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Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2016 | June | Q#8

Hits: 11   Question The gradient, , at the point (x,y) on a curve is given by a.                        i.               Find                   ii.               The curve passes through the point . Verify that the curve has a minimum point at P. b.                       i.               Show that at the points on the curve where y is decreasing                   ii.               Solve the inequality Solution a.   We are given;                     i. […]

Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2016 | June | Q#7

Hits: 19   Question The diagram shows the sketch of a curve and the tangent to the curve at P. The curve has equation  and the point P(-2,24) lies on the curve. The tangent at P  crosses the x-axis at Q. a.                       i.               Find the equation of the […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#11

Hits: 340   Question A curve has equation , where k is a non-zero constant.      i.       Find the x-coordinates of the stationary points in terms of k, and determine the nature of each  stationary point, justifying your answers.    ii.   The diagram shows part of the curve for the case when k = 1. Showing all necessary working, […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#10

Hits: 296   Question A curve is such that  , where a is a positive constant. The point A(a2, 3) lies on the  curve. Find, in terms of a,      i.       the equation of the tangent to the curve at A, simplifying your answer,    ii.       the equation of the curve.  It is now given that B(16,8) also lies on the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/13) | Q#4

Hits: 247   Question The function f is such that  for , where n is an integer. It is given that  f is an increasing function. Find the least possible value of n. Solution We are given function; We are also given that it is an increasing function. To test whether a function  is increasing or decreasing at a particular […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#7

Hits: 410 Question The equation of a curve is .     i.       Obtain an expression for    ii.       Explain why the curve has no stationary points.  At the point P on the curve, x = 2.   iii.       Show that the normal to the curve at P passes through the origin.   iv.       A point moves along the curve in such a way that its x-coordinate […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#11

Hits: 297 Question The point P(3,5) lies on the curve  .      i.       Find the x-coordinate of the point where the normal to the curve at P intersects the x-axis.    ii.       Find the x-coordinate of each of the stationary points on the curve and determine the nature of  each stationary point, justifying your answers. Solution      i.   The […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/11) | Q#10

Hits: 336 Question A curve has equation  and it is given that . The point A is the only point  on the curve at which the gradient is −1.      i.       Find the x-coordinate of A.    ii.       Given that the curve also passes through the point (4,10), find the y-coordinate of A, giving  your answer as a fraction. Solution      […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#7

Hits: 439 Question The point P(x,y) is moving along the curve  in such a way that the rate of change  of y is constant. Find the values of x at the points at which the rate of change of x is equal to half the rate of change of y. Solution We are required to find values of x […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#5

Hits: 228 Question A curve has equation . Find the values of x at which the curve has a stationary  point and determine the nature of each stationary point, justifying your answers. Solution Coordinates of stationary point on the curve  can be found from the derivative of equation of the  curve by equating it with ZERO. This results in […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/13) | Q#3

Hits: 284 Question A curve is such that  and passes through the point P(1,9). The gradient of the curve at P is 2. i.       Find the value of the constant k. ii.       Find the equation of the curve. Solution i.   We are given that; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#10

Hits: 402 Question The diagram shows the part of the curve  for , and the minimum point M.      i.         Find expressions for ,  and .    ii.       Find the coordinates of M and determine the coordinates and nature of the stationary point on          the part of the curve for which .   […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#8

Hits: 452 Question A curve has equation  and passes through the points A(1,-1) and B(4,11). At each of  the points C and D on the curve, the tangent is parallel to AB. Find the equation of the  perpendicular bisector of CD. Solution We are required to find the equation of perpendicular bisector of CD. To find the equation of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#5

Hits: 630 Question A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures x m by y m and is fully enclosed by metal  fencing. The farmer uses 480m of fencing. i.       Show that the total area of land used for the sheep pens, A m2, is […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/11) | Q#4

Hits: 357 Question A curve is such that .     i.       A point P moves along the curve in such a way that the x-coordinate is increasing at a  constant rate of 0.3 units per second. Find the rate of change of the y-coordinate as P crosses the  y-axis. The curve intersects the y-axis where  .    ii.       Find the equation of the curve. […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#10

Hits: 314 Question The diagram shows part of the curve , which touches the x-axis at the point P. The  point Q(3,4) lies on the curve and the tangent to the curve at Q crosses the x-axis at R.      i.       State the x-coordinate of P.  Showing all necessary working, find by calculation ii.       the x-coordinate of R, iii.    the area of […]

Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#6

Hits: 389 Question A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius  is r cm and the internal height is h cm. The volume of the flask is 1000 cm3. A flask is most efficient when the total internal surface area, A cm2, is a minimum.      i.       Show that […]