# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#11

Question The function  is defined for .      i.       Find the exact value of  for which .    ii.       State the range of .   iii.       Sketch the graph of .   iv.       Find an expression for . Solution i.   We are given that; We can write it as; We are given that , therefore; Using calculator; ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#10

Question A curve is such that . The curve has a stationary point at  where .      i.            State, with a reason, the nature of this stationary point.    ii.              Find an expression for .   iii.       Given that the curve passes through the point , find the coordinates of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is . The coordinates of A, B and C are ,  and  respectively.      i.       Find the equation of AD.    ii.       Find, by calculation, the coordinates of D. The point E is such that ABCE is a parallelogram.   iii.       Find the length […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#7

Question The diagram shows a pyramid OABCX. The horizontal square base OABC has side 8 units and the  centre of the base is D. The top of the pyramid, X, is vertically above D and XD = 10 units. The mid- point of OX is M. The unit vectors  and  are parallel to  and  respectively and the unit vector   is vertically upwards. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#6

Question The equation of a curve is , where  and  are constants.      i.       In the case where the curve has no stationary point, show that .    ii.       In the case where  and , find the set of values of  for which  is a decreasing               function of . Solution i.   We are given; Gradient […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#5

Question      i.       Show that the equation  can be expressed as    ii.       Hence solve the equation  for . Solution i.   We are given; We know that; We can write the given equation as; We have the trigonometric identity; We can rearrange this identity as; Substituting in above equation; ii.   From (i) we know that we can write the given equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#4

Question A curve has equation .      i.       Find .   A point moves along this curve. As the point passes through A, the x-coordinate is increasing           at a rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per            second.   ii.  Find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#2

Question The diagram shows a triangle AOB in which OA is 12 cm, OB is 5 cm and angle AOB is a right  angle. Point P lies on AB and OP is an arc of a circle with centre A. Point Q lies on AB and OQ is an arc of a circle with centre B. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#1

Question The diagram shows part of the curve . Find the volume obtained when the shaded region  is rotated through  about the y-axis. Solution Expression for the volume of the solid formed when the shaded region under the curve  is rotated completely about the y-axis is; We are given; We can rearrange it to change the subject; We can see that curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#3

Question      i.       Find the first 3 terms, in ascending powers of , in the expansion of . The coefficient of  in the expansion of  is 95.    ii.       Use the answer to part (i) to find the value of the positive constant . Solution i.   Expression for the Binomial expansion of  is: In the given case: Hence; ii.   To evaluate […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2014 | Oct-Nov | (P1-9709/12) | Q#8

Question a)   The sum, , of the first  terms of an arithmetic progression is given by . Find the  first term and the common difference. b)   A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the  first two terms is 12.8. Find the first term of the progression. Solution […]