Question The equation of a curve is 2×3+y3=24 i. Express in terms of x and y, and show that the gradient of the curve is never positive. ii. Find the coordinates of the two points on the curve at which the gradient is −2. Solution i. Gradient (slope) of the curve […]
Question The diagram shows the part of the curve for , and the stationary point M. i. Find the equation of the tangent to the curve at the origin. ii. Find the coordinates of M, giving each coordinate correct to 3 decimal places. Solution i. We are given that […]
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 with […]
Question The variables x and y satisfy the equation y = Kxp, where K and p are constants. The graph of ln y against ln x is a straight line passing through the points (1.28, 3.69) and (2.11, 4.81), as shown in the diagram. Find the values of K and p correct to 2 decimal […]
Question The diagram shows the curve with parametric equations for . i. Show that can be expressed in the form ii. Find the equation of the normal to the curve at the point where the curve crosses the positive y-axis. Give your answer in the form y = mx +c, where the constants […]
Question The equation of a curve is 3×2 + 4xy + y2 = 24. Find the equation of the normal to the curve at the point (1, 3), giving your answer in the form ax +by +c = 0 where a, b and c are integers. Solution i. We are given equation of the […]
Question The variables x and y satisfy the equation y = Aepx, where A and p are constants. The graph of ln y against x is a straight line passing through the points (5, 3.17) and (10, 4.77), as shown in the diagram. Find the values of A and p correct to 2 decimal places. Solution […]
Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q. i. Show that . ii. Find the coordinates of M. iii. Find the gradient of the curve at P and at Q. Solution i. […]
Question The diagram shows the curve with parametric equations for . The minimum point is M and the curve crosses the x-axis at points P and Q. i. Show that . ii. Find the coordinates of M. iii. Find the gradient of the curve at P and at Q. Solution i. […]
Question Solve the equation . Solution SOLVING EQUATION: PIECEWISE Let, . We can write it as; We have to consider two separate cases; When When We have the equation; It can be written as; We have to consider two separate cases; When ; When ; Hence, the only solutions for the given equation are; […]
Question The curve C has equation , where k is a constant. a. Find . The point P, where x=-2, lies on C. The tangent to C at the point P is parallel to the line with equation 2y – 17x – 1=0. Find b. the value of k. c. the value of y-coordinate of P. d. the equation of the tangent […]
Question The points P(0, 2) and Q(3, 7) lie on the line , as shown in Figure. The line is perpendicular to , passes through Q and crosses the x-axis at the point R, as shown in Figure. Find a. an equation for , giving your answer in the form ax + by + c = […]
Question The straight line with equation y = 3x – 7 does not cross or touch the curve with equation y = 2px2 – 6px + 4p, where p is a constant. a. Show that 4p2 – 20p + 9 < 0 b. Hence find the set of possible values of p. Solution a. We are given […]
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 is such that , where a is a positive constant. The point A(a2, 3) lies on the curve. Find, in terms of a, i. the equation of the tangent to the curve at A, simplifying your answer, ii. the equation of the curve. It is now given that B(16,8) also lies on the curve. iii. Find […]
Question Three points, A, B and C, are such that B is the mid-point of AC. The coordinates of A are (2,m) and the coordinates of B are (n,-6), where m and n are constants. i. Find the coordinates of C in terms of m and n. The line y =x + 1 passes through C and is […]
Question Find the set of values of k for which the curve and the line do not meet. Solution We can find the coordinates of intersection point of a curve and line. However, here we are required to show that given curve and line do not meet that means there is no point of intersection of the two. […]