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#3

Hits: 342     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: 171     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: 77     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 […]

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

Hits: 434   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 […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#8

Hits: 372 Question A curve  has a stationary point at . It is given that, where k is a constant. i.       Show that  and hence find the x-coordinate of the other stationary point, Q.    ii.       Find  and determine the nature of each of the stationary points P and Q.   iii.       Find . Solution i.   A stationary point […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#8

Hits: 462 Question The equation of a curve is . Find     i.       an expression for  and the coordinates of the stationary point on the curve,    ii.       the volume obtained when the region bounded by the curve and the x-axis is rotated through  about the x-axis. Solution i.   Gradient (slope) of the curve is the derivative of equation of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#10

Hits: 608 Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.     i.       Find the coordinates of B and C.    ii.       Find the equation of the normal to the curve at C.   iii.       Find the volume obtained when the shaded region is rotated through  about […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#7

Hits: 1173 Question The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48m.     i.       Find an expression for  in terms of .    ii.       Given that the area of the garden is A m2, show that .   iii.       Given that  can vary, find the maximum area of the garden, showing that this is […]

PPast Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#2

Hits: 1813 Question A curve has equation . Show that the gradient of the curve is never negative. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: For the given case; Rule for differentiation of  is: Therefore; Rule for differentiation of  is: Rule for differentiation of […]

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#9

Hits: 429 Question A curve is such that  and P(9, 5) is a point on the curve.     i.       Find the equation of the curve.    ii.       Find the coordinates of the stationary point on the curve.   iii.       Find an expression for  and determine the nature of the stationary point.   iv.       The normal to the curve at P makes an angle of  with the positive […]