# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#6

Question The diagram shows the part of the curve  for , and its minimum point M.      i.       Find the coordinates of M.    ii.       Use the trapezium rule with 2 intervals to estimate the value of Giving your answer correct to 1 decimal place.   iii.       State, with a reason, whether the […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#5

Question The equation of a curve is 3×2 + 2xy + y2 = 6. It is given that there are two points on the curve where the tangent is parallel to the x-axis. i.       Show by differentiation that, at these points, y = −3x. ii.       Hence find the coordinates of the two […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | Oct-Nov | (P2-9709/02) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . We can write it as; We have to consider both moduli separately and it leads to following cases; When If then above four intervals translate to following with their corresponding inequality; When When When If then above four intervals translate to following […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2006 | May-Jun | (P2-9709/02) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let  and , then; We have to consider two separate cases; When When We have the inequality; It can be written as; We have to consider two separate cases; When When Therefore the inequality will hold for ; SOLVING INEQUALITY: ALGEBRAICALLY Let, . Since given […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | June | Q#10

Question The curve C with equation y=f(x), x ≠ 0, passes through the point . Given that a.   find f(x). b.   Verify that f(–2) = 5. c.   Find an equation for the tangent to C at the point (–2, 5), giving your answer in the form ax + by +  c = 0, where a, b […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | January | Q#9

Question Figure 2 shows part of the curve C with equation The curve cuts the x-axis at the points P, (1, 0) and Q, as shown in Figure 2. a.   Write down the x-coordinate of P, and the x-coordinate of Q. b.   Show that . c.   Show that y=x+7 is an equation of the tangent to […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | January | Q#3

Question The line L has equation y=5 – 2x. a.   Show that the point P (3, –1) lies on L. b.   Find an equation of the line perpendicular to L, which passes through P. Give your answer in the form ax + by + c = 0, where a, b and c are integers. […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | June | Q#7

Question A circle has equation . a.   Find                            i.       the coordinates of the centre of the circle;                          ii.       the radius of the circle in the form , where  is an integer. b.   A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to  this chord. c.   A line has equation , where  is a constant.                            i.       Show […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | June | Q#3

Question A curve has equation . a.   Find b.   Find an equation for the tangent to the curve at the point where . c.   Determine whether  is increasing or decreasing when . Solution a.   We have the equation; Rule for differentiation is of  is: Rule for differentiation is of  is: Rule for differentiation is of  is: b. […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | June | Q#1

Question The point  has coordinates  and the point  has coordinates . a.                          i.    Find the gradient of the line .                    ii.    Hence, or otherwise, show that the line  has equation b.         The line  intersects the line with equation   at the point . Find the coordinates of  . c.           Find an equation of the line through which is perpendicular to  . Solution a. […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#8

Question The diagram shows the curve with equation  and the line  . The points  and  have coordinates  and  respectively. The curve touches the x-axis at  the origin  and crosses the x-axis at the point .  The line  cuts the curve at the point   where  and touches the curve at  where . a.   Find the area of the rectangle . b.                                i.      […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#3

Question a.                                i.       Express  in the form , where   and  are integers.                          ii.       Hence, or otherwise, describe the coordinates of the minimum point of the curve with                       equation . b.   The line  has equation  and the curve  has the equation .                             i.       Show that the x-coordinates of the points of […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | January | Q#2

Question The point  has coordinates  and the point  has coordinates . The line  has equation . a.                                i.       Show that .                          ii.       Hence find the coordinates of the mid-point of . b.   Find the gradient of . c.   Line  is perpendicular to the line .                            i.       Find the gradient of .                          ii.       Hence find the equation of the line .                        iii.       Given that point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#7

Question The diagram shows the curve y = x(x − 1)(x − 2), which crosses the x-axis at the points O(0, 0), A(1, 0) and B(2, 0).     i.       The tangents to the curve at the points A and B meet at the point C. Find the x-coordinate of C.    ii.       Show by integration that the area of the shaded region […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | Oct-Nov | (P1-9709/01) | Q#5

Question The three points A(1, 3), B(13, 11) and C(6, 15) are shown in the diagram. The perpendicular from C to AB meets AB at the point D. Find i.       the equation of CD,    ii.       the coordinates of D. Solution      i.   To find the equation of the line either we need coordinates of the two points on the line […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#9

Question A curve is such that  and P (1, 8) is a point on the curve. i.       The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the coordinates of the mid-point of QR.    ii.   Find the equation of the curve. Solution      i.   To find the mid-point […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2006 | May-Jun | (P1-9709/01) | Q#5

Question The curve  intersects the line  at two points. Find the distance between the two points. Solution Expression to find distance between two given points  and is: So first we need to find the coordinates of points of intersection of the curve and the line. If two lines (or a line and a curve) intersect each other […]