Question The line meets the the curve at the points A and B as shown in the figure. a. Find the coordinates of A and the coordinates of B. b. Find the distance AB in the form where r is a rational number. Solution a. We are required to find the coordinates of the points […]
Question The line has equation 4x + 2y – 3 = 0. a. Find the gradient of . The line is perpendicular to and passes through the point (2,5). b. Find an equation of in the form y = mx + +c, where m and c are constants. Solution a. We are given equation of line ; We are […]
Question Figure shows a sketch of the curve H with equation; , a. Give the coordinates of the point where H crosses the x-axis. b. Give the equations of the asymptotes to H. c. Find an equation for the normal to H at the point P(–3, 3). This normal crosses the x-axis at A […]
Question Figure 1 shows a sketch of the curve with equation y = f(x) where , The curve crosses the x-axis at (1, 0), touches it at (–3, 0) and crosses the y-axis at (0, –9). a. In the space below, sketch the curve C with equation y=f(x+2) and state the coordinates of the points where the […]
Question The straight line L1 passes through the points (–1, 3) and (11, 12). a. Find an equation for L1 in the form ax + by + c = 0, where a, b and c are integers. The line L2 has equation 3y + 4x – 30 = 0. b. Find the coordinates of the point of intersection […]
Question The curve C has equation , a. Find , giving each term in its simplest form. The point P on C has x-coordinate equal to . b. Find the equation of the tangent to C at P, giving your answer in the form y = ax + b, where a and b are constants. The […]
Question Figure 1 shows a sketch of the curve with equation , x ≠ 0. The curve C has equation , x ≠ 0, and the line has equation y = 4x + 2. a. Sketch and clearly label the graphs of C and on a single diagram. On your diagram, show clearly the coordinates of the […]
Question The line L1 has equation y = -2x+3. The line L2 is perpendicular to L1 and passes through the point (5, 6). a. Find an equation for L2 in the form ax + by + c = 0, where a, b and c are integers. The line L2 crosses the x-axis at the point A and the y-axis […]
Question A curve has equation . The point P with coordinates (-1,6) lies on the curve. a. Find the equation of the tangent to the curve at the point P, giving your answer in the form . b. The point Q with coordinates (2,k) lies on the curve. i. Find the value of k. ii. Verify that Q also […]
Question The line AB has equation . a. The point with coordinates (p,p+2) lies on the line AB. Find the value of the constant p. b. Find the gradient of AB. c. The point A has coordinates (1,2). The point C(-5,k) is such that AC is perpendicular to AB. Find the value of k. d. The line AB […]
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. The curve and the line intersect at two distinct points. i. Show that . ii. Find the possible values of k. Solution a. […]
Question A circle with centre C(-3,2) has equation a. Find the y-coordinates of the points where the circle crosses the y-axis. b. Find the radius of the circle. c. The point P(2,5) lies outside the circle. i. Find the length of CP, giving your answer in the form , where n is an integer. ii. The point Q lies […]
Question The gradient, , of a curve at the point (x,y) is given by The curve passes through the point P(1,4). a. Find the equation of the tangent to the curve at the point P, giving your answer in the form . b. Find the equation of the curve. Solution a. We are required to find the […]
Question a. i. Express in the form . ii. Use the result from part (a)(i) to show that the equation has no real solutions. b. A curve has equation . i. Find the coordinates of the vertex of the curve. ii. Sketch the curve, […]
Question The point A has coordinates (-3,2) and the point B has coordinates (7,k). The line AB has equation . a. i. Show that . ii. Hence find the coordinates of the midpoint of AB. b. Find the gradient of AB. c. A line which passes through the point A is perpendicular to the line AB. Find an equation […]