Question The diagram shows the graph of for . i. Find the values of , and . ii. Find the smallest value of in the interval for which . Solution i. We are given that; When we observe given graph of this function, it is evident that y-intercept of the graph, i.e. when , is 3. Hence for […]
Question The diagram shows the curve for . The curve has a maximum point at A and a minimum point on the x-axis at B. The normal to the curve at C (2, 2) meets the normal to the curve at B at the point D. i. Find the coordinates of A and B. ii. Find the equation of the normal […]
Question The diagram shows part of the curve . i. Find the gradient of the curve at the point where . ii. Find the volume obtained when the shaded region is rotated through 360o about the x-axis, giving your answer in terms of . Solution i. Gradient (slope) of the curve at the particular point is the derivative of […]
Question The diagram shows points A, B and C lying on the line . The point A lies on the y-axis and AB = BC. The line from D(10, −3) to B is perpendicular to AC. Calculate the coordinates of B and C. Solution i. First we find the coordinates of point B. We recognize that point […]
Question Relative to an origin , the position vectors of points and are given by i. Find the value of and hence state whether angle AOB is acute, obtuse or a right angle. ii. The point is such that . Find the unit vector in the direction of OX. Solution i. We recognize that is angle between and . Hence […]
Question The diagram shows a circle with centre O. The circle is divided into two regions, and , by the radii OA and OB, where angle radians. The perimeter of the region is equal to the length of the major arc AB. i. Show that . ii. Given that the area of region is 30 cm2, find the area of […]
Question Find the set of values of for which the line intersects the curve at two distinct points. Solution If two lines (or a line and a curve) intersect each other at a point then that point lies on both lines i.e. coordinates of that point have same values on both lines (or on the line and the […]
Question Prove the identity Solution We are given the equation; We can rewrite it as; We have the trigonometric identity; We can rewrite it as; Therefore; We have the trigonometric relation; Therefore;
Question The function is defined by for , where is a constant. i. Express in the form , where , and are constants. ii. State the value of for which the graph of has a line of symmetry. iii. When has this value, find the range of . The function g is defined by for . iv. Explain why g […]
Question i. Find the first three terms in the expansion of in ascending powers of . ii. Hence find the value of the constant for which there is no term in in the expansion of . Solution i. Expression for the Binomial expansion of is: In the given case: Hence; ii. To find the value of the constant for which […]
Question a) Find the sum to infinity of the geometric progression with first three terms 0.5, 0.53 and 0.55. b) The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200. Find the sum of all the terms in […]