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, find the […]
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. iii. Find […]
Question A curve is such that . The point (2,5) lies on the curve. Find the equation of the curve. Solution We are given that; We can find equation of the curve from its derivative through integration; Therefore; Rule for integration of is: If a point lies on the curve , we can find out value of . We […]
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 i. […]
Question The diagram shows parts of the curves and , intersecting at points A and B. i. State the coordinates of A. ii. Find, showing all necessary working, the area of the shaded region. Solution i. It is evident that point A is the intersection point of the two curves given by equations; It is also […]
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 with respect to […]
Question The diagram shows part of the curve and the point P(2,3) lying on the curve. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360o about the x-axis. Solution Expression for the volume of the solid formed when the shaded region under the curve is rotated completely about the x-axis is; Therefore, for […]
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 . iii. Find the […]
Question A curve is such that . Given that the curve passes through the point , find the equation of the curve. Solution We are given that curve passes through the point and we are required to find the equation of the curve. We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule […]
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. Solution i. […]
Question The diagram shows part of the curve . The shaded region is bounded by the curve, the y- axis and the lines y = 1 and y = 2. Showing all necessary working, find the volume, in terms of , when this shaded region is rotated through about the y-axis. Solution Expression for the volume of the solid formed when […]
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 the shaded […]
Question A curve for which passes through . Find the equation of the curve. Solution i. We can find equation of the curve from its derivative through integration; We are given that; Therefore; Rule for integration of is: Rule for integration of is: If a point lies on the curve , we can find out value of […]