# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#10

Question A curve is such that  , where a is a positive constant. The point A(a2, 3) lies on the  curve. Find, in terms of a,      i.       the equation of the tangent to the curve at A, simplifying your answer,    ii.       the equation of the curve.  It is now given that B(16,8) also lies on the curve.   iii.       Find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#6

Question Three points, A, B and C, are such that B is the mid-point of AC. The coordinates of A are (2,m) and  the coordinates of B are (n,-6), where m and n are constants. i.       Find the coordinates of C in terms of m and n. The line y =x + 1 passes through C and is […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#1

Question Find the set of values of k for which the curve  and the line  do not meet. Solution We can find the coordinates of intersection point of a curve and line. However, here we are required  to show that given curve and line do not meet that means there is no point of intersection   of the two. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#7

Question The equation of a curve is .     i.       Obtain an expression for    ii.       Explain why the curve has no stationary points.  At the point P on the curve, x = 2.   iii.       Show that the normal to the curve at P passes through the origin.   iv.       A point moves along the curve in such a way that its x-coordinate is decreasing […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#5

Question The line , where a and b are positive constants, intersects the x- and y-axes at the points A  and B respectively. The mid-point of AB lies on the line  and the distance .  Find the values of a and b. Solution We need to work through the problem statement very carefully to glean the information scattered  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#3

Question A curve has equation .     i.       Find the set of values of  for which .    ii.       Find the value of the constant  for which the line  is a tangent to the curve. Solution i.   We are required to find the set of values of x for which . We are given that; Therefore; We solve the following equation to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#11

Question The point P(3,5) lies on the curve  .      i.       Find the x-coordinate of the point where the normal to the curve at P intersects the x-axis.    ii.       Find the x-coordinate of each of the stationary points on the curve and determine the nature of  each stationary point, justifying your answers. Solution      i.   The point where […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#7

Question The diagram shows parts of the curves  and , intersecting at points A and  B.      i.       State the coordinates of A.    ii.       Find, showing all necessary working, the area of the shaded region. Solution      i.   It is evident that point A is the intersection point of the two curves given by equations; It is also […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#4

Question C is the mid-point of the line joining A(14,−7) to B(−6,3). The line through C perpendicular to AB  crosses the y-axis at D.      i.       Find the equation of the line CD, giving your answer in the form .     ii.       Find the distance AD. Solution i.   We are required to write equation of the line CD. To find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#11

Question Triangle ABC has vertices at A(−2,−1), B(4,6) and C(6,−3). i.       Show that triangle ABC is isosceles and find the exact area of this triangle. ii.    The point D is the point on AB such that CD is perpendicular to AB. Calculate the x-coordinate of  D. Solution      i.   An isosceles triangle is a triangle with (at least) two equal sides. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#11

Question The function  is defined by  for .      i.       Find the set of values of x for which f(x) ≤ 3.    ii.       Given that the line y=mx+c is a tangent to the curve y = f(x), show that The function g is defined by  for x ≥ k, where k is a constant.   iii.       Express   in the form , where a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#8

Question Three points have coordinates A(0,7), B(8,3) and C(3k,k). Find the value of the constant k for which       i.       C lies on the line that passes through A and B,    ii.       C lies on the perpendicular bisector of AB. Solution i.   If point C lies on line AB, then coordinates of point C must satisfy equation of line AB. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#8

Question A curve has equation  and passes through the points A(1,-1) and B(4,11). At each of  the points C and D on the curve, the tangent is parallel to AB. Find the equation of the  perpendicular bisector of CD. Solution We are required to find the equation of perpendicular bisector of CD. To find the equation of the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#6

Question a.   Find the values of the constant m for which the line  is a tangent to the curve . b.   The function f is defined for  by , where a and b are constants. The solutions of the equation  are x = 1 and x = 9. Find i.       the values of a and b,  ii.       y=the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#10

Question The diagram shows part of the curve , which touches the x-axis at the point P. The  point Q(3,4) lies on the curve and the tangent to the curve at Q crosses the x-axis at R.      i.       State the x-coordinate of P.  Showing all necessary working, find by calculation ii.       the x-coordinate of R, iii.    the area of the shaded […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#5

Question Two points have coordinates A(5,7) and B(9,-1).      i.       Find the equation of the perpendicular bisector of AB. The line through C(1,2) parallel to AB meets the perpendicular bisector of AB at the point X.    ii.       Find, by calculation, the distance BX. Solution a.   We are required to write equation of perpendicular bisector of AB with points A(5,7) and B(9,-1). […]