# 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#5

Question i.   Sketch, on the same diagram, the curves  and  for  . ii.     Hence state the number of solutions, in the interval , of the equations                      a.                          b.   Solution i.   We are required to sketch  and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#10

Question The diagram shows part of the curve  and the point  on the curve. The tangent at A cuts the x-axis at B and the normal at A cuts the y-axis at C.      i.       Find the coordinates of B and C.  ii.   Find the distance AC, giving your answer in the form  , where a and b are integers.   iii.       Find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#9

Question A curve has equation  and is such that .      i.       By using the substitution  , or otherwise, find the values of  for which the curve  has stationary points.    ii.     Find  and hence, or otherwise, determine the nature of each stationary point.   iii.       It is given that the curve  passes through the point . Find . Solution      i.   A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#7

Question A curve has equation  and a line has equation , where  is a constant.      i.       For the case where , the curve and the line intersect at the points A and B. Find the coordinates of the mid-point of AB.    ii.       Find the non-zero value of  for which the line is a tangent to the curve, and find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#3

Question In the diagram, OAB is a sector of a circle with centre O and radius 8 cm. Angle BOA  is  radians. OAC is a semicircle with diameter OA. The area of the semicircle  OAC is twice the area of the sector OAB.      i.       Find  in terms of .    ii.       Find the perimeter of the complete figure in terms […]

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

Question It is given that  , for . Show that  is an increasing function. Solution We are given the function; To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point. If  , the function  is increasing. If  , the function  is decreasing. If  , the test is inconclusive.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#6

Question Relative to an origin , the position vectors of three points A, B and C are given by ,         and         where  and  are constants.      i.       Show that  is perpendicular to  for all non-zero values of  and .    ii.       Find the magnitude of  in terms of  and .   iii.       For the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#5

Question      i.       Show that    ii.       Hence solve the equation For . Solution      i.   We have the equation; We have the trigonometric identity; Therefore;    ii.   To solve this equation   for ,  as demonstrated in (i), we can rewrite the equation as; We have the trigonometric identity; We can rewrite the identity as; Therefore; Now […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#8

Question      i.     Express  in the form , where ,  and  are constants.    ii.       The function  is defined by  for , where  is a constant. It is given that f is a one-one function. State the smallest possible value of . The value of  is now given to be 7.   iii.       Find the range of .   iv.       Find an expression for  and state the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#2

Question      i.       In the expression ,  is non-zero constant. Find the first 3 terms in the expansion of , in ascending powers of .    ii.       It is given that the coefficient of  in the expansion of  is zero. Find the value of . Solution i.   First rewrite the given expression in standard form. Expression for the Binomial […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | May-Jun | (P1-9709/11) | Q#4

Question The third term of a geometric progression is −108 and the sixth term is 32. Find      i.       the common ratio,    ii.       the first term,   iii.     the sum to infinity. Solution From the given information, we can compile following data about Geometric Progression (G.P); i.   Expression for the general term  in the Geometric […]