Question The diagram shows the part of the curve for , and its minimum point M. i. Find the coordinates of M. ii. Use the trapezium rule with 2 intervals to estimate the value of Giving your answer correct to 1 decimal place. iii. State, with a reason, whether the […]
Question The equation of a curve is 3×2 + 2xy + y2 = 6. It is given that there are two points on the curve where the tangent is parallel to the x-axis. i. Show by differentiation that, at these points, y = −3x. ii. Hence find the coordinates of the two […]
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 given […]
Question The curve C with equation y=f(x), x ≠ 0, passes through the point . Given that a. find f(x). b. Verify that f(–2) = 5. c. Find an equation for the tangent to C at the point (–2, 5), giving your answer in the form ax + by + c = 0, where a, b […]
Question Figure 2 shows part of the curve C with equation The curve cuts the x-axis at the points P, (1, 0) and Q, as shown in Figure 2. a. Write down the x-coordinate of P, and the x-coordinate of Q. b. Show that . c. Show that y=x+7 is an equation of the tangent to […]
Question The line L has equation y=5 – 2x. a. Show that the point P (3, –1) lies on L. b. Find an equation of the line perpendicular to L, which passes through P. Give your answer in the form ax + by + c = 0, where a, b and c are integers. […]
Question A circle has equation . a. Find i. the coordinates of the centre of the circle; ii. the radius of the circle in the form , where is an integer. b. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord. c. A line has equation , where is a constant. i. Show […]
Question A curve has equation . a. Find b. Find an equation for the tangent to the curve at the point where . c. Determine whether is increasing or decreasing when . Solution a. We have the equation; Rule for differentiation is of is: Rule for differentiation is of is: Rule for differentiation is of is: b. […]
Question The point has coordinates and the point has coordinates . a. i. Find the gradient of the line . ii. Hence, or otherwise, show that the line has equation b. The line intersects the line with equation at the point . Find the coordinates of . c. Find an equation of the line through which is perpendicular to . Solution a. […]
Question The diagram shows the curve with equation and the line . The points and have coordinates and respectively. The curve touches the x-axis at the origin and crosses the x-axis at the point . The line cuts the curve at the point where and touches the curve at where . a. Find the area of the rectangle . b. i. […]
Question a. i. Express in the form , where and are integers. ii. Hence, or otherwise, describe the coordinates of the minimum point of the curve with equation . b. The line has equation and the curve has the equation . i. Show that the x-coordinates of the points of […]
Question The point has coordinates and the point has coordinates . The line has equation . a. i. Show that . ii. Hence find the coordinates of the mid-point of . b. Find the gradient of . c. Line is perpendicular to the line . i. Find the gradient of . ii. Hence find the equation of the line . iii. Given that point […]
Question The diagram shows the curve y = x(x − 1)(x − 2), which crosses the x-axis at the points O(0, 0), A(1, 0) and B(2, 0). i. The tangents to the curve at the points A and B meet at the point C. Find the x-coordinate of C. ii. Show by integration that the area of the shaded region […]
Question The three points A(1, 3), B(13, 11) and C(6, 15) are shown in the diagram. The perpendicular from C to AB meets AB at the point D. Find i. the equation of CD, ii. the coordinates of D. Solution i. To find the equation of the line either we need coordinates of the two points on the line […]
Question A curve is such that and P (1, 8) is a point on the curve. i. The normal to the curve at the point P meets the coordinate axes at Q and at R. Find the coordinates of the mid-point of QR. ii. Find the equation of the curve. Solution i. To find the mid-point […]
Question The curve intersects the line at two points. Find the distance between the two points. Solution Expression to find distance between two given points and is: So first we need to find the coordinates of points of intersection of the curve and the line. If two lines (or a line and a curve) intersect each other […]