Question The diagram shows part of the curve , and the lines x = 1 and x = 3. The point A on the curve has coordinates (2, 3). The normal to the curve at A crosses the line x = 1 at B. (i) Show that the normal AB has equation . (ii) […]
Question Relative to an origin O, the position vectors of points A, B and X are given by and i.Find and show that AXB is a straight line. The position vector of point C is given by . ii.Show that CX is perpendicular to AX. iii.Find the area of triangle ABC. Solution i. First, we […]
Question The first, second and third terms of a geometric progression are , and respectively. (i) Show that satisfies the equation 7k2 − 48k + 36 = 0. (i) Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of k. (ii) One of […]
Question A function is defined for and is such that . i. Find the set of values of for which f is decreasing. ii. It is now given that . Find . Solution i. We are given derivative of the function as; We are also given that it is a decreasing […]
Question i. Show that the equation can be expressed as Where ii. Hence solve the equation for . Solution i. We are given the equation; We have the trigonometric identity; From this we can substitute in above equation; Let ; ii. We are required to solve the equation for […]
Question A line has equation and a curve has equation , where k is a constant. i.Find the set of values of for which the line and curve meet at two distinct points. i.For each of two particular values of , the line is a tangent to the curve. Show that these two tangents meet […]
Question The dimensions of a cuboid are x cm, 2x cm and 4x cm, as shown in the diagram. i.Show that the surface area S cm2 and the volume V cm3 are connected by the relation ii.When the volume of the cuboid is 1000 cm3 the surface area is increasing at 2 cm2 s−1. Find […]
Question The diagram shows a semicircle ACB with centre O and radius . Arc OC is part of a circle with centre A. (i)Express angle CAO in radians in terms of . (ii)Find the area of the shaded region in terms of , and , simplifying your answer. Solution (i) We are required to […]
Question The equation of a curve is . The curve has no stationary points in the interval . Find the least possible value of and the greatest possible value of . Solution We are given; We are given that curve has no stationary point. A stationary point on the curve is the point where gradient […]
Question The function g is defined by for . By first completing the square, find an expression for and state the domain of . Solution We are given that; We use method of “completing square” to obtain the desired form. We complete the square for the terms which involve . We have the algebraic formula; […]
Question (i) Expand in ascending powers of y as far as the term in y2. (ii) In the expansion of the coefficient of x2 is 48. Find the value of the positive constant . Solution i. We are required to expand . Expression for the Binomial expansion of is: In the given […]