# 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#11

Question The diagram shows the graph of , where  for .      i.       Find an expression, in terms of , for  and explain how your answer shows that  is a decreasing function.    ii.       Find an expression, in terms of , for  and find the domain of .   iii.       Copy the diagram and, on your copy, sketch the graph of , making clear the relationship between […]

# 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 | May-Jun | (P1-9709/01) | Q#9

Question Relative to an origin , the position vectors of points  and  are given by i. Given that  is the point such that  , fFind the unit vector in the direction of . The position vector of the point  is given by , where  is a constant, and it is given that  , where  and  are constants. ii. Find the values of ,  and . […]

# 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#10

Question The equation of a curve is  . i.       Obtain expressions for  and .    ii.       Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.   iii.       Show that the normal to the curve at the point (−2, −2) intersects the x-axis at the point (−10, 0).   iv.       Find the area of the region enclosed […]

# 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#6

Question The diagram shows a rectangle ABCD. The point A is (2, 14), B is (−2, 8) and C lies on the x-axis. Find     i.       the equation of BC    ii.       the coordinates of C and D. Solution      i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form […]

# 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#4

Question Find the real roots of the equation Solution Multiplying the entire equation with ; Let ; Now we have two options; Since ; Imaginary roots. Real roots. Hence real roots of the given equation are;

# 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;

# 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#2

Question The diagram shows the curve .  The shaded region is bounded by the curve, the x-axis and the lines  and . Find the volume of the solid obtained when this shaded region is rotated completely about the x-axis, giving your answer in terms of . Solution Expression for the volume of the solid formed when the shaded […]

# 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#1

Question Find the value of the constant  for which the line  is a tangent to the curve . Solution Since line is tangent to the curve i.e. both intersect each other at a single point. To find that point; If two lines (or a line and a curve) intersect each other at a point then that point lies on […]

# 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#7

Question The second term of a geometric progression is 3 and the sum to infinity is 12.     i.       Find the first term of the progression. An arithmetic progression has the same first and second terms as the geometric progression.    ii.       Find the sum of the first 20 terms of the arithmetic progression. Solution i.   From the given information, we can […]