# 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, […]

# 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#9

Question A curve has equation  and a line has equation , where  is a constant. i.Show that, for all values of k, the curve and the line meet. ii.State the value of k for which the line is a tangent to the curve and find the coordinates of the  point where the line touches the […]

# 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#8

Question A curve passes through (0, 11) and has an equation for which , where a and b are  constants.      i.      Find the equation of the curve in terms of a and b.    ii.       It is now given that the curve has a stationary point at (2, 3). Find the values […]

# 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#4

Question Two points A and B have coordinates  and respectively. The line BC is perpendicular to  AB and intersects the x-axis at C.     i.       Find the equation of BC and the x-coordinate of C.    ii.       Find the distance AC, giving your answer correct to 3 decimal places. Solution i.   We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#10

Question The equation of a curve is  and the equation of a line is , where k is a constant. i.       Find the set of values of k for which the line does not meet the curve. In the case where k = 15, the curve intersects the line at points A and B. ii. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#3

Question The diagram shows part of the curve  and the line , intersecting at the origin O and the point R. Point P lies on the line  between O and R and the x-coordinate of P is .  Point Q lies on the curve and PQ is parallel to the y-axis.      i.       Express the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#10

Question A curve has equation  . The point A on the curve has coordinates .     i.                      a.   Find and simplify the equation of the normal through A.              b.   Find the x-coordinate of the point where this normal meets the curve again. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#7

Question The diagram shows part of the curve with equation y = k(x3 − 7×2 + 12x) for some constant k. The  curve intersects the line y = x at the origin O and at the point A(2, 2).     i.       Find the value of k.    ii.       Verify that the curve meets the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#3

Question Two points A and B have coordinates (3a, -a) and (-a, 2a) respectively, where a is a positive constant.      i.               Find the equation of the line through the origin parallel to AB.    ii.               The length of the line AB is  units. Find the value of a.   Solution i.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/11) | Q#2

Question A line has equation y = x + 1 and a curve has equation y = x2 + bx + 5. Find the set of values of the  constant b for which the line meets the curve. Solution If two lines (or a line and a curve) intersect each other at a point then […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#11

Question The diagram shows part of the curve intersecting the x-axis at the origin O  and at . The line AB intersects the y-axis at B and has equation .     i.       Show that AB is the tangent to the curve at A.    ii.       Show that the area of the shaded region can be […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#9

Question A curve has equation  and a line has equation , where  is a constant. i.Find the set of values of  for which the curve and the line meet. ii.The line is a tangent to the curve for two particular values of . For each of these values find  the x-coordinate of the point at […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#4

Question A straight line cuts the positive x-axis at A and the positive y-axis at B(0, 2). Angle radians, where O is the origin.      i.       Find the exact value of the x-coordinate of A.    ii.       Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx […]

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

Question The one-one function f is defined by  for , where c is a  constant.   i.       State the smallest possible value of c. In parts (ii) and (iii) the value of c is 4.    ii.       Find an expression for  and state the domain of .   iii.       Solve the equation , giving your […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#8

Question i.       The tangent to the curve y = x3 − 9×2 + 24x − 12 at a point A is parallel to the line  y = 2 − 3x. Find the equation of the tangent at A.    ii.       The function f is defined by f(x) = x3 − 9×2 + 24x − 12 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/13) | Q#6

Question The coordinates of points A and B are (−3k – 1, k + 3) and (k + 3, 3k + 5)  respectively, where k is a constant (k ≠ −1). i.       Find and simplify the gradient of AB, showing that it is independent of k.  ii.       Find and simplify the equation of the perpendicular […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#11

Question The diagram shows part of the curve . The line y = 4 intersects the curve at  the points P and Q. i. Show that the tangents to the curve at P and Q meet at a point on the line y = x. ii. Find, showing all necessary working, the volume obtained when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#9

Question A curve is such that  and (2,5) is a point on the curve.     i.       Find the equation of the curve.    ii.       A point P moves along the curve in such a way that the y-coordinate is  increasing at a constant rate of 0.06 units per second. Find the rate of change of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | May-Jun | (P1-9709/12) | Q#8

Question Points A and B have coordinates (h, h) and (4h + 6, 5h) respectively. The equation of  the perpendicular bisector of AB is 3x + 2y = k. Find the values of the constants h  and k. Solution We are given that line AB has coordinates of the two points  and . We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/11) | Q#2

Question The equation of a curve is  , where  is a constant.     i.       Find the set of values of  for which the whole of the curve lies above the x-axis.    ii.       Find the value of  for which the line y + 2x = 7 is a tangent to the curve. Solution i.   […]