations Find the exact gradient of the curve at the point for which . Solution      i.   Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point. Gradient (slope) of the curve at a particular […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#5

Hits: 612  Question i.    Sketch, on the same diagram, the graphs of  and  for . ii.       Verify that  is a root of the equation , and state the other root of this equation for which .  iii.       Hence state the set of values of , for , for which Solution i.   We are required to sketch  and  for . First we sketch  for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#5

Hits: 1038   Question The diagram represents a metal plate , consisting of a sector  of a circle with centre  and radius , together with a triangle  which is right-angled at C. Angle  radians and  is perpendicular to . i.       Find an expression in terms of  and  for the perimeter of the plate. ii.    For the case where  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#9

Hits: 1117   Question The function f is such that , for , where  is a constant.      i.       In the case where ,         a.   Find the range of ,        b.   Find the exact solutions of the equation .    ii.       In the case where ,        a.   Sketch the graph of  ,     […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/12) | Q#4

Hits: 933   Question A curve has equation  and P(2, 2) is a point on the curve.     i.       Find the equation of the tangent to the curve at P.    ii.       Find the angle that this tangent makes with the x-axis. Solution i.   To find the equation of the line either we need coordinates of the two points on the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#5

Hits: 389 Question      i.       Given that  Show that, for real values of      ii.       Hence solve the equation  for . Solution i.   We have the equation; We have the trigonometric identity; We can rewrite it as; Thus Now we have two options; NOT POSSIBLE So we are left with ONLY option;      ii.   To solve the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 1739 Question In the diagram, ABCD is a parallelogram with AB=BD=DC=10cm and angle ABD= 0.8 radians. APD and BQC are arcs of circles with centres B and D respectively.     i.       Find the area of the parallelogram ABCD.    ii.       Find the area of the complete figure ABQCDP.   iii.       Find the perimeter of the complete figure ABQCDP. Solution i.   Expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#6

Hits: 1273 Question The diagram shows a circle  touching a circle  at a point . Circle  has centre A and radius 6cm, and circle  has centre B and radius 10 cm. Points D and E lie on  and  respectively and DE is parallel to AB. Angle  radians and angle  radians.     i.       By considering the perpendicular distances of D […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#3

Hits: 555 Question      i.       Sketch, on a single diagram, the graphs of  and  for .    ii.       Write down the number of roots of the equation  in the interval .   iii.       Deduce the number of roots of the equation  in the interval . Solution     i.        ii.   If two lines (or a line and a curve) intersect each other […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#8

Hits: 382 Question i.  Prove the identity    ii.  Hence solve the equation   , for .   Solution i.   We have the equation; We have the relation , therefore, We have the trigonometric identity; We can rewrite it as; Therefore; Using formula;      ii.   To solve the equation  , for , as demonstrated in (i), we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#7

Hits: 1503 Question In the diagram, AB is an arc of a circle, centre O and radius 6 cm, and angle  radians. The line AX is a tangent to the circle at A, and OBX is a straight line.     i.       Show that the exact length of AX is  cm.  Find, in terms of  and ,    ii.       the area of the shaded […]