# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#10

Question A curve has equation .      i.       Find the set of values of x for which y > 9.    ii.      Express  in the form , where a, b and c are constants, and  state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of x by  and  , where k is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows part of the curve  and three points A, B and C on the  curve with x-coordinates 1, 2 and 5 respectively. i.       A point P moves along the curve in such a way that its x-coordinate increases at  a constant rate of 0.04 units per second. Find the rate at which the y-coordinate of P  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#8

Question The function  is defined by  for .      i.       Solve the equation .    ii.       Find the range of .   iii.       Sketch the graph of . The function  is defined by  for .   iv.       State the maximum value of for which  has an inverse.    v.       Obtain an expression for . Solution i.   We have the function; We can write it as; We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#6

Question In the diagram, S is the point  and T is the point . The point Q lies on ST,  between S and T, and has coordinates . The points P and R lie on the x-axis  and y-axis respectively and OPQR is a rectangle.      i.       Show that the area, A, of the rectangle OPQR is given by […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#5

Question The diagram shows a rectangle ABCD in which point A is  and point B is . The diagonal AC has equation . Find, by calculation, the coordinates of  C and D. Solution We are required to find coordinates of the points C and D. First we find the coordinates of point C. It is evident from the diagram that point C […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#3

Question The equation of a curve is  .      i.    Find the gradient of the curve at the point where x = 2.    ii.   Find  and hence evaluate . Solution      i.   We are required to find gradient of the curve at a given point. Gradient (slope) of the curve at the particular point is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#2

Question Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6  cm and the height is 8 cm. The paper is cut from A to O and opened out to form the  sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#4

Question Relative to an origin , the position vectors of three points A and B are given by and      i.       In the case where , find the unit vector in the direction of .    ii.       Find the values of  for which angle . Solution      i.   We are given that; In the case where , A unit vector […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#1

Question Given that , where  is an acute angle in degrees, find, in terms of ,      i.           ii.          iii.        Solution      i.   We have; Since we are required to write it in terms of  where  , so we first write the given  expression in terms of  . We have the trigonometric identity; We can rewrite […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2013 | Oct-Nov | (P1-9709/12) | Q#7

Question a)   An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that  each mile takes 12 seconds longer than the preceding mile.            i.       Given that the nth mile takes 9 minutes, find the value of n.          ii.      Assuming […]