Question The diagram shows the curve y=(4−x)ex and its maximum point M. The curve cuts the x-axis at A and the y-axis at B. i. Write down the coordinates of A and B. ii. Find the x-coordinate of M. iii. The point P on the curve has x-coordinate p. The tangent to the curve at P passes through the […]
Question i. By sketching a suitable pair of graphs, for x < 0, show that exactly one root of the equation is negative. ii. Verify by calculation that this root lies between -1.0 and -0.5. iii. Use the iterative formula to determine the root correct to 2 decimal places, showing the result of each iteration. Solution […]
Question Two variable quantities x and y are related by the equation where a and k are constants. Four pairs of values of x and y are measured experimentally. The result of plotting ln y against x is shown in the diagram. Use the diagram to estimate the values of a and k. Solution We […]
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 in standard form as; We have to consider two separate cases; When When Therefore […]
Question The parametric equations of a curve are i. Show that ii. Find the equation of the tangent to the curve at the point where . iii. For the part of the curve where , find the coordinates of the points where the tangent is parallel to the x-axis. Solution i. We are required to show that […]
Question i. By sketching a suitable pair of graphs, show that the equation Has exactly one root. ii. Verify by calculation that this root lies between 1.0 and 1.4. iii. Use the iterative formula to determine the root correct to 2 decimal places, showing the value of […]
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 following with […]
Question i. By sketching a suitable pair of graphs, show that there is only one value of x in the interval that is a root of the equation ii. Verify by calculation that this root lies between 1 and 1.5. iii. Show that this value of x is also a root of the equation iv. Use the iterative […]
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 following […]
Question i. Sketch the graph of the curve , for . The straight line , where is a constant, passes through the maximum point of this curve for . ii. Find the value of k in terms of . iii. State the coordinates of the other point, apart from the origin, where the line and the curve intersect. Solution i. […]
Question The diagram shows points and on the curve . The tangent to the curve at B crosses the x-axis at C. The point D has coordinates (2, 0). i. Find the equation of the tangent to the curve at B and hence show that the area of triangle BDC is ii. Show that the volume of the solid formed when […]
Question The diagram shows a trapezium ABCD in which BC is parallel to AD and angle . The coordinates of A, B and D are (2, 0), (4, 6) and (12, 5) respectively. i. Find the equations of BC and CD. ii. Calculate the coordinates of C. Solution i. First we find the equation of . To […]
Question A curve is such that . The curve passes through the point (1, 5). i. Find the equation of the curve. ii. Find the set of values of for which the gradient of the curve is positive. Solution i. To find the equation of the curve; We can find equation of the curve from its […]
Question Find the coordinates of the points of intersection of the line and the curve . 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 and the curve). Therefore, […]
Question The line has equation . The line passes through the point and is perpendicular to . i. Find the equation of . ii. Given that the lines and intersect at the point B, find the length of AB. Solution i. To find the equation of the line either we need coordinates of the two points on […]