# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#7

Question The diagram shows the part of the curve y = sin2 x for  .      i.       Show that    ii.       Hence find the x-coordinates of the points on the curve at which the gradient of the curve is  0.5.   iii.       By expressing sin2 x in terms of cos […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#6

Question A curve is such that . The point (0, 1) lies on the curve.      i. Find the equation of the curve.    ii. The curve has one stationary point. Find the x-coordinate of this point and determine whether it  is a maximum or a minimum point. Solution      i.   We […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | Oct-Nov | (P2-9709/02) | Q#4

Question The equation of the curve is .     i.       Show that    ii.       Find the equation of the tangent to the curve at the point (2, 4), giving your answer in the form ax+by=c. Solution      i.   We are given that; Therefore; Rule for differentiation of  is: If  and  are functions […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2005 | May-Jun | (P2-9709/02) | Q#5

Question i.By differentiating , show that if then . ii. The parametric equations of a curve are x = 1 + tanθ , y = secθ , for . Show that .   iii.Find the coordinates of the point on the curve at which the gradient of the curve is . Solution      i.   […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | January | Q#9

Question The gradient of the curve C is given by The point P(1,4) lies on C. a)   Find an equation of the normal to C at P. b)  Find an equation for the curve C in the form . c)   Using , show that there is no point on C which the tangent is […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | January | Q#7

Question The curve C has equation , . The point P on C has x-coordinate 1.  a)   Show that the value of  at P is 3. b)  Find an equation of the tangent to C at P. This tangent meets the x-axis at the point (k,0). c)   Find the value of k. Solution a)     We need […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | January | Q#2

Question Given that , find a.   b.   c.   Find Solution a.   We are given; We are required to find . Rule for differentiation of  is: Rule for differentiation of  is: b.   We are required to find . Second derivative is the derivative of the derivative. If we have derivative of the curve   as  , then  […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | June | Q#10

Question The curve C has equation . The point P has coordinates (3, 0). a)   Show that P lies on C. b)  Find the equation of the tangent to C at P, giving your answer in the form y=mx+c, where m and c  are constants. Another point Q also lies on C. The tangent to C […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2005 | June | Q#2

Question Given that , , a.   Find b.   Find Solution a.   We are given; We are required to find . Rule for differentiation of  is: Rule for differentiation of  is: b.   We are given; We are required to find . Rule for integration of  is: Rule for integration of  is: Rule for integration of  is:

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | June | Q#4

Question The curve with equation  is sketched below. The curve touches the x-axis at the point  and cuts the x-axis at the point  . a.                               i.       Use the factor theorem to show that  is a factor  of                          ii.       Hence find the coordinates of B b. The point , shown on  the  diagram, is  a   minimum point  of  the  curve with equation                            i.       Find  . […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | January | Q#6

Question The diagram below shows a rectangular sheet of metal 24 cm by 9 cm. A square of side x cm is cut from each corner and the metal is then folded along the broken lines to  make an open box with a rectangular base and height x cm. a.   Show that the volume,  cm3, of liquid the […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | January | Q#2

Question A curve has equation   . a.   Find . b.   The point  on the curve has coordinates .                            i.       Show that the gradient of the curve at  is 5.                          ii.       Hence find an equation of the normal to the curve at P, expressing your answer in the form ax + by = c , where a, b and c are integers. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#10

Question A curve is such that  and (1, 4) is a point on the curve. i.       Find the equation of the curve.    ii.        A line with gradient  is a normal to the curve. Find the equation of this normal, giving your answer in the form .   iii.       Find the area of the region enclosed by the curve, the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#9

Question The equation of a curve is  and the equation of a line  is , where  is a constant. i.       In the case where , find the coordinates of the points of intersection of  and the curve.    ii.       Find the set of values of  for which  does not intersect the curve.   iii.       In the case where , one of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#5

Question The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm, and the cylinder has radius  cm and height cm. The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone. i.       Express h in terms of r and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#10

Question The equation of a curve is . i.       Show that the whole of the curve lies above the x-axis.    ii.       Find the set of values of x for which  is a decreasing function of x. The equation of a line is , where k is a constant.   iii.       In the case where k = 6, find the coordinates of the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#9

Question A curve has equation  . i.       The normal to the curve at the point (4, 2) meets the x-axis at P and the y-axis at Q. Find the length of PQ, correct to 3 significant figures.    ii.       Find the area of the region enclosed by the curve, the x-axis and the lines x = 1 and […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | May-Jun | (P1-9709/01) | Q#2

Question Find the gradient of the curve  at the point where x = 3. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: In the given case: We can rewrite the equation as; Rule for differentiation of  is: Rule for differentiation of  is: Rule for differentiation of […]