Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#11

Question A curve has equation , where k is a non-zero constant.      i.       Find the x-coordinates of the stationary points in terms of k, and determine the nature of each  stationary point, justifying your answers.    ii.   The diagram shows part of the curve for the case when k = 1. Showing all necessary working, find  the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#10

Question A curve is such that  , where a is a positive constant. The point A(a2, 3) lies on the  curve. Find, in terms of a,      i.       the equation of the tangent to the curve at A, simplifying your answer,    ii.       the equation of the curve.  It is now given that B(16,8) also lies on the curve.   iii.       Find […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#9

Question a.   Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and  second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r  respectively. Find the value of the common sum to infinity. b.   The first term of an arithmetic progression […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#8

Question      i.      Express  in the form , where a, b and c are constants.    ii. Functions f and g are both defined for . It is given that  and     . Find .   iii.       Find  and give the domain of .   Solution      i.   We have the expression; We use method of “completing square” […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#7

Question The diagram shows a triangular pyramid ABCD. It is given that i.       Verify, showing all necessary working, that each of the angles DAB, DAC and CAB is 90o.    ii.       Find the exact value of the area of the triangle ABC, and hence find the exact value of the volume of the pyramid.  [The volume V of a pyramid of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#6

Question Three points, A, B and C, are such that B is the mid-point of AC. The coordinates of A are (2,m) and  the coordinates of B are (n,-6), where m and n are constants. i.       Find the coordinates of C in terms of m and n. The line y =x + 1 passes through C and is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#5

Question The diagram shows a major arc AB of a circle with centre O and radius 6 cm. Points C and D on OA  and OB respectively are such that the line AB is a tangent at E to the arc CED of a smaller  circle also with centre O. Angle COD = 1.8 radians. i.       Show that […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#4

Question The function f is such that  for , where n is an integer. It is given that  f is an increasing function. Find the least possible value of n. Solution We are given function; We are also given that it is an increasing function. To test whether a function  is increasing or decreasing at a particular point , […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#3

Question Showing all necessary working, solve the equation  for . Solution We are given equation; We know that ; therefore, Now we have two options. Using calculator we can find that; We have following properties of . Properties of Domain Range Periodicity Odd/Even Translation/ Symmetry We utilize the periodicity property of   to find other solutions (roots) of . Therefore; For; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#2

Question The coefficient of  in the expansion of  is 100. Find the value of the constant  a. Solution We can find the coefficient of  in the expansion of given expression by finding coefficients of  in  the expansion of individual terms of expression and then adding them. Let us first find the coefficient of  in the expansion of . We are given […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#1

Question Find the set of values of k for which the curve  and the line  do not meet. Solution We can find the coordinates of intersection point of a curve and line. However, here we are required  to show that given curve and line do not meet that means there is no point of intersection   of the two. […]