# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#11

Question The diagram shows part of the curve  and a point P(6,5) lying on the curve. The line  PQ intersects the x-axis at Q(8,0).      i.       Show that PQ is a normal to the curve.   ii.       Find, showing all necessary working, the exact volume of revolution obtained when the shaded  region is rotated through 360o about the x-axis. [In part (ii) you […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#10

Question The diagram shows a cuboid OABCPQRS with a horizontal base OABC in which AB = 6 cm and  OA = a cm, where a is a constant. The height OP of the cuboid is 10 cm. The point T on BR is such that BT = 8 cm, and M is the mid-point of AT. Unit vectors […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#9

Question      i.       Express  in the form , where a, b and c are constants. The function  is defined for , where  is a constant.    ii.       State the smallest value of  for which  is one-one.   iii.       For the case where , find n expression for and state the domain of . Solution i.   We have the expression; We use method of “completing square” to obtain […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#8

Question The first term of a progression is  and the second term is . i.       For the case where the progression is arithmetic with a common difference of 12, find the  possible values of  and the corresponding values of the third term.    ii.       For the case where the progression is geometric with a sum to infinity of 8, find the third term. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#7

Question The diagram shows a circle with centre A and radius r. Diameters CAD and BAE are perpendicular  to each other. A larger circle has centre B and passes through C and D.      i.       Show that the radius of the larger circle is .    ii.       Find the area of the shaded region in terms of . Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#6

Question A curve has equation  and a line has equation , where  is a constant.      i.       Show that the x-coordinates of the points of intersection of the line and the curve are given by  the equation .    ii.       For the case where the line intersects the curve at two points, it is given that the x-coordinate  of one of the points of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#5

Question A curve has equation  .      i.         Find  and     ii.        Find the coordinates of the stationary points and state, with a reason, the nature of each                   stationary point. Solution i.   We are given that; Gradient (slope) of the curve is the derivative of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#4

Question      i.       Show that the equation  can be expressed as     ii.       Hence solve the equation  for . Solution i.   We are given that; Since ; We have the trigonometric identity; From this we can write; Therefore; ii.   We are required to solve the equation  for . From (i), we know that; Therefore; Let ; Since; We have two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#3

Question Solve the equation . Solution i.   We are given that; Using calculator; Let ; Now we have two options. Since ;  is not possible

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#2

Question The function f is such that  and . Find . Solution i. We are given that; We are also given that . 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; Rule for integration of  is: Rule for integration of  is: Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2015 | Oct-Nov | (P1-9709/11) | Q#1

Question In the expansion of , where a is a non-zero constant, show that the coefficient of  is zero. Solution Expression for the Binomial expansion of  is: We are given; In the given case: Hence; It is evident that to get the terms containing  in the product of  we need; This will result in terms containing ; Hence coefficient of in the expansion […]