Hits: 66

# 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

Hits: 66     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 […]

# 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

Hits: 100     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. […]

# 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

Hits: 39     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; […]

# 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

Hits: 135     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 […]

# 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

Hits: 341     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 […]

# 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

Hits: 170     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 […]

# 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

Hits: 76     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 […]

# 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

Hits: 53     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 […]

# 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

Hits: 103     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 […]

# 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

Hits: 106     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 […]

# 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

Hits: 160     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 […]

# 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

Hits: 98   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 […]

# 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

Hits: 153   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 […]