# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#7

Question      i.       Given that y = tan 2x, find .    ii.       Hence, or otherwise, show that and, by using an appropriate trigonometrical identity, find the exact value of   iii.       Use the identity cos 4x ≡ 2cos2 2x − 1 to find the exact value of Solution      […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#5

Question The diagram shows a chord joining two points, A and B, on the circumference of a circle with centre  O and radius r. The angle AOB is radians, where . The area of the shaded  segment is one sixth of the area of the circle.     i.       Show that  satisfies the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#4

Question i. Prove the identity    ii. Hence, solve the equation Solution      i.   We are given that; Utilizing addition formulae; We have algebraic formulae; We have double angle formula as; From this we can write; Substituting in above equation;    ii.   We are required to solve the equation; As demonstrated in […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#3

Question The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the  curve for 0 ≤ x ≤ 2π, and determine the nature of each of these stationary points. Solution We are required to find the x-coordinates of stationary points of the curve. A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#2

Question i. Prove the identity    ii. Hence solve the equation For . Solution      i.   We are given that; We utilize following two addition formulae;    ii.   We are required to solve; As demonstrated in (i); Therefore; Using calculator we can find; Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry All other […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#6

Question In the diagram, ABC is a triangle in which AB=4 cm, BC=6 cm and angle ABC=150◦. The line CX is perpendicular to the line ABX.      i.       Find the exact length of BX and show that    ii.       Show that the exact length of AC is  cm. Solution i.   It is evident that BCX is a right-angled triangle. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#2

Question Given that  , find the exact value of      i.           ii.        Solution i.   We are given that; We can rewrite it as; We have the trigonometric identity; We can also rewrite it as; For the given case; Substituting   in this equation; ii.   We have the trigonometric relation; Taking square of both sides; For the given case; Substituting values […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#2

Question Solve the equation  for . Solution First we need to manipulate the given equation to write it in a single trigonometric ratio i.e. ; Dividing the entire equation with ; Using the relation ; 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows an open container constructed out of 200 cm2 of cardboard. The two vertical end pieces are isosceles triangles with sides  cm,  cm and  cm, and the two side pieces are rectangles of length  cm and width  cm, as shown. The open top is a horizontal rectangle.      i.       Show that      ii.       Show that the volume,  cm3, of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#3

Question In the diagram, AOB is a sector of a circle with centre O and radius 12 cm. The point A lies on the side CD of the rectangle OCDB. Angle  radians. Express the area of the shaded region in the form  , stating the values of the integers  and . Solution From the given information we can compile […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#6

Question In the diagram, ABC is a triangle in which  cm,  cm and angle . The line CX is perpendicular to the line ABX. i.       Find the exact length of BX and show that .    ii.       Show that the exact length of AC is Solution From the given information we can compile following data; i.   To find  we consider the […]