# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#8

Question The equation of a curve is 2×2 − 3x − 3y + y2 = 6.      i.       Show that    ii.       Find the coordinates of the two points on the curve at which the gradient is −1. Solution      i.   We are given equation of the curve as; We are required […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#7

Question The diagram shows the curve . The curve has a gradient of 3 at the point P.      i.       Show that the x-coordinate of P satisfies the equation    ii.       Verify that the equation in part (i) has a root between x = 3.1 and x = 3.3.   iii.       Use the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#1

Question Find the gradient of the curve y = ln(5x + 1) at the point where x = 4. Solution 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 natural logarithmic function , for  is; Gradient (slope) […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#6

Question The parametric equations of a curve are x = 1 + 2 sin2θ , y = 4 tanθ , i. Show that    ii. Find the equation of the tangent to the curve at the point where , giving your answer in  the form y = mx + c. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#3

Question The diagram shows the curve y= x − 2 ln x and its minimum point M.      i. Find the x-coordinates of M.    ii. Use the trapezium rule with three intervals to estimate the value of giving your answer correct to 2 decimal places.   iii. State, with a reason, whether […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#7

Question The parametric equations of a curve are  ,  , i.       Show that . ii.       Show that the tangent to the curve at the point (1, 3) is parallel to the x-axis. iii.       Find the exact coordinates of the other point on the curve at which the tangent is parallel to the  x-axis. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#3

Question The diagram shows the part of the curve  for . Find the x-coordinates of the  points on this part of the curve at which the gradient is 4. Solution We are required to find the x-coordinate of the points on the curve where gradient is 4. Therefore first we need to find […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | Q#6

Question The curve y = 4×2 ln x has one stationary point.      i.       Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.    ii.       Determine whether this point is a maximum or a minimum point. Solution      i.   We are required to find the coordinates […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | Q#5

Question A curve has equation x2+2y2+5x+6y =10. Find the equation of the tangent to the curve at the point (2,-1). Give your answer in the form ax+by+c=0, wher a,b and c are integers. Solution We are required to find equation of tangent to the curve at the point (2,-1). To find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/22) | Q#6

Question The curve y = 4×2 ln x has one stationary point.      i.       Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.    ii.       Determine whether this point is a maximum or a minimum point. Solution      i.   We are required to find the coordinates […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/22) | Q#5

Question A curve has equation x2+2y2+5x+6y =10. Find the equation of the tangent to the curve at the point (2,-1). Give your answer in the form ax+by+c=0, wher a,b and c are integers. Solution We are required to find equation of tangent to the curve at the point (2,-1). To find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#5

Question Find the value of  when  for each of the following cases: i.       . ii.        . Solution      i.   We are given that; We are required to find the value of when , therefore, first we need to find . If  and  are functions of , and if , then; If , then; […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#2

Question A curve has parametric equations Find the exact gradient of the curve at the point for which . Solution      i.   Gradient (slope) of the curve at the particular point is the derivative of equation of the curve at that  particular point. Gradient (slope) of the curve at a particular point can […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#10

Question The curve C has equation y = (x +1)(x + 3)2 a.   Sketch C, showing the coordinates of the points at which C meets the axes. b.   Show that . The point A, with x-coordinate -5, lies on C. c.   Find the equation of the tangent to C at A, giving your answer in the […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#2

Question Given that  , , find, in their simplest form, a.   b.   . 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  is: Therefore; Rule for differentiation is of  is: Rule for differentiation is of […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#11

Question The curve C has equation  , x>0 a.   Find b.   Show that the point P(4,−8) lies on C. c.   Find an equation of the normal 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. […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | June | Q#3

Question The volume, V m3 , of water in a tank after time t seconds is given by a.   Find . b.                         i.       Find the rate of change of volume, in m3 s-1, when .                   ii.       Hence determine, with a reason, whether the volume is increasing or decreasing when . c.                         i.       Find the positive value of t […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | January | Q#4

Question The curve sketched below passes through the point A(-2,0) . The curve has equation and the point P(1,12) lies on the curve. a.                         i.       Find the gradient of the curve at the point P.                   ii.       Hence find the equation of the tangent to the curve at the point P, giving your answer in the  […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | January | Q#1

Question The curve with equation  passes through the point P where . a.   Find .  b.    Show that the point P is a stationary point of the curve and find the other value of x where the  curve has a stationary point. c.                             […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#10

Question Functions  and  are defined by ,        i.       Evaluate .  ii.       Sketch in a single diagram the graphs of  and , making clear the relationship between the graphs.   iii.      Obtain an expression for  and use your answer to explain why  has an inverse.   iv.       Express each of  and , in terms of . Solution i.   We have functions; We […]