# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined by  for .      i.       Find the set of values of x for which f(x) ≤ 3.    ii.       Given that the line y=mx+c is a tangent to the curve y = f(x), show that The function g is defined by  for x ≥ k, where k is a constant.   iii.       Express   in the form , where a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#10

Question The diagram shows the part of the curve  for , and the minimum point M.      i.         Find expressions for ,  and .    ii.       Find the coordinates of M and determine the coordinates and nature of the stationary point on          the part of the curve for which .   iii.       Find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#9

Question A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that  each day a greater amount of water is lost.      i.       On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.          a.   How many […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#8

Question Three points have coordinates A(0,7), B(8,3) and C(3k,k). Find the value of the constant k for which       i.       C lies on the line that passes through A and B,    ii.       C lies on the perpendicular bisector of AB. Solution i.   If point C lies on line AB, then coordinates of point C must satisfy equation of line AB. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#7

Question      i.       Prove the identity .      ii.       Hence solve, for , the equation Solution i.   We are given the identity; We have the trigonometric identity; It can be rearranged as; Therefore; We have ; ii.   We are required to solve the equation; We have found in (i) that; Therefore; Using calculator we can find the value of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#6

Question The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at  P and angle  radians. The line OT meets the circle at Q.      i.       Express the perimeter of the shaded region PQT in terms of r and .   ii.       In the case where  and , find the area of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#5

Question In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle    radians and angle  radians. Denoting the lengths of BM and MC by x,        i.       find AM in terms of x,    ii.       show that .Solution i.   We are required to find AM. Consider right angled triangle AMC. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#4

Question Find the term independent of  in the expansion of      i.             ii.        Solution i.   Expression for the general term in the Binomial expansion of  is: In the given case : Hence; Since we are looking for the coefficient of the term independent of  i.e. , so we can  equate Hence, substituting ; Becomes; Hence coefficient of the term […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#3

Question Relative to an origin , the position vectors of points A and B are given by and The point C is such that . Find the unit vector in the direction of . Solution A unit vector in the direction of  is; Therefore for the given case; Therefore, we need to find . We are given that; A vector in […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#2

Question A curve is such that  . Given that the curve passes through the point , 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#1

Question Functions  and  are defined by , ,, Solve the equation  Solution   First we find left hand side of required equation . We are given; It can be written as; Therefore; Now we find right hand side of required equation . We are given; These can be written as; Therefore; Now for  substitute  ; Finally, equating left […]