Question The water level in a reservoir rises and falls during a four-hour period of heavy rainfall. The height, h cm, of water above its normal level, t hours after it starts to rain, can be modelled by the equation , a. Find the rate of change of the height of water, in cm per hour, 3 hours […]
Question The diagram shows the right-angled corner AFE of a building and four sections of fencing running parallel to the walls of the building. Each of the sections of fencing AB and DE has length x metres and each of the sections of wall AF and FE has length y metres. The total length of the four sections of […]
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. We […]
Question a. A curve has equation . i. Find the values of x where the curve crosses the x-axis, giving your answer in the form , where m and n are integers. ii. Sketch the curve, giving the value of the y-intercept. b. A line has equation , where k is a constant. i. Show that the x-coordinates of any points of intersection […]
Question a. i. Express in the form , where p and q are rational numbers. ii. Hence write down the minimum value of . a. Describe the geometrical transformation which maps the graph of onto the graph of . Solution a. i. We have the expression; We use method of “completing square” to obtain the desired form. First we complete the […]
Question A curve has equation and a line has equation , where k is a constant. a. Show that the x-coordinate of any point of intersection of the curve and the line satisfies the equation b. Given that the line and the curve do not intersect:. i. Show that . […]
Question a. Sketch the curve with equation . b. The polynomial is given by . i. Find the remainder when is divided by . ii. Use the Factor Theorem to show that is a factor of . iii. Express in the form , where B and c are integers. iv. Hence show that the equation has exactly one real root and state its value. Solution […]
Question a. Express in the form , where p and q are rational numbers. b. A curve has equation . i. Use the result from part (a) to write down the coordinates of the vertex of the curve. ii. State the equation of the line of symmetry of the curve. c. The curve with equation is translated by vector . Find the equation of […]
Question A circle with center C has the equation . a. Express this equation in the form b. i. State the coordinates of C. ii. Find the radius of the circle, giving your answer in the form . c. i. The point P with coordinates (4,k) lies on the circle. Find the possible values of k. ii. The points […]
Question Solve each of the following inequalities: a. b. . Solution a. b. 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 are & . Standard form of quadratic equation is; The graph of quadratic equation is […]
Question a. i. Express in the form where p and q are integers. ii. Hence write down the maximum value of . b. i. Factorise . ii. Sketch the curve with equation , stating the values of x where the curve crosses the x-axis and the value of […]
Question a. The polynomial f(x) is given by . i. Use the Factor Theorem to show that (x-1) is a factor of f(x). ii. Express f(x) in the form , where p and q are integers. iii. Hence show that the equation f(x)=0 has exactly one real root and state its value. b. The curve with equation is sketched below. The […]
Question a. The polynomial f(x) is given by . i. Use the Factor Theorem to show that x+3 is a factor of f(x). ii. Express f(x) in the form , where p and q are integers. b. A curve has equation . i. Find . ii. Show that the x-coordinates of any stationary points of the curve satisfy the equation iii. Use the […]
Question The polynomial is given by . a. Use the Remainder Theorem to find the remainder when is divided by . b. Use the Factor Theorem to show that is a factor of . c. i. Express in the form , where b and c are integers. ii. Hence show that the equation has exactly one real root. Solution a. Remainder theorem states […]
Question a. Express in the form , where p and q are rational numbers. b. A curve has equation . i. Find the coordinates of the vertex of the curve. ii. State the equation of the line of symmetry of the curve. iii. Sketch the curve, stating the value of the intercept on […]
Question a. The polynomial is given by i. Find the remainder when is divided by . ii. Use the Factor Theorem to show that is a factor of . iii. Express in the form , where b and c are integers. iv. The equation has one root equal to -2. Show that equation has no other real roots. […]
Question a. i. Express in the form , where p and q are integers. ii. Hence show that is always positive. b. A curve has equation . i. Write down the coordinates of the minimum point of the curve. ii. Sketch the curve, showing the value of the intercept on the y-axis. […]
Question a. Factorise . b. Show that can be written as . c. A curve has equation . i. Write down the equation of its line of symmetry. ii. Find the coordinates of its vertex. iii. Sketch the curve, indicating the values of the intercepts on the x-axis and the y-axis. Solution a. b. c. i. We are given […]
Question The quadratic equation has real roots. (a) Show that . (b) Find the possible values of k. Solution a. We are given the quadratic equation; For a quadratic equation , the expression for solution is; Where is called discriminant. If , the equation will have two distinct roots. If , the equation will have two identical/repeated […]
Question a. i. Express in the form , where p and q are rational numbers. ii. Hence write down the minimum value of . b. The point A has coordinates (-3,5) and the point B has coordinates (x,3x+9). i. Show that . ii. Use your result from part (a)(ii) to find the minimum […]