# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#9

Question The functions  and  are defined by   for     for       i.       Express  in terms of  and solve the equation .    ii.       On the same diagram sketch the graphs of  and , showing the coordinates of their point of intersection and the relationship between the graphs.   iii.       Find the set of values of  which satisfy . Solution i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#10

Question The diagram shows the line  and the curve , meeting at  and .     i.       Find the area of the shaded region.    ii.       Find the volume obtained when the shaded region is rotated through 360◦ about the y-axis. Solution     i.   It is evident from the diagram that; To find the area of region under the curve , we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#8

Question A curve  has a stationary point at . It is given that, where k is a constant. i.       Show that  and hence find the x-coordinate of the other stationary point, Q.    ii.       Find  and determine the nature of each of the stationary points P and Q.   iii.       Find . Solution i.   A stationary point  on the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#7

Question i.   A straight line passes through the point (2, 0) and has gradient . Write down the equation of the line. ii.  Find the two values of  for which the line is a tangent to the curve . For each value of , find the coordinates of the point where the line touches the curve.   iii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#6

Question Relative to an origin , the position vectors of points A and B are  and , respectively.      i.       Use a scalar product to calculate angle BOA. The point C is the mid-point of AB. The point D is such that .    ii.       Find . Solution      i.   We recognize that angle BOA is between  and . Therefore, we need […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#5

Question      i.       Given that  Show that, for real values of      ii.       Hence solve the equation  for . Solution i.   We have the equation; We have the trigonometric identity; We can rewrite it as; Thus Now we have two options; NOT POSSIBLE So we are left with ONLY option;      ii.   To solve the equation , for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#4

Question In the diagram, ABCD is a parallelogram with AB=BD=DC=10cm and angle ABD= 0.8 radians. APD and BQC are arcs of circles with centres B and D respectively.     i.       Find the area of the parallelogram ABCD.    ii.       Find the area of the complete figure ABQCDP.   iii.       Find the perimeter of the complete figure ABQCDP. Solution i.   Expression for the area […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#3

Question The diagram shows the curve  and the line  intersecting at points A, O and B.     i.       Show that the x-coordinates of A and B satisfy the equation .    ii.       Solve the equation  and hence find the coordinates of A and B, giving your answers in an exact form. Solution i.   We are given that points A & B are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#1

Question The coefficient of  in the expansion of , is 30. Find the value of . Solution Expression for the Binomial expansion of  is: In the given case: Hence; Since the coefficient of  in the expansion of , is 30:

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#2

Question The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10  terms given that the progression is     i.       an arithmetic progression    ii.       a geometric progression. Solution i.   From the given information, we can compile following data about Arithmetic Progression (A.P); Expression for the sum of  number of terms in […]