# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#6

Question The parametric equations of a curve are x = 1 + 2 sin2θ , y = 4 tanθ , i. Show that    ii. Find the equation of the tangent to the curve at the point where , giving your answer in  the form y = mx + c. Solution      i. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | Q#3

Question The diagram shows the curve y= x − 2 ln x and its minimum point M.      i. Find the x-coordinates of M.    ii. Use the trapezium rule with three intervals to estimate the value of giving your answer correct to 2 decimal places.   iii. State, with a reason, whether […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#7

Question The parametric equations of a curve are  ,  , i.       Show that . ii.       Show that the tangent to the curve at the point (1, 3) is parallel to the x-axis. iii.       Find the exact coordinates of the other point on the curve at which the tangent is parallel to the  x-axis. […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/23) | Q#5

Question A curve has equation x2+2y2+5x+6y =10. Find the equation of the tangent to the curve at the point (2,-1). Give your answer in the form ax+by+c=0, wher a,b and c are integers. Solution We are required to find equation of tangent to the curve at the point (2,-1). To find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/22) | Q#5

Question A curve has equation x2+2y2+5x+6y =10. Find the equation of the tangent to the curve at the point (2,-1). Give your answer in the form ax+by+c=0, wher a,b and c are integers. Solution We are required to find equation of tangent to the curve at the point (2,-1). To find the equation […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/23) | Q#2

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 | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/22) | 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 | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | Oct-Nov | (P2-9709/21) | Q#1

Question Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let, . It can be written as; We have to deal with two separate cases; When ; When Therefore; Therefore; Hence, Hence, We have the inequality; It can be written as; We have to consider two separate cases; When When Therefore the inequality will […]

# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2011 | May-Jun | (P2-9709/21) | Q#1

Question Solve the inequality . Solution SOLVING EQUATION: PIECEWISE Let, . We have to consider both moduli separately and it leads to following cases;  OR We have the equation; We have to consider both moduli separately and it leads to following cases; Hence, the only solution for the given equation is; SOLVING EQUATION: […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#10

Question The curve C has equation y = (x +1)(x + 3)2 a.   Sketch C, showing the coordinates of the points at which C meets the axes. b.   Show that . The point A, with x-coordinate -5, lies on C. c.   Find the equation of the tangent to C at A, giving your answer in the […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | June | Q#3

Question The points P and Q have coordinates (–1, 6) and (9, 0) respectively. The line  is perpendicular to PQ and passes through the mid-point of PQ. Find an equation for , giving your […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#11

Question The curve C has equation  , x>0 a.   Find b.   Show that the point P(4,−8) lies on C. c.   Find an equation of the normal to C at the point P, giving your answer in the form ax + by + c = 0  , where a, b and c are integers. Solution a. […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#10

Question a.   On the axes below, sketch the graphs of i.        ii.     showing clearly the coordinates of all the points where the curves cross the coordinate axes. b.   Using your sketch state, giving a reason, the number of real solutions to the equation Solution a.   i.   We are required to sketch; We need to expand it […]

# Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2011 | January | Q#9

Question The line L1 has equation 2y − 3x − k = 0, where k is a constant. Given that the point A (1, 4) lies on L1, find a.   the value of k, b.   the gradient of L1. The line L2 passes through A and is perpendicular to L1. c.   Find an equation of L2 giving […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | June | Q#4

Question a.   Express  in the form  , where p and q are rational numbers. b.   A curve has equation .      i.   Find the coordinates of the vertex of the curve.     ii.   State the equation of the line of symmetry of the curve.    iii.   Sketch the curve, stating the value of the intercept on […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | June | Q#8

Question A circle has centre C(3,-8) and radius 10 . a.   Express the equation of the circle in the form b.   Find the x-coordinates of the points where the circle crosses the x-axis. c.   The tangent to the circle at the point A has gradient . Find an equation of the line CA, giving your answer […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | June | Q#3

Question The volume, V m3 , of water in a tank after time t seconds is given by a.   Find . b.                         i.       Find the rate of change of volume, in m3 s-1, when .                   ii.       Hence determine, with a reason, whether the volume is increasing or decreasing when . c.                         i.       Find the positive value of t […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | June | Q#1

Question The line AB has equation  . a.   Find the gradient of AB. b.   The point C has coordinates .                     i.       Find an equation of the line which passes through the point C and which is parallel to AB.                   ii.       The point  is the mid-point of AC . Find the coordinates of the point A.  c.   The line […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | January | Q#7

Question a.                         i.       Express  in the form .                   ii.       Hence write down the equation of the line of symmetry of the curve with equation  . b.   The curve C has equation  and the line L has equation , where k is a constant.                     i.       Show that the x-coordinates of any points of intersection of the curve C with the line L […]

# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2011 | January | Q#6

Question A circle has centre C(-3,1) and radius . a.                         i.       Express the equation of the circle in the form                   ii.       Hence find the equation of the circle in the form where m, n and p are integers. b.   The circle cuts the y-axis at the points A and B. Find the distance AB. c.                         i.       Verify that the […]