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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#6

Hits: 12     Question The diagram shows the part of the curve  for . The curve cuts the x-axis at A and  its maximum point is M.      i.       Write down the coordinates of A.    ii.       Show that the x-coordinate of M is e, and write down the y-coordinate of M […]

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#7

Hits: 2     Question      i.       Given that y = tan 2x, find .    ii.       Hence, or otherwise, show that and, by using an appropriate trigonometrical identity, find the exact value of   iii.       Use the identity cos 4x ≡ 2cos2 2x − 1 to find the exact value of […]

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

Hits: 8     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, […]

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#7

Hits: 11 Question      i.       Differentiate ln(2x + 3).    ii.       Hence, or otherwise, show that   iii.       Find the quotient and remainder when 4×2 + 8x is divided by 2x + 3.   iv.       Hence show that Solution      i.   We are required to find; If we define , […]

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

Hits: 6   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 […]

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#3

Hits: 23    Question The equation of a curve is y = x + 2cos x. Find the x-coordinates of the stationary points of the  curve for 0 ≤ x ≤ 2π, and determine the nature of each of these stationary points. Solution We are required to find the x-coordinates of stationary points of the […]

Past Papers’ Solutions | Edexcel | AS & A level | Mathematics | Core Mathematics 1 (C1-6663/01) | Year 2006 | June | Q#5

Hits: 4   Question Differentiate with respect to x; a.   b.   Solution a.   We are required to differentiate; We are required to find . Rule for differentiation of  is: Rule for differentiation of  is: b.   We are required to differentiate; We are required to find . Rule for differentiation of  is: Rule for differentiation is of  is: […]

Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2006 | June | Q#5

Hits: 2   Question The curve with equation  is sketched below. The curve crosses the x-axis at the origin  and the point  lies on the curve.  a.                                i.       Find                          ii.       Hence show that the curve has a stationary  point when  and find the x-coordinate  of the other stationary point. b.                             […]

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

Hits: 1   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 […]

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

Hits: 17   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#7

Hits: 2   Question The volume, , of water in a tank at time  seconds is given by; , for  a.   Find:                            i.                                  ii.        b.   Find the rate of change of the volume of water in the tank, in , when . c.                                 i.       Verify that   has a stationary value when .                           ii.       Determine whether this is a maximum or minimum […]

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#9

Hits: 301 Question The diagram shows an open container constructed out of 200 cm2 of cardboard. The two vertical end pieces are isosceles triangles with sides  cm,  cm and  cm, and the two side pieces are rectangles of length  cm and width  cm, as shown. The open top is a horizontal rectangle.      i.       Show that      ii.       Show that the volume, […]

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#10

Hits: 98 Question The diagram shows the curve , where  is a constant. The curve has a minimum point on the x-axis.      i.       Find the value of .    ii.       Find the coordinates of the maximum point of the curve.   iii.       State the set of values of  for which  is a decreasing function of .   iv.       Find the area of the […]

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#8

Hits: 149 Question The equation of a curve is  . i.       Calculate the gradient of the curve at the point where x = 1.    ii.       A point with coordinates (x, y) moves along the curve in such a way that the rate of increase of y has a constant value of 0.02 units per second. Find the rate of increase of […]

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

Hits: 101   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 […]

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

Hits: 95 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 […]

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#1

Hits: 80 Question A curve has equation . . Given that the gradient of the curve is −3 when x = 2, find the value of the constant k. Solution Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve  with respect to  is: In this case; Therefore; We can rewrite the expression; […]