Question The equation of the curve is . i. Show that ii. Find the coordinates of each of the points on the curve where the tangent is parallel to the x- axis. Solution i. We are given; We are required to find . To find from an implicit […]
Question It is given that the curve has one stationary point. i. Find the x-coordinates of this point. ii. Determine whether this point is a maximum or a minimum point. Solution i. We are required to find the coordinates of stationary point of the curve; A stationary […]
Question The variables x and y satisfy the equation y = A(b-x), where A and b are constants. The graph of ln y against ln x is a straight line passing through the points (0, 1.3) and (1.6, 0.9), as shown in the diagram. Find the values of A and b, correct to 2 […]
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 Solve the inequality . Solution SOLVING INEQUALITY: PIECEWISE Let and , then; We have to consider two separate cases; When When We have the inequality; It can be written as; We have to consider two separate cases; When When Therefore the inequality will hold for ; SOLVING INEQUALITY: ALGEBRAICALLY Let, . Since […]
Question The points Q (1, 3) and R (7, 0) lie on the line , as shown in Figure. The length of QR is . a. Find the value of a. The line l2 is perpendicular to , passes through Q and crosses the y-axis at the point P, as shown in Figure. Find b. an […]
Question The curve C has equation , where k is a constant. a. Find . Point A with x-coordinate lies on C. The tangent to C at A is parallel to the line with equation . Find b. The value of k. c. The value of y-coordinate of A. Solution a. Gradient (slope) of the curve […]
Question The curve C has equation y=(x + 3)(x −1)2 . a. Sketch C showing clearly the coordinates of the points where the curve meets the coordinate axes. b. Show that the equation of C can be written in the form y = x3 + x2 − 5x + k, where k is a positive integer, and […]
Question The curve C has equation y = f(x), x > 0, and . Given that the point P(4,1) lies on C, a. find f(x) and simplify your answer. b. Find an equation of the normal to C at the point P(4, 1). Solution a. We are required to find f(x), when; We are also given that […]
Question The point A(–6, 4) and the point B (8, –3) lie on the line L. a. Find an equation for L in the form ax + by + c = 0, where a, b and c are integers. b. Find the distance AB, giving your answer in the form , where k is an integer.` Solution a. […]
Question The straight line L has equation and the curve C has equation a. Sketch on the same axes the line L and the curve C, showing the values of the intercepts on the x-axis and the y-axis. b. Show that the x-coordinates of the points of intersection of L and C satisfy the equation . c. Hence find the coordinates […]
Question The circle S has centre C(8,13) and touches the x-axis, as shown in the diagram. a. Write down an equation for S, giving your answer in the form b. The point P with coordinates (3,1) lies on the circle. i. Find the gradient of the straight line passing through P and C. ii. Hence […]
Question The curve with equation is sketched below. The points A(-2,0) , B(2,0) and C(1,15) lie on the curve. a. Find an equation of the straight line AC . b. i. Find . ii. Hence calculate the area of the shaded region bounded by the curve and the line AC . Solution a. We are required to […]
Question The curve C has equation . The line L has equation , where k is a constant. a. Show that the x-coordinates of any points of intersection of the line L with the curve C satisfy the equation b. The curve C and the line L intersect in two distinct points. Show that c. Solve the inequality […]
Question A circle with centre C has equation . a. By completing the square, express this equation in the form b. Write down: i. the coordinates of C; ii. the radius of the circle, leaving your answer in surd form. c. A line has equation . i. Show that the x-coordinate of any point of intersection of […]
Question The triangle ABC has vertices A(-2,3), B(4,1) and C(2,-5). a. Find the coordinates of the mid-point of BC . b. i. F ind the gradient of AB, in its simplest form. ii. Hence find an equation of the line AB , giving your answer in the form , where q and r are […]
Question The equation of a curve is . i. Show that the equation of the normal to the curve at the point is This normal meets the curve again at the point Q. ii. Find the coordinates of Q. iii. Find the length of PQ. Solution i. We are required to find the equation of the normal to the curve. To […]
Question In the diagram, the points A and C lie on the x- and y-axes respectively and the equation of AC is . The point B has coordinates (2, 2). The perpendicular from B to AC meets AC at the point X. i. Find the coordinates of X. The point D is such that the quadrilateral ABCD has AC as a […]
Question The equation of a curve C is and the equation of a line L is . i. Find the x-coordinates of the points of intersection of L and C. ii. Show that one of these points is also the stationary point of C. Solution i. If two lines (or a line and a curve) […]