# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#8

Question The equation of a curve is x2 + 2xy − y2 + 8 = 0.      i.       Show that the tangent to the curve at the point (-2,2) is parallel to the x-axis.    ii.       Find the equation of the tangent to the curve at the other point on the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve and its maximum point M.      i.       Find the exact coordinates of M.    ii.       Use the trapezium rule with three intervals to estimate the value of giving your answer correct to 2 decimal places. Solution      i.   We are required to find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#8

Question The diagram shows the curve , for . The point  lies on the curve. i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on a line (or […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/22) | Q#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#8

Question The diagram shows the curve , for . The point  lies on the curve.     i.       Show that the normal to the curve at Q passes through the point .    ii.       Find .   iii.       Hence evaluate Solution      i.   If a point P(x,y) lies on a line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | Oct-Nov | (P2-9709/21) | Q#4

Question The parametric equations of a curve are for t > 2.      i.       Show that in terms of t.      ii.       Find the coordinates of the only point on the curve at which the gradient of the curve is equal  to 0. Solution      i.   We are required to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/23) | Q#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve has […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/22) | Q#5

Question The equation of a curve y=x3e-x.      i.       Show that the curve has a stationary point where x = 3.    ii.       Find the equation of the tangent to the curve at the point where x = 1. Solution      i.   We are required to show that the curve […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2010 | May-Jun | (P2-9709/21) | Q#8

Question      i.       By differentiating , show that if y = cot x then    ii.       By expressing in terms of and using the result of part (i), show that   iii.       Express cos 2x in terms of sin2 x and hence show that can be expressed as .  Hence using the […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2010 | June | Q#11

Question The curve C has equation y=f(x), x>0, where Given that point P(4,5) lies on C, find a.   f(x) b.   an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0,  where a, b and c are integers. Solution a.     We […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2010 | June | Q#7

Question Given that  , x>0 find . Solution We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: Therefore; Rule for differentiation is of  is: Rule for differentiation is of  is: Rule for differentiation is of  is:

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2010 | January | Q#6

Question The curve C has equation  , x>0 a.  Find  in its simplest form. b.  Find an equation of the tangent to C at the point where x=2. Solution a.   We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2010 | January | Q#1

Question Given that  , find . Solution We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: Therefore; Rule for differentiation is of  is: Rule for differentiation is of  is: Rule for differentiation is of  is:

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | June | Q#6

Question The diagram shows a block of wood in the shape of a prism with triangular cross-section. The end  faces are right-angled triangles with sides of lengths 3x cm, 4x cm and 5x cm, and the length of the  prism is y cm, as shown in the diagram. The total surface area of the five faces is 144 […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | June | Q#4

Question The curve with equation  is sketched below. The point  lies on the curve. a.                                i.       Find                          ii.       Hence find the area of the shaded region bounded by the curve and the line  . b.   The point  lies on the curve with equation   .                            i.       Find the gradient of the curve at the point A.                          ii.       Hence find an equation of the tangent to […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | January | Q#6

Question The curve with equation  is sketched below. a.                       i.       Find the gradient of the curve with equation  at the point A.                   ii.       Hence find the equation of the normal to the curve at the point A, giving your answer in the  form  , where p and q are integers.   b.                       i.       Find the value of                   ii.       Hence determine […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | January | Q#5

Question a.   Express  in the form  , where p and q are integers. b.                        i.       Sketch the graph of  , stating the coordinates of the minimum point  and the point where the graph crosses the y-axis.                   ii.       Write down an equation of the tangent to the graph of  at its vertex. c.   […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2010 | January | Q#3

Question The depth of water, y metres, in a tank after time t hours is given by ,  a.   Find:                             i.                                  ii.        b.   Verify that y has a stationary value when  and determine whether it is a maximum value or a minimum value. c.                                 i.       Find the rate of change of the depth of water, in metres […]

# 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#6

Question A curve has equation . It is given that .     i.       Find the set of values of  for which  is an increasing function.    ii.       Given that the curve passes through (1, 3), find . Solution i.   To test whether a function  is increasing or decreasing at a particular point , we take derivative of a function at that point. […]