# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#8

Question      i.                     a.   Prove the identity                  b.   Hence prove that    ii.       By differentiating , show that if  then .     iii.       Using the results of parts (i) and (ii), find the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | Oct-Nov | (P2-9709/02) | Q#6

Question Find the exact coordinates of the point on the curve   at which . Solution We are given that; We are required to find; Second derivative is the derivative of the derivative. If we have derivative of the curve   as , then  expression for the second derivative of the curve  is; Therefore […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | May-Jun | (P2-9709/02) | Q#7

Question The equation of the curve is .     i.       Show that    ii.       Find the coordinates of each 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 […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2008 | May-Jun | (P2-9709/02) | Q#6

Question It is given that the curve has one stationary point.      i.       Find the x-coordinates of this point.    ii.       Determine whether this point is a maximum or a minimum point. Solution      i.   We are required to find the coordinates of stationary point of the curve; A stationary […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | June | Q#9

Question The curve C has equation , where k is a constant. a.   Find . Point A with x-coordinate  lies on C. The tangent to C at A is parallel to the line with equation  . Find b.   The value of k. c.   The value of y-coordinate of A. Solution a.   Gradient (slope) of the curve […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | June | Q#4

Question , a.   Differentiate to find  . Given that , b.   Find the value of x. Solution a.   We are required to differentiate; Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation is of  is: b.   We are given that; We have found in (a) that; We are given that x>0, therefore;

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | January | Q#10

Question The curve C has equation y=(x + 3)(x −1)2 . a.   Sketch C showing clearly the coordinates of the points where the curve meets the coordinate  axes. b.   Show that the equation of C can be written in the form y = x3 + x2 − 5x + k, where k is a positive integer, and […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | January | Q#9

Question The curve C has equation y = f(x), x > 0, and . Given that the point P(4,1) lies on C, a.   find f(x) and simplify your answer. b.   Find an equation of the normal to C at the point P(4, 1). Solution a.   We are required to find f(x), when; We are also given that […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2008 | January | Q#5

Question a.   Write  in the form  where p and q are constants. Given that  , x > 0, b.   find  , simplifying the coefficient of each term. Solution a.   We are given; b.   We are given; We are required to find; Therefore; Rule for differentiation of  is: As demonstrated in (a); Therefore; Rule for differentiation of  is: […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | January | Q#6

Question a.   The polynomial  is given by .                   i.       Use the Factor Theorem to show that  is a factor of  .                 ii.       Express  as the product of three linear factors. b.   The curve with equation  is sketched below. The curve cuts the x-axis at the point A and the points […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | June | Q#5

Question The curve with equation  is sketched below. The points A(-2,0) , B(2,0) and C(1,15) lie on the curve.  a.   Find an equation of the straight line AC . b.                          i.       Find .                   ii.       Hence calculate the area of the shaded region bounded by the curve and the line AC . Solution a.   We are required to […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | June | Q#3

Question Two numbers, x and y, are such that  , where  and . It is given that  .  a.   Show that  . b.                         i.       Show that , and state the value of the integer k.                   ii.       Hence find the two values of x for which c.    Find d.                         i.       Find the value of  for each of the two values […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2008 | January | Q#2

Question The curve with equation  has a single stationary point, M. a.   Find b.   Hence find the x-coordinate of M. c.                         i.       Find                   ii.       Hence, or otherwise, determine whether M is a maximum or a minimum point.  d.    Determine whether the curve is increasing or decreasing at the point on the curve where . Solution a.   We are […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#8

Question The equation of a curve is . i.       Show that the equation of the normal to the curve at the point  is This normal meets the curve again at the point Q.    ii.       Find the coordinates of Q.   iii.       Find the length of PQ. Solution i.   We are required to find the equation of the normal to the curve. To […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | Oct-Nov | (P1-9709/01) | Q#7

Question A wire, 80 cm long, is cut into two pieces. One piece is bent to form a square of side x cm and the other piece is bent to form a circle of radius r cm (see diagram). The total area of the square and the circle is Acm2. i.       Show that    ii.       Given that x and r […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2008 | May-Jun | (P1-9709/01) | Q#4

Question The equation of a curve C is  and the equation of a line L is . i. Find the x-coordinates of the points of intersection of L and C.    ii. Show that one of these points is also the stationary point of C. Solution i.   If two lines (or a line and a curve) […]