Question The curve C has equation The point P has coordinates (2, 7). a) Show that P lies on C. b) Find the equation of the tangent to C at P, giving your answer in the form y=mx+c, where m and c are constants. The point Q also lies on C. Given that the tangent to […]
Question a) Factorise completely x3 – 6×2 + 9x b) Sketch the curve with equation y = x3 – 6×2 + 9x showing the coordinates of the points at which the curve meets the x-axis. Using your answer to part (b), or otherwise, c) (c) sketch, on a separate diagram, the curve with equation y = (x […]
Question , x>0 a) Show that , where A and B are constants to be found. b) Find c) Evaluate . Solution a) We are given; Utilizing algebraic identity; b) We are given; We are required to find . We can write the given equation/function as; Gradient (slope) of the curve is the derivative of […]
Question The points A and B have coordinates (6, 7) and (8, 2) respectively. The line passes through the point A and is perpendicular to the line AB, as shown in Figure 1. a) Find an equation for in the form ax + by + c = 0, where a, b and c are integers. Given that […]
Question A sequence is defined by , , Where is a constant. a) Write down an expression for in terms of k. b) Show that . Given that , c) Find the value of k. Solution a) We are given that sequence is defined by We are required to find . We can utilize the given expression […]
Question The equation x2 + 3px + p = 0, where p is a non-zero constant, has equal roots. Find the value of p. Solution We are given that; We are given that given equation has equal roots. For a quadratic equation , the expression for solution is; Where is called discriminant. If , the equation will […]
Question A 40-year building programme for new houses began in Oldtown in the year 1951 (Year 1) and finished in 1990 (Year 40). The numbers of houses built each year form an arithmetic sequence with first term a and common difference d. Given that 2400 new houses were built in 1960 and 600 new houses were built […]
Question Find the set of values of x for which a. b. c. both and Solution a. We are given; b. We are required to solve the inequality; We solve the following equation to find critical values of ; Now we have two options; Hence the critical points on the curve for the given condition […]
Question Given that , , find a. b. , simplifying each term. Solution a. We are given; We are required to find . Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is: Therefore; Rule for differentiation is of is: Rule for differentiation is of is: […]
Question Given that , find the value of a. Solution We are given;
Question Simply a. b. Solution a. We are given; 63 b. We are given; Since ;