# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#10

Question The function f is defined by    ,      i.       Sketch, in a single diagram, the graphs of  and , making clear the relationship between the two graphs. The function g is defined by    ,    ii.       Express  in terms of , and hence show that the maximum value of  is 9. The function h is defined by     , […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#5

Question The function f is such that  for , where  and  are positive constants. The maximum value of  is 10 and the minimum value is −2.      i.       Find the values of  and .    ii.       Solve the equation .   iii.       Sketch the graph of . Solution i.   We have the function; We know that; For ; For ; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows the curve  and the points P(0,1) and Q(1,2) on the curve. The shaded region is bounded by the curve, the y-axis and the line y = 2.     i.       Find the area of the shaded region.    ii.       Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. Tangents are drawn to the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#8

Question The equation of a curve is . i.       Show that the equation of the normal to the curve at the point  is This normal meets the curve again at the point Q.    ii.       Find the coordinates of Q.   iii.       Find the length of PQ. Solution i.   We are required to find the equation of the normal to the curve. To […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#7

Question A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side x cm and the other piece is bent to form a circle of radius r cm (see diagram). The total area of the square and the circle is Acm2. i.       Show that    ii.       Given that x and r […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#6

Question In the diagram, the circle has centre O and radius 5 cm. The points P and Q lie on the circle, and the arc length PQ is 9 cm. The tangents to the circle at P and Q meet at the point T. Calculate i.       angle POQ in radians,    ii.       the length of PT,   iii.       the area of the shaded region. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#4

Question The diagram shows a semicircular prism with a horizontal rectangular base . The vertical ends   and  are semicircles of radius 6 cm. The length of the prism is 20 cm. The mid-point of  is the origin , the mid-point of  is  and the mid-point of  is . The points  and  are the  highest points of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#2

Question Prove the identity Solution We are given the equation; We can rewrite it as; We have the trigonometric identity; We can rewrite it as; Therefore;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/11) | Q#1

Question Find the value of the coefficient of  in the expansion of Solution Expression for the general term in the Binomial expansion of  is: In the given case: Hence; Since we are looking for the term with  , we can  equate Finally substituting  in: Therefore the coefficient of  is .