# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2020 | Feb-Mar | (P1-9709/12) | Q#12

Question A diameter of a circle C1 has end-points at (−3, −5) and (7, 3). a)   Find an equation of the circle C1. The circle C1 is translated by  to give circle C2, as shown in the diagram. b)  Find an equation of the circle C2.  The two circles intersect at points R and S. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows part of the curve , and the lines x = 1 and x = 3. The point A  on the curve has coordinates (2, 3). The normal to the curve at A crosses the line x = 1 at B. (i)       Show that the normal AB has equation . (ii)    […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/13) | Q#6

Question A line has equation  and a curve has equation , where k is a constant. i.Find the set of values of  for which the line and curve meet at two distinct points. i.For each of two particular values of , the line is a tangent to the curve. Show that these two tangents meet […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#10

Question The diagram shows part of the curve . The curve intersects the x-axis at A. The normal to the curve at A intersects the y-axis at B. i.       Obtain expressions for  and ii.    Find the coordinates of B. iii. Find, showing all necessary working, the area of the shaded region. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/12) | Q#2

Question The point M is the mid-point of the line joining the points (3, 7) and (−1, 1). Find the equation of the  line through M which is parallel to the line .   Solution We are required to find the equation of the line which passes through the point M and is parallel  to […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#9

Question A curve for which passes through the point (2,3).      i.       Find the equation of the curve.     ii.      Find .  iii.      Find the coordinates of the stationary point on the curve and, showing all necessary working,  determine the nature of this stationary point. Solution i.   We can find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#6

Question A straight line has gradient  and passes through the point (0, −2). Find the two values of for  which the line is a tangent to the curve y = x2 − 2x + 7 and, for each value of , find the coordinates  of the point where the line touches the curve. Solution We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Oct-Nov | (P1-9709/11) | Q#3

Question The line y = ax + b is a tangent to the curve y = 2×3 − 5×2 − 3x + c at the point (2, 6). Find the  values of the constants a, b and c. Solution We are given equation of the line as; We are given equation of the curve as; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#10

Question The diagram shows part of the curve with equation  and the tangent to the curve at the  point A. The x-coordinate of A is 4.      i.      Find the equation of the tangent to the curve at A.    ii.       Find, showing all necessary working, the area of the shaded region.  iii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/13) | Q#7

Question The coordinates of two points A and B are (1, 3) and (9, −1) respectively and D is the mid-point of  AB. A point C has coordinates (x, y), where x and y are variables. i.State the coordinates of D. ii.It is given that CD2 = 20. Write down an equation relating x and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#11

Question The diagram shows part of the curve and the minimum point M. i.Find the expressions for and ii.Find the coordinates of M. The shaded region is bounded by the curve, the y-axis and the line through M parallel to the x-axis. iii.Find, showing all necessary working, the area of the shaded region. Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#9

Question The curve C1 has equation y = x2− 4x + 7. The curve C2 has equation y2 = 4x + k, where k is a constant. The tangent to C1 at the point where x = 3 is also the tangent to C2 at the point P. Find the  value of k and the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#6

Question The equation of a curve is and the equation of a line is . i.State the smallest and largest values of y for both the curve and the line for . ii.Sketch, on the same diagram, the graphs of and for . iii.State the number of solutions of the equation for . Solution i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/12) | Q#2

Question Two points A and B have coordinates (1, 3) and (9, −1), respectively. The perpendicular bisector of  AB intersects the y-axis at the point C. Find the coordinates of C. Solution Coordinates of point C which is y-intercept of the perpendicular bisector of AB. The point at which curve (or line) intercepts y-axis, the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#11

Question The diagram shows part of the curve  and a point P(2, 1) lying on the curve. The normal  to the curve at P intersects the x-axis at Q.     i.       Show that the x-coordinate of Q is .    ii.       Find, showing all necessary working, the area of the shaded region. Solution i.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#4

Question The diagram shows a trapezium ABCD in which the coordinates of A, B and C are (4, 0), (0, 2) and  (h, 3h) respectively. The lines BC and AD are parallel, angle ABC = 90o and CD is parallel to the x- axis.      i.       Find, by calculation, the value of h.    ii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | May-Jun | (P1-9709/11) | Q#2

Question The line , where  is a constant, is a tangent to the curve  at the point  on the curve. i.Find the value of . ii.Find the coordinates of . Solution i. We are given equation of the line as; We are given equation of the curve as; It is given that line is tangent […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#9

Question The diagram shows part of the curve with equation . The shaded region is bounded by  the curve, the x-axis and the line x = 3.      i.       Find, showing all necessary working, the volume obtained when the shaded region is rotated  through 360O about the x-axis.    ii.       P is the point on […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2019 | Feb-Mar | (P1-9709/12) | Q#4

Question A curve has equation .     i.       Find  and  .     ii.       Find the x-coordinates of the stationary points and, showing all necessary working, determine  the nature of each stationary point. Solution i.   We are required to find and  . Therefore, we find the derivative of equation of the curve. Gradient (slope) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/13) | Q#10

Question The diagram shows part of the curve  and the lines  and . The curve and the line  intersect at point A. i.       Find, showing all necessary working, the volume obtained when the shaded region is rotated  through 3600 about the x-axis.    ii.       Find the equation of the normal to the curve at A, […]