Hits: 139

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

Hits: 368     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 […]

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

Hits: 187     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 […]

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

Hits: 112     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 […]

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

Hits: 172     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 […]

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

Hits: 49   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 […]

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

Hits: 65   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 […]

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

Hits: 70     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 […]

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

Hits: 150     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; […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#7

Hits: 342 Question i.   A straight line passes through the point (2, 0) and has gradient . Write down the equation of the line. ii.  Find the two values of  for which the line is a tangent to the curve . For each value of , find the coordinates of the point where the line touches the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/13) | Q#3

Hits: 567 Question The diagram shows the curve  and the line  intersecting at points A, O and B.     i.       Show that the x-coordinates of A and B satisfy the equation .    ii.       Solve the equation  and hence find the coordinates of A and B, giving your answers in an exact form. Solution i.   We are given that points A & […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#9

Hits: 1076 Question The diagram shows a quadrilateral ABCD in which the point A is , the point B is  and the point C is . The diagonals AC and BD intersect at M. Angle   and . Calculate     i.        the coordinates of M and D,    ii.       the ratio . Solution     i.   First we find […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 476 Question A curve is such that   . The line  is the normal to the curve at the point  on the curve. Given that the x-coordinate of  is positive, find     i.       the coordinates of P,    ii.       the equation of the curve. Solution i.   We are given that equation of the line is; We can rearrange the equation of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/12) | Q#4

Hits: 455 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 the line and the curve.    ii.       Find the value of  for which the line is a tangent to the curve. Solution     i.   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#10

Hits: 628 Question The diagram shows the curvemeeting the x-axis at A and the y-axis at B. The y-coordinate of the point C on the curve is 3.     i.       Find the coordinates of B and C.    ii.       Find the equation of the normal to the curve at C.   iii.       Find the volume obtained when the shaded region is rotated through  about […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#9

Hits: 503 Question A line has equation  and a curve has equation , where  is a constant.     i.       For the case where , the line and the curve intersect at points A and B. Find the distance AB and the coordinates of the mid-point of AB.    ii.       Find the two values of  for which the line is a tangent to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | Oct-Nov | (P1-9709/11) | Q#3

Hits: 576 Question      i.       Sketch, on a single diagram, the graphs of  and  for .    ii.       Write down the number of roots of the equation  in the interval .   iii.       Deduce the number of roots of the equation  in the interval . Solution     i.        ii.   If two lines (or a line and a curve) intersect each other […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#9

Hits: 445 Question A curve is such that  and P(9, 5) is a point on the curve.     i.       Find the equation of the curve.    ii.       Find the coordinates of the stationary point on the curve.   iii.       Find an expression for  and determine the nature of the stationary point.   iv.       The normal to the curve at P makes an angle of  with the positive […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#3

Hits: 470 Question The line  , where a and b are positive constants, meets the x-axis at P and the y-axis at Q. Given that   and that the gradient of the line PQ is  , find the values of a and b. Solution We are given that points P and Q are x and y intercepts, respectively. First we […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2011 | May-Jun | (P1-9709/13) | Q#2

Hits: 507 Question Find the set of values of  for which the line  intersects the curve  at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line […]