# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#4

Question i.       Sketch the curve  for . ii.       By adding a suitable straight line to your sketch, determine the number of real roots of the equation State the equation of the straight line. Solution i.   We are required to sketch  for ; We can sketch the graph of  for  as follows. We can find the points of the graph as follows. ii.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#6

Question A curve has equation  and a line has equation , where  is a non-zero constant.     i.       Find the set of values of  for which the curve and the line have no common points.    ii.       State the value of  for which the line is a tangent to the curve and, for this case, find the coordinates of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#8

Question The diagram shows a rhombus ABCD in which the point A is (−1, 2), the point C is (5, 4) and the point B lies on the y-axis. Find      i.       the equation of the perpendicular bisector of AC,    ii.       the coordinates of B and D,   iii.       the area of the rhombus. Solution i.   Consider the diagram below. It is evident that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows parts of the curves  and  and their points of intersection  and . The x-coordinates of  and  are and  respectively.     i.       Show that  and  are roots of the equation . Solve this equation and hence state the value of  and the value of .    ii.       Find the area of the shaded region between the two curves. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/13) | Q#2

Question Points A, B and C have coordinates (2, 5), (5, −1) and (8, 6) respectively.     i.       Find the coordinates of the mid-point of AB.    ii.       Find the equation of the line through C perpendicular to AB. Give your answer in the form ax + by + c = 0. Solution i.   To find the mid-point of a line we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#11

Question The diagram shows part of the curve  . The curve cuts the y-axis at A and the line  at B.     i.       Show that the equation of the line AB is .    ii.       Find the volume obtained when the shaded region is rotated through 360o about the x-axis. Solution     i.   To find the equation of the line either we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#8

Question The diagram shows part of the curve  and the line . The curve and the line meet at points A and B. i.       Find the coordinates of A and B.    ii.       Find the length of the line AB and the coordinates of the mid-point of AB. Solution i.   If two lines (or a line and a curve) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#11

Question The equation of a curve is  .     i.        Find an expression for  and determine, with a reason, whether the curve has any stationary points.    ii.       Find the volume obtained when the region bounded by the curve, the coordinate axes and the line  is rotated through 360o about the x-axis.   iii.       Find the set of values of  for which […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/11) | Q#10

Question The equation of a curve is .     i.       Show that the equation of the normal to the curve at the point (3, 6) is .    ii.       Given that the normal meets the coordinate axes at points A and B, find the coordinates of the mid-point of AB.   iii.       Find the coordinates of the point at which the normal meets […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#9

Question The diagram shows part of the curve  which has a minimum point at M. The line  intersects the curve at the points A and B.     i.        Find the coordinates of A, B and M.    ii.       Find the volume obtained when the shaded region is rotated through 360◦ about the x-axis. Solution i.   It is evident from […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/13) | Q#5

Question The equation of a curve is such that . Given that the curve passes through the point , find     i.       the equation of the normal to the curve at P    ii.       the equation of the curve. Solution     i.   To find the equation of the line either we need coordinates of the two points on the line (Two-Point form of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#10

Question The equation of a curve is .     i.        Find .    ii.      Find the equation of the tangent to the curve at the point where the curve intersects the y-axis.   iii.       Find the set of values of  for which  is an increasing function of . Solution     i.   Rule for differentiation of  is: Rule […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#9

Question The diagram shows the curve  and the line , which intersect at points A and B. Find the area of the shaded region. Solution It is evident from the diagram that; However, we need the limits to integrate the equations of line & curve.  It is evident that these limits are from point A to point B. Therefore we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/12) | Q#4

Question In the diagram,  is the point  and  is the point . The line  passes through  and is parallel to . The line  passes through  and is perpendicular to . The lines  and  meet at . Find the coordinates of . Solution It is evident from the diagram that point C is the intersection of lines AC […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#8

Question The diagram shows a triangle  in which  is  and  is . The gradients of ,  and  are ,  and  respectively, where  is a positive constant. i.       Find the gradient of  and deduce the value of m.    ii.       Find the coordinates of C. The perpendicular bisector of  meets  at .   iii.       Find the coordinates of D. Solution i.   Expression for […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#7

Question The diagram shows part of the curve  , which crosses the x-axis at A and the y-axis at B. The normal to the curve at A crosses the y-axis at C.     i.       Show that the equation of the line AC is .    ii.       Find the length of BC. Solution i.   To find the equation of the line either we […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | May-Jun | (P1-9709/11) | Q#4

Question The diagram shows the curve  and the line . Find the area of the shaded region. Solution      i.   Consider the diagram below. It is evident that; Therefore, first we find area under the curve. To find the area of region under the curve , we need to integrate the curve from point  to  along x-axis. It is […]