Question The diagram shows part of the curve and the lines and . The curve and the line intersect at point A. i. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 3600 about the x-axis. ii. Find the equation of the normal to the curve at A, […]
Question A curve has equation and a line has equation , where is a constant. i.Show that, for all values of k, the curve and the line meet. ii.State the value of k for which the line is a tangent to the curve and find the coordinates of the point where the line touches the […]
Question A curve passes through (0, 11) and has an equation for which , where a and b are constants. i. Find the equation of the curve in terms of a and b. ii. It is now given that the curve has a stationary point at (2, 3). Find the values […]
Question Two points A and B have coordinates and respectively. The line BC is perpendicular to AB and intersects the x-axis at C. i. Find the equation of BC and the x-coordinate of C. ii. Find the distance AC, giving your answer correct to 3 decimal places. Solution i. We are required […]
Question The equation of a curve is and the equation of a line is , where k is a constant. i. Find the set of values of k for which the line does not meet the curve. In the case where k = 15, the curve intersects the line at points A and B. ii. […]
Question The diagram shows part of the curve and the line , intersecting at the origin O and the point R. Point P lies on the line between O and R and the x-coordinate of P is . Point Q lies on the curve and PQ is parallel to the y-axis. i. Express the […]
Question A curve has equation . The point A on the curve has coordinates . i. a. Find and simplify the equation of the normal through A. b. Find the x-coordinate of the point where this normal meets the curve again. […]
Question The diagram shows part of the curve with equation y = k(x3 − 7×2 + 12x) for some constant k. The curve intersects the line y = x at the origin O and at the point A(2, 2). i. Find the value of k. ii. Verify that the curve meets the line […]
Question Two points A and B have coordinates (3a, -a) and (-a, 2a) respectively, where a is a positive constant. i. Find the equation of the line through the origin parallel to AB. ii. The length of the line AB is units. Find the value of a. Solution i. We are […]
Question A line has equation y = x + 1 and a curve has equation y = x2 + bx + 5. Find the set of values of the constant b for which the line meets the curve. Solution If two lines (or a line and a curve) intersect each other at a point then […]
Question The diagram shows part of the curve intersecting the x-axis at the origin O and at . The line AB intersects the y-axis at B and has equation . i. Show that AB is the tangent to the curve at A. ii. Show that the area of the shaded region can be […]
Question A curve has equation and a line has equation , where is a constant. i.Find the set of values of for which the curve and the line meet. ii.The line is a tangent to the curve for two particular values of . For each of these values find the x-coordinate of the point at […]
Question A straight line cuts the positive x-axis at A and the positive y-axis at B(0, 2). Angle radians, where O is the origin. i. Find the exact value of the x-coordinate of A. ii. Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx […]
Question The one-one function f is defined by for , where c is a constant. i. State the smallest possible value of c. In parts (ii) and (iii) the value of c is 4. ii. Find an expression for and state the domain of . iii. Solve the equation , giving your […]
Question i. The tangent to the curve y = x3 − 9×2 + 24x − 12 at a point A is parallel to the line y = 2 − 3x. Find the equation of the tangent at A. ii. The function f is defined by f(x) = x3 − 9×2 + 24x − 12 […]
Question The coordinates of points A and B are (−3k – 1, k + 3) and (k + 3, 3k + 5) respectively, where k is a constant (k ≠ −1). i. Find and simplify the gradient of AB, showing that it is independent of k. ii. Find and simplify the equation of the perpendicular […]
Question The diagram shows part of the curve . The line y = 4 intersects the curve at the points P and Q. i. Show that the tangents to the curve at P and Q meet at a point on the line y = x. ii. Find, showing all necessary working, the volume obtained when […]
Question A curve is such that and (2,5) is a point on the curve. i. Find the equation of the curve. ii. A point P moves along the curve in such a way that the y-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of […]
Question Points A and B have coordinates (h, h) and (4h + 6, 5h) respectively. The equation of the perpendicular bisector of AB is 3x + 2y = k. Find the values of the constants h and k. Solution We are given that line AB has coordinates of the two points and . We are […]
Question The equation of a curve is , where is a constant. i. Find the set of values of for which the whole of the curve lies above the x-axis. ii. Find the value of for which the line y + 2x = 7 is a tangent to the curve. Solution i. […]