# 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#10

Question The diagram shows a cube  in which the length of each side is 4 units. The vectors ,  and  are parallel to ,  and  respectively. The mid-points of  and  are  and  respectively and  is the centre of the square face . i.       Express each of the vectors  and  in terms of ,  and .    ii.       Use a scalar […]

# 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#11

Question The function  is defined by  for .      i.       Express  in the form , where ,  and  are constants.    ii.       State the range of .   iii.       Explain why  does not have an inverse. The function g is defined by   for , where  is a constant.   iv.       State the largest value of  for which  has an inverse.    v.       When  has this value, obtain an expression, in […]

# 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#9

Question A curve is such that  and the point P(2, 9) lies on the curve. The normal to the curve at P meets the curve again at Q. Find i.       the equation of the curve,    ii.       the equation of the normal to the curve at P   iii.       the coordinates of Q. Solution      i.   We are given the equation; 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#8

Question The equation of a curve is .     i.       Express  and  in terms of .    ii.       Find the  coordinates of the two stationary points and determine the nature of each stationary point. Solution      i.   We are given the equation; For the given case; Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation of  is: […]

# 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 | Oct-Nov | (P1-9709/01) | Q#6

Question The three points A (3, 8), B (6, 2) and C (10, 2) are shown in the diagram. The point D is such that the line DA is perpendicular to AB and DC is parallel to AB. Calculate the coordinates of D. Solution It is evident that point D is the intersection of lines AD & DC. Therefore to […]

# 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#2

Question Find the area of the region enclosed by the curve , the x-axis and the lines x = 1 and x = 4. Solution To find the area of region under the curve , we need to integrate the curve from point  to  along x-axis. For the given case; Rule for integration of  is:

# 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#1

Question Determine the set of values of the constant k for which the line  does not intersect the curve . Solution First we find the conditions when line and the curve intersect. This we can do by equating equations of line and the curve. If line and the curve do not intersect then above equation has no real […]

# 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#3

Question i.       Find the first three terms in the expansion of in ascending powers of .    ii.       Use the substitution in your answer to part (i) to find the coefficient of  in the expansion of . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   Substituting  in  Gives us We are interested only in […]

# 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#4

Question The 1st term of an arithmetic progression is  and the common difference is , where .     i.       Write down expressions, in terms of  and , for the 5th term and the 15th term.  The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms  of a geometric progression.    ii.    […]