# 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#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 with […]

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

Question Given that ,, a)   Express  in the form , where a and b are integers. The curve C with equation y = f(x), , meets the y-axis at P and has a minimum point at Q.  b)  In the space provided on page 19, sketch the graph of C, showing the coordinates of P and Q. The […]

# 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#8

Question The points A(1,7), B(20,7) and C(p,q) form the vertices of a triangle ABC, as shown in Figure 2. The  point D(8, 2) is the mid-point of AC. a)   Find the value of p and the value of q. The line l, which passes through D and is perpendicular to AC, intersects AB at E. b)  Find […]

# 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 | 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#8

Question The line  passes through the point (9, – 4) and has gradient . a)   Find an equation for  in the form ax+by+c=0, where a, b and c are integers. The line  passes through the origin O and has gradient –2. The lines  and  intersect at the point  P. b)  Calculate the coordinates of P. Given that […]

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

Question a.   The function f is defined for all values of  by  .                            i.       Find the remainder when  is divided by .                          ii.       Given that  and  , write down two linear factors of .                        iii.       Hence express  as the product of three linear factors. b.   The curve with equation  is sketched below.                            i.       The curve intersects the y-axis at the point . Find the […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | June | Q#8

Question A line has equation   , where  is a constant. A curve has equation  . a.   Show that the x-coordinate of any point of intersection of the line and the curve satisfies the  equation b.   Find the values of  for which the equation   has equal roots. c.   Describe geometrically the situation when  takes either of the values found in part (b). Solution […]

# 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 | June | Q#3

Question A circle has center  and radius 5. The point  has coordinates . a.   Write down the equation of the circle. b.   Verify that point  lies on the circle. c.   Find the gradient of the line .   d.                                 i.  Find the gradient of the […]

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

Question a.   Express  in the form . b.   A curve has equation  .  Using your answer form part (a), or otherwise:                       i.       Find the coordinates of the vertex (minimum point) of the curve.                     ii.       Sketch the curve, indicating the value where the curve crosses the y-axis. c.   Describe geometrically the […]

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

Question The point  has coordinates  and the point  has coordinates . a.   Find the coordinates of mid-point of . b.   Show that  has length , where  is an integer. c.                                i.       Find the gradient of line .                          ii.       Hence, or otherwise, show that the line  has equation d.    The line  intersects the line with equation  at the point . Find the  coordinates of . Solution a. […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2005 | January | Q#3

Question A circle has equation .  a.   By completing  the square, express the equation in the form b.   Write down:                            i.       the coordinates of the center of the circle;                          ii.       the radius of the circle c.   The line with equation  intersects the circle at the points P and Q.                            i.       Show that the x-coordinates of P and Q satisfy the equation […]

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

Question The point  has coordinates  and the point  has coordinates . a.                 i.    Find the gradient of .           ii.    Hence, or otherwise, show that the line  has equation . b.   The line with equation   intersects the line  at the point . Find the coordinates of . Solution a.   i.   Expression […]

# 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#3

Question In the diagram, ABED is a trapezium with right angles at E and D, and CED is a straight line. The lengths of AB and BC are  and  respectively, and angles BAD and CBE are and  respectively. i.       Find the length of CD in terms of .    ii.       Show that angle CAD = Solution From the given information we can […]

# 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 […]