Question The diagram shows part of the curve The shaded region is bounded by the curve and the two axes. i. Show that can be expressed in the form where the values of the constants and are to be determined. ii.Find the exact area of the shaded region. Solution i. […]
Question i. Solve the inequality . ii. Hence find the largest integer y satisfying the inequality . Solution SOLVING INEQUALITY: PIECEWISE i. 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 […]
Question The parametric equations of a curve are , i. Find in terms of t and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. ii. Find the gradient of the normal to the curve at the point where the curve crosses the x-axis. Solution […]
Question The equation of a curve is x2+4xy+2y2=7 i. Find the equation of the tangent to the curve at the point (-1, 3). Give your answer in the form ax +by +c = 0, where a, b and c are integers. ii. Show that there is no point on the curve at which […]
Question The diagram shows the curve y=4e−2x and a straight line. The curve crosses the y-axis at the point P. The straight line crosses the y-axis at the point (0, 9) and its gradient is equal to the gradient of the curve at P. The straight line meets the curve at two points, one of […]
Question The equation of a curve is . The curve has a stationary point M in the interval . Find the coordinates of M, giving each coordinate correct to 3 significant figures. Solution We are required to find the x-coordinates of stationary point of the curve. A stationary point on the curve is the point […]
Question The parametric equations of a curve are , i. Find in terms of t and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. ii. Find the gradient of the normal to the curve at the point where the curve crosses the x-axis. Solution […]
Question The diagram shows the curve with parametric equations for . The end-points of the curve are (1, 4) and (3, 3). i. Show that . ii. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures. iii. Find the exact gradient of the normal to the curve […]
Question Find the equation of the tangent to the curve at the point on the curve for which x = 0. Give your answer in the form ax+by +c = 0 where a, b and c are integers. Solution We are given that curve with equation and we are required to find the equation of […]
Question The diagram shows the curve with parametric equations for . The end-points of the curve are (1, 4) and (3, 3). i. Show that . ii. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures. iii. Find the exact gradient of the normal to the curve […]
Question Find the equation of the tangent to the curve at the point on the curve for which x = 0. Give your answer in the form ax+by +c = 0 where a, b and c are integers. Solution We are given that curve with equation and we are required to find the equation […]
Question The diagram shows the curve with equation The curve crosses the x-axis at the point P and has a minimum point M. i. Find the gradient of the curve at the point P. ii. Find the exact coordinates of the point M. Solution i. We are required to find the […]
Question The parametric equations of a curve are i. Find and use division to show that can be written in the form , where a and b are constants to be found. ii. The straight line x − 2y + 9 = 0 is the normal to the curve at the point P. […]
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 It is given that the variable x is such that and Find the set of possible values of x, giving your answer in the form a < x < b where the constants a and b are correct to 3 significant figures. Solution First we find the value of x for; Taking natural […]
Question It is given that x satisfies the equation . Find the possible values of 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 […]
Question It is given that the variable x is such that and Find the set of possible values of x, giving your answer in the form a < x < b where the constants a and b are correct to 3 significant figures. Solution First we find the value of x for; Taking natural […]
Question The diagram shows the curve and points A(1,0) and B(5,2) lying on the curve. i. Find the equation of the line AB, giving your answer in the form y=mx+c. ii. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to AB. iii. Find the perpendicular distance between the line AB and the […]
Question The diagram shows parts of the graphs of and intersecting at points A and B. i. Find by calculation the x-coordinates of A and B. ii. Find, showing all necessary working, the area of the shaded region. Solution i. We are required to find the x-coordinates of points A and B which are intersection points of a line and […]
Question Find the set of values of a for which the curve and the straight line meet at two distinct points. Solution We need to find the equation that satisfies the x-coordinates of the points of intersection of given curve and line. If two lines (or a line and a curve) intersect each other at a point then that point […]