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. […]
Question The equation of a curve is . i. Obtain an expression for ii. Explain why the curve has no stationary points. At the point P on the curve, x = 2. iii. Show that the normal to the curve at P passes through the origin. iv. A point moves along the curve in such a way that its x-coordinate is decreasing […]
Question The line , where a and b are positive constants, intersects the x- and y-axes at the points A and B respectively. The mid-point of AB lies on the line and the distance . Find the values of a and b. Solution We need to work through the problem statement very carefully to glean the information scattered […]
Question A curve has equation . i. Find the set of values of for which . ii. Find the value of the constant for which the line is a tangent to the curve. Solution i. We are required to find the set of values of x for which . We are given that; Therefore; We solve the following equation to find […]
Question The point P(3,5) lies on the curve . i. Find the x-coordinate of the point where the normal to the curve at P intersects the x-axis. ii. Find the x-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. Solution i. The point where […]
Question The diagram shows parts of the curves and , intersecting at points A and B. i. State the coordinates of A. ii. Find, showing all necessary working, the area of the shaded region. Solution i. It is evident that point A is the intersection point of the two curves given by equations; It is also […]
Question C is the mid-point of the line joining A(14,−7) to B(−6,3). The line through C perpendicular to AB crosses the y-axis at D. i. Find the equation of the line CD, giving your answer in the form . ii. Find the distance AD. Solution i. We are required to write equation of the line CD. To find […]
Question Triangle ABC has vertices at A(−2,−1), B(4,6) and C(6,−3). i. Show that triangle ABC is isosceles and find the exact area of this triangle. ii. The point D is the point on AB such that CD is perpendicular to AB. Calculate the x-coordinate of D. Solution i. An isosceles triangle is a triangle with (at least) two equal sides. […]
Question The function is defined by for . i. Find the set of values of x for which f(x) ≤ 3. ii. Given that the line y=mx+c is a tangent to the curve y = f(x), show that The function g is defined by for x ≥ k, where k is a constant. iii. Express in the form , where a […]
Question Three points have coordinates A(0,7), B(8,3) and C(3k,k). Find the value of the constant k for which i. C lies on the line that passes through A and B, ii. C lies on the perpendicular bisector of AB. Solution i. If point C lies on line AB, then coordinates of point C must satisfy equation of line AB. […]
Question A curve has equation and passes through the points A(1,-1) and B(4,11). At each of the points C and D on the curve, the tangent is parallel to AB. Find the equation of the perpendicular bisector of CD. Solution We are required to find the equation of perpendicular bisector of CD. To find the equation of the line […]
Question a. Find the values of the constant m for which the line is a tangent to the curve . b. The function f is defined for by , where a and b are constants. The solutions of the equation are x = 1 and x = 9. Find i. the values of a and b, ii. y=the […]
Question The diagram shows part of the curve , which touches the x-axis at the point P. The point Q(3,4) lies on the curve and the tangent to the curve at Q crosses the x-axis at R. i. State the x-coordinate of P. Showing all necessary working, find by calculation ii. the x-coordinate of R, iii. the area of the shaded […]
Question Two points have coordinates A(5,7) and B(9,-1). i. Find the equation of the perpendicular bisector of AB. The line through C(1,2) parallel to AB meets the perpendicular bisector of AB at the point X. ii. Find, by calculation, the distance BX. Solution a. We are required to write equation of perpendicular bisector of AB with points A(5,7) and B(9,-1). […]