# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#6

Question The equation of a curve is        i.       Show, by differentiation, that the gradient of the curve is always negative.    ii.       Use the trapezium rule with 2 intervals to estimate the value of giving your answer correct to 2 significant figures.     iii.   The diagram shows a […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#7

Question      i.       By differentiating  , show that if y = cot x then    ii.       Hence, show that   By using appropriate trigonometrical identities, find the exact value of     iii.     iv.   Solution      i.   We are given; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | Oct-Nov | (P2-9709/02) | Q#6

Question The diagram shows the curve y=(4−x)ex and its maximum point M. The curve cuts the x-axis at A  and the y-axis at B.     i.       Write down the coordinates of A and B.    ii.       Find the x-coordinate of M.   iii.       The point P on the curve has x-coordinate p. The tangent to the curve at P passes through the  […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2003 | May-Jun | (P2-9709/02) | Q#7

Question The parametric equations of a curve are      i. Show that    ii. Find the equation of the tangent to the curve at the point where .   iii. For the part of the curve where , find the coordinates of the points where the tangent  is parallel to the x-axis. Solution      i.   We are required to show that […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2002 | Oct-Nov | (P2-9709/02) | Q#7

Question The equation of a curve is      i.       Show that    ii.       Find the coordinates 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, differentiate each term with respect to , using the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#9

Question The diagram shows points  and  on the curve . The tangent to the curve at B crosses the x-axis at C. The point D has coordinates (2, 0). i.       Find the equation of the tangent to the curve at B and hence show that the area of triangle BDC is    ii.       Show that the volume of the solid formed when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | Oct-Nov | (P1-9709/01) | Q#8

Question A solid rectangular block has a base which measures  cm by  cm. The height of the block is  cm and the volume of the block is 72 cm3.      i.       Express  in terms of  and show that the total surface area, A cm2, of the block is given by  Given that x can vary,    ii.       find the value of  for which A […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#10

Question The equation of a curve is      i.      Calculate the gradient of the curve at the point where .    ii.       A point with coordinates  moves along the curve in such a way that the rate of increase of  has the constant value 0.03 units per second.  Find the rate of increase of   at the instant when […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2003 | May-Jun | (P1-9709/01) | Q#3

Question a)   Differentiate  with respect to . b)  Find Solution a)   First write the expression in equation form. We can rewrite the equation without fractions: In the given case: Rule for differentiation of  is: Rule for differentiation of  is: b)   The given case is: We can rewrite the expression without fractions: Rule for integration of  is: Rule for […]