# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#11

Question The diagram shows part of the curve  and a point  and  which lie on the curve. The tangent to the curve at B intersects the line  at the point C.      i. Find the coordinates of C.    ii. Find the area of the shaded region. Solution i.   We are required to find the coordinates of point C. It is evident […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#7

Question The diagram shows three points ,  and . The point X lies on  AB, and CX is perpendicular to AB. Find, by calculation,       i.       the coordinates of X,    ii.       the ratio AX : XB. Solution i.   We are required to find the coordinates of point X. It is evident from the diagram that point X is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#6

Question The non-zero variables x, y and u are such that . Given that , find the stationary value of u and determine whether this is a maximum or a  minimum value. Solution We are given that; We are also given; Substituting this value of  in the equation of . A stationary point  on the curve  is the point where gradient of the curve is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#3

Question   i.      Express the equation  as a quadratic equation in .  ii.       Solve the equation  for , giving solutions in terms of . Solution i.   We are given that; We know that; Therefore; We have the trigonometric identity; It can be rearranged as follows; Substituting this expression of  in the above equation; It can be rewritten as; This clearly is a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#2

Question The diagram shows a circle C with centre O and radius 3 cm. The radii OP and OQ  are extended to S and R respectively so that ORS is a sector of a circle with centre  O. Given that PS = 6 cm and that the area of the shaded region is equal to the area  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#1

Question A curve is such that  and    is a point on the curve. Find the equation of the curve. Solution We are given that curve  passes through the point  and we are required to find the equation of the curve. We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule for integration […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#8

Question The diagram shows a parallelogram OABC in which relative to an origin , the position vectors of three points A, B and C are given by and i.       Use a scalar product to find angle BOC. ii.       Find a vector which has magnitude 35 and is parallel to the vector . Solution      i.   We recognize that angle BOC […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#10

Question A function  is defined by , for , where  is a constant.      i.       In the case where , solve the equation . The function  is defined by ,  .    ii.        Find the set of values of  for which the equation   has no real solutions. The function h is defined by ,  .   iii.       Find an expression for . Solution i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#4

Question      i.       Find the first three terms in the expansion of  in ascending powers of .    ii.       Given that coefficient of  in the expansion of is 240, find the possible values of . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   To find the value of  in the expansion of  First we know […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/13) | Q#9

Question a)   In an arithmetic progression, the sum, , of the first  terms is given by  . Find the first term and the common difference of the progression. b)   The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the  geometric progression are also the 1st term, the 9th term […]