Question The function is defined for the domain . i. Solve the equation . ii. Sketch the graph of . iii. Find the set of values of for which the equation has no solution. The function is defined for the domain . iv. State the largest value of for which g has an inverse. v. For […]
Question The equation of a curve is . i. Find . ii. Find the equation of the tangent to the curve at the point where the curve intersects the y-axis. iii. Find the set of values of for which is an increasing function of . Solution i. Rule for differentiation of is: Rule […]
Question The diagram shows the curve and the line , which intersect at points A and B. Find the area of the shaded region. Solution It is evident from the diagram that; However, we need the limits to integrate the equations of line & curve. It is evident that these limits are from point A to point B. Therefore we […]
Question A solid rectangular block has a square base of side cm. The height of the block is cm and the total surface area of the block is 96 cm2. i. Express in terms of and show that the volume, cm3, of the block is given by Given that can vary, ii. find the stationary value of , […]
Question Relative to an origin , the position vectors of the points and are given by and i. Find the value of the for which is perpendicular to . ii. Find the value of the for which magnitude of is 7. Solution i. If and & , then and are perpendicular. Therefore we need to find the scalar/dot product […]
Question In the diagram, is the point and is the point . The line passes through and is parallel to . The line passes through and is perpendicular to . The lines and meet at . Find the coordinates of . Solution It is evident from the diagram that point C is the intersection of lines AC […]
Question The diagram shows part of the curve , where a is a positive constant. Given that the volume obtained when the shaded region is rotated through about the x-axis is , find the value of . Solution Expression for the volume of the solid formed when the shaded region under the curve is rotated completely about the x-axis […]
Question i. Show that the equation can be written in the form ii. Solve the equation , for . Solution i. We have the equation; We have the relation , therefore, ii. To solve the equation , for , as demonstrated in (i), we can write the given equation as; To solve this equation for , Using calculator […]
Question The functions and are defined for by i. Find the range of . ii. Find the value of the constant for which the equation has equal roots. Solution i. Standard form of quadratic equation is; The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum […]
Question i. Find the first 3 terms in the expansion of in ascending powers of . ii. Given that there is no term in in the expansion of , find the value of the constant . iii. For this value of , find the coefficient of in the expansion of . Solution i. Expression for […]
Question a) Find the sum of all the multiples of 5 between 100 and 300 inclusive. b) A geometric progression has a common ratio of and the sum of the first 3 terms is 35. Find i. the first term of the progression, ii. the sum to infinity. Solution a) It is evident that each next multiple of […]