Hits: 408

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/22) | Q#7

Hits: 408   Question The diagram shows the curve y = x2 cos x, for , and its maximum point M.     i.       Show by differentiation that the x-coordinate of M satisfies the equation    ii.       Verify by calculation that this equation has a root (in radians) between 1 and 1.2.   iii. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/22) | Q#4

Hits: 177     Question The parametric equations of a curve are  ,  ,     i.       Show that .    ii.       Hence find the exact value of t at the point on the curve at which the gradient is 2. Solution      i.   We are given that; We are required to show […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/21) | Q#8

Hits: 130     Question The equation of a curve is y2 + 2xy − x2 = 2.      i.       Find the coordinates of the two points on the curve where x = 1.    ii.       Show by differentiation that at one of these points the tangent to the curve is parallel to […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | Oct-Nov | (P2-9709/21) | Q#6

Hits: 130     Question The curve with equation y = x ln x has one stationary point. i.       Find the exact coordinates of this point, giving your answers in terms of e. ii.       Determine whether this point is a maximum or a minimum point. Solution      i.   We are required […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | May-Jun | (P2-9709/02) | Q#8

Hits: 239     Question a)   Find the equation of the tangent to the curve at the point where . b)                  i.       Find the value of the constant A such that           ii.       Hence show that Solution a.     We are given that curve with […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2009 | May-Jun | (P2-9709/02) | Q#7

Hits: 217     Question The diagram shows the curve and its minimum point M.      i.       Find the exact coordinates of M.    ii.       Show that the curve intersects the line y = 20 at the point whose x-coordinate is the root of  the equation   iii.       Use the iterative formula […]

Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2009 | June | Q#3

Hits: 13   Question Given that  , , find a.   b.  , simplifying each term. Solution a.   We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: Therefore; Rule for differentiation is of  is: Rule for differentiation is […]

Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2009 | January | Q#6

Hits: 60   Question a.  The polynomial  is given by .                     i.       Use the Factor Theorem to show that  is a factor of .                   ii.       Express  in the form  , where a and b are constants. b. The curve C with equation  , sketched below, crosses the x-axis at the point .                     i.       Find the gradient of the curve C at the point Q.                   ii.       Hence […]

Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2009 | June | Q#3

Hits: 31   Question The curve with equation  passes through the point P, where  . a.   Find                          i.                                  ii.        b.   Verify that the point P is a stationary point of the curve. c.                                i.       Find the value of  at the point P.                           ii.       Hence, or otherwise, determine whether […]

Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2009 | January | Q#5

Hits: 54   Question A model car moves so that its distance, x centimeters, from a fixed point O after time t seconds is given by  , a.   Find:                            i.                                  ii.        b.   Verify that x has a stationary value when  and determine whether this stationary value is a  maximum value or a minimum value. c.   Find the rate […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#10

Hits: 733 Question i.       The diagram shows the line  and the curve , which intersect at the points A and B. Find a.   the x-coordinates of A and B, b.   the equation of the tangent to the curve at B, c.   the acute angle, in degrees correct to 1 decimal place, between this tangent and the line . […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 1091 Question A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is q radians (see diagram). i.       Express  in terms of r and show that the area, Acm2, of the sector is given by    […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#7

Hits: 838 Question The equation of a curve is . i.       Obtain an expression for .    ii.       Find the equation of the normal to the curve at the point P(1, 3).   iii.       A point is moving along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#6

Hits: 246 Question A curve is such that , where k is a constant.     i.       Given that the tangents to the curve at the points where  and  are perpendicular, find the value of .    ii.       Given also that the curve passes through the point (4, 9), find the equation of the curve. Solution      i.   If two lines […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2009 | Oct-Nov | (P1-9709/11) | Q#4

Hits: 223 Question The equation of a curve is .     i.       Find the coordinates of the stationary point on the curve and determine its nature.    ii.       Find the area of the region enclosed by the curve, the x-axis and the lines    and . Solution      i.   Coordinates of stationary point on the curve  can be found from the derivative […]