# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#7

Question The equation of a curve is        i.       Show that    ii.       Find the coordinates of each of the points on the curve where the tangent is parallel to the x- axis. Solution      i.   We are given; We are required to find . To find from an implicit equation, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/23) | Q#4

Question The parametric equations of a curve are for t < 0.      i.       Show that in terms of t.    ii.       Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution      i.   We are required to find  for the parametric equations given below; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/21) | Q#4

Question The parametric equations of a curve are for t < 0.      i.       Show that in terms of t.    ii.       Find the exact coordinates of the only point on the curve at which the gradient is 3. Solution      i.   We are required to find  for the parametric equations given below; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#5

Question The parametric equations of a curve are      i.       Find an expression for in terms of t.      i.       Find the equation of the normal to the curve at the point for which t = 0. Give your answer in  the form ax + by + c = 0, where a, b and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/23) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/22) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | Oct-Nov | (P2-9709/21) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following with […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/23) | Q#1

Question Solve the equation . Solution SOLVING EQUATION: PIECEWISE Let, . We can write it as; We have to consider two separate cases; When When We have the equation; We have to consider two separate cases; When ; When ; Hence, the only solutions for the given equation are; SOLVING EQUATION: ALGEBRAICALLY Let, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/22) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2012 | May-Jun | (P2-9709/21) | Q#1

Question Solve the equation . Solution SOLVING EQUATION: PIECEWISE Let, . We can write it as; We have to consider two separate cases; When When We have the equation; We have to consider two separate cases; When ; When ; Hence, the only solutions for the given equation are; SOLVING EQUATION: ALGEBRAICALLY Let, […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#11

Question The diagram shows the curve with equation . The minimum point on the curve has coordinates  and the x-coordinate of the maximum point is , here  and  are constants.      i.       State the value of .    ii.       Find the value of .   iii.       Find the area of the shaded region.   iv.       The gradient, , of the curve has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#10

Question A straight line has equation , where  is a constant, and a curve has equation .      i.       Show that the x-coordinates of any points of intersection of the line and curve are given by the equation .    ii.          Find the two values of  for which the line is a tangent to the curve.   […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/13) | Q#7

Question      i.       The diagram shows part of the curve  and part of the straight line  meeting at the point , where  and  are positive constants. Find the values of  and .    ii.             The function f is defined for the domain  by Express  in a similar way. Solution i.   It is evident from the diagram that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#9

Question The diagram shows part of the curve  , crossing the y-axis at the point B(0, 3). The point A on the curve has coordinates (3, 1) and the tangent to the curve at A crosses the y-axis at C.     i.       Find the equation of the tangent to the curve at A.    ii.    Determine, showing all necessary working, whether C […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#5

Question The diagram shows a triangle ABC in which A has coordinates (1, 3), B has coordinates (5, 11) and angle ABC is  . The point X (4, 4) lies on AC. Find     i.       the equation of BC,    ii.     the coordinates of C. Solution i.   We need to find the equation of BC. 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 2012 | Oct-Nov | (P1-9709/12) | Q#4

Question The line  , where  is a constant, is a tangent to the curve  at the point P. Find i.          the value of k,  ii.       the coordinates of P. Solution i.   We are given that line is tangent to the curve, therefore, line and the curve intersect at one point only i.e. P. If two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/12) | Q#3

Question The diagram shows a plan for a rectangular park ABCD, in which AB = 40m and AD = 60m. Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a triangular playground. The length of DY is  m and the length of XC is  m.     i.       Show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#11

Question The diagram shows the curve  and the point A(1, 2) which lies on the curve. The tangent to the curve at A cuts the y-axis at B and the normal to the curve at A cuts the x-axis at C. i.           Find the equation of the tangent AB and the equation of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | Oct-Nov | (P1-9709/11) | Q#8

Question The diagram shows the curve  and the straight line . The curve and straight line intersect at  and , where  is a constant. i.      Show that . ii.   Find, showing all necessary working, the area of the shaded region. Solution i.   It is evident from the diagram that line and the curve intersect at two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2012 | May-Jun | (P1-9709/13) | Q#10

Question The equation of a line is , where is a constant, and the equation of a curve is .     i.       In the case where, the line intersects the curve at the points A and B. Find the equation of the perpendicular bisector of the line AB.    ii.       Find the set of values of  for which the line  intersects the […]