# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#4

Question The diagram shows the graph of  for .      i.       Find the values of ,  and .    ii.      Find the smallest value of  in the interval  for which . Solution i.   We are given that; When we observe given graph of this function, it is evident that y-intercept of the graph, i.e. when , is 3. Hence for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#11

Question The diagram shows the curve  for . The curve has a maximum point at A and a minimum point on the x-axis at B. The normal to the curve at C (2, 2) meets the normal to the curve at B at the point D. i.       Find the coordinates of A and B.    ii.       Find the equation of the normal […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#9

Question The diagram shows part of the curve . i.       Find the gradient of the curve at the point where .    ii.       Find the volume obtained when the shaded region is rotated through 360o about the x-axis, giving your answer in terms of . Solution      i.   Gradient (slope) of the curve at the particular point is the derivative of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#8

Question The diagram shows points A, B and C lying on the line . The point A lies on the y-axis and AB = BC. The line from D(10, −3) to B is perpendicular to AC. Calculate the coordinates of B and C. Solution      i.   First we find the coordinates of point B. We recognize that point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#6

Question Relative to an origin , the position vectors of points  and  are given by i.      Find the value of  and hence state whether angle AOB is acute, obtuse or a right angle.    ii.       The point  is such that . Find the unit vector in the direction of OX. Solution i.   We recognize that  is angle between  and  . Hence […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#5

Question The diagram shows a circle with centre O. The circle is divided into two regions,  and , by the radii OA and OB, where angle  radians. The perimeter of the region  is equal to the length of the major arc AB. i.       Show that .    ii.       Given that the area of region  is 30 cm2, find the area of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#2

Question Find the set of values of  for which the line  intersects the curve  at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#1

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; We have the trigonometric relation; Therefore;

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#10

Question The function  is defined by  for , where  is a constant.      i.       Express  in the form , where ,  and  are constants.    ii.       State the value of  for which the graph of  has a line of symmetry.   iii.        When  has this value, find the range of . The function g is defined by  for .   iv.       Explain why g […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#3

Question i.       Find the first three terms in the expansion of in ascending powers of .    ii.       Hence find the value of the constant  for which there is no term in   in the expansion of . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   To find the value of the constant  for which […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | May-Jun | (P1-9709/01) | Q#7

Question a) Find the sum to infinity of the geometric progression with first three terms 0.5, 0.53 and 0.55. b) The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200. Find the sum of all the terms in […]