# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | May-Jun | (P2-9709/02) | Q#7

Question The diagram shows the part of the curve y=ex cos x for . The curve meets the y-axis at the  point A. The point M is a maximum point. i. Write down the coordinates of A. ii. Find the x-coordinate of M. iii. Use the trapezium rule with three intervals to estimate […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | Oct-Nov | (P2-9709/02) | Q#7

Question      i.       Prove the identity    ii.       Using the identity, or otherwise, find the exact value of Solution      i.   We are given that; We have algebraic formula; Since we know that ; We have the trigonometric identity; From this we can write ; Since we know that ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | Oct-Nov | (P2-9709/02) | Q#6

Question i.       Express  in the form , where  and , giving the  exact value of R and the value of  correct to 2 decimal places.    ii.       Hence solve the equation Giving all solutions in the interval . Solution      i.   We are given the expression; We are required to write […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | May-Jun | (P2-9709/02) | Q#6

Question      i.       Express cos2 x in terms of cos 2x.    ii.       Hence show that   iii.       By using an appropriate trigonometrical identity, deduce the exact value of Solution      i.   We are required to show   in terms of . From this we can write; Therefore;    ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2007 | May-Jun | (P2-9709/02) | Q#5

Question      i. By sketching a suitable pair of graphs, show that the equation Where x is in radians, has only one root in the interval .    ii. Verify by calculation that this root lies between 1.0 and 1.2.   iii. Show that this root also satisfies the equation   iv. Use the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | May-Jun | (P1-9709/01) | Q#8

Question The function f is defined by  for . It is given that  and  .      i.       Find the values of  and .    ii.       Find the x-coordinates of the points where the curve  intersects the x-axis.   iii.       Sketch the graph of . Solution i.   We have the function; We are given that; Adding both equations gives us; Substituting  in anyone of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | Oct-Nov | (P1-9709/01) | Q#5

Question      i.       Show that the equation  can be written as .    ii.       Hence solve the equation   for . Solution i.   We are given the equation; We have the trigonometric relation; Substituting  in the above equation; We have the trigonometric identity; We  can rewrite it as; Substituting it in above equation; ii.   To solve  for , as demonstrated in (i) we can […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | Oct-Nov | (P1-9709/01) | Q#7

Question In the diagram, AB is an arc of a circle, centre O and radius r cm, and angle  radians. The point X lies on OB and AX is perpendicular to OB. i.       Show that the  area, Acm2, of the shaded region AXB is given by     ii.       In the case where  and , find the perimeter of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | May-Jun | (P1-9709/01) | Q#5

Question In the diagram, OAB is a sector of a circle with centre O and radius 12 cm. The lines AX and BX are tangents to the circle at A and B respectively. Angle  radians. i.       Find the exact length of AX, giving your answer in terms of .    ii.       Find the area of the shaded region, giving your answer […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2007 | May-Jun | (P1-9709/01) | Q#3

Question Prove the identity Solution We have the trigonometric relation; Therefore we can write the given expression as; We have the trigonometric identity; Therefore we can write the above expression as; We have the trigonometric identity; We can also rewrite the identity as; Therefore we can write; As;