Question A curve and the line AB are sketched below. The curves has the equation and the points A(-1,9) and B(2,12) lie on the curve. a. Find the equation of the normal to the curve at the point A, giving your answer in the form . b. i. Find . ii. Hence find the area of the shaded region […]
Question The point A has coordinates (-2,5) and the point B has coordinates (8,-6). a. Find and equation for the straight line AB, giving your answer in the form , where p, q and r are integers. b. The point C has coordinates (k,k+1). Given that the angle ACB is a right angle, find the two possible values […]
Question The diagram shows the sketch of a curve and the tangent to the curve at P. The curve has equation and the point P(-2,24) lies on the curve. The tangent at P crosses the x-axis at Q. a. i. Find the equation of the tangent to the […]
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 circle with center C(5,-3) passes through the point A(-2,1). a. Find the equation of the circle in the form b. Given that AB is a diameter of the circle, find the coordinates of the point B. c. Find an equation of the tangent to the circle at the point A, giving your answer in the […]
Question The line AB has equation . a. The line AB is parallel to the line with equation . Find the value of m. b. The line AB intersects the line with equation at the point B. Find the coordinates of B. c. The point with coordinates lies on the line AB. Find the value of k. Solution […]
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 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 The diagram shows a sketch of a curve and a line. The curve has equation . The points A(-1,6) and B(2,30) lie on the curve. a. Find an equation of the tangent to the curve at the point A. b. i. Find ii. Calculate the area of the shaded region bounded by the curve and the line AB. […]
Question The line AB has equation . a. i. Find the gradient of AB. ii. Find an equation of the line that is perpendicular to the line AB and which passes through the point (-2,-3) . Express your answer in the form , where p, q and r are […]
Question a. The function f is defined for all values of by . i. Find the remainder when is divided by . ii. Given that and , write down two linear factors of . iii. Hence express as the product of three linear factors. b. The curve with equation is sketched below. i. The curve intersects the y-axis at the point . Find the […]
Question A circle with center C has the equation . The point A(3,-2) lies on the circle. a. Express the equation of the circle in the form b. i. Write down the coordinates of C. ii. Show that the circle has radius , where n is an integer. c. i. Find the equation of the tangent to the circle […]
Question The diagram shows a curve and a line which intersect at the points A, B and C. The curve has equation and the straight line has equation . The point B has coordinates (0,7). a. i. Show that the x-coordinates of the points A and […]
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 curve has equation . a. Find: i. ; ii. b. The point on the curve where is P. i. Determine whether y is increasing or decreasing at P, giving a reason for your answer. ii. Find an equation of the tangent to the curve at […]
Question The point A has coordinates (-1,2) and the point B has coordinates (3,-5). a. i. Find the gradient of AB. ii. Hence find an equation of the line AB, giving your answer in the form , where p, q and r are integers. b. Midpoint of AB is M. i. Find the coordinates of M. ii. Find an equation […]
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 circle with center C has equation . a. Write down: i. the coordinates of C; ii. the radius of the circle b. The point D has coordinates (7,-2). i. Verify that the circle passes through the origin O. ii. Given that the circle also passes through the points (10,0) and (0,p), sketch the circle and find the value of p. […]
Question a. The polynomial is given by . i. Use the Factor Theorem to show that is a factor of . ii. Express in the form , where a and b are constants. b. The curve C with equation , sketched below, crosses the x-axis at the point . i. Find the gradient of the curve C at the point Q. ii. Hence find an […]
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 […]