Question The function is defined by for . i. State the range of . ii. Find the coordinates of the points at which the curve intersects the coordinate axes. iii. Sketch the graph of . iv. Obtain an expression for , stating both the domain and range of . Solution i. […]
Question Relative to an origin O, the position vectors of points A, B and C are given by and respectively, where k is a constant. i. Find the value of k in the case where angle AOB=90o. ii. Find the possible values of k for which the lengths of AB and OC are equal. The point D is such that is […]
Question a. The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression. b. The first three terms of an arithmetic progression are , and respectively, where x is an acute angle. i. Show that . ii. Find the sum of the […]
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 In the diagram, AOB is a quarter circle with centre O and radius r. The point C lies on the arc AB and the point D lies on OB. The line CD is parallel to AO and angle radians. i. Express the perimeter of the shaded region in terms of r, and . ii. For the case […]
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 A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures x m by y m and is fully enclosed by metal fencing. The farmer uses 480m of fencing. i. Show that the total area of land used for the sheep pens, A m2, is given by […]
Question A curve is such that . i. A point P moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the y-coordinate as P crosses the y-axis. The curve intersects the y-axis where . ii. Find the equation of the curve. Solution i. […]
Question The diagram shows part of the curve . The shaded region is bounded by the curve, the y- axis and the lines y = 1 and y = 2. Showing all necessary working, find the volume, in terms of , when this shaded region is rotated through about the y-axis. Solution Expression for the volume of the solid formed when […]
Question Solve the equation for . Solution We are given; We have the trigonometric identity; From this identity we can have; Substituting in given equation; Let ; Now we have two options. Since; Using calculator we can find the values of . NOT POSSIBLE We utilize the periodic property of to find other solutions (roots) of : Symmetry Property Hence; […]
Question Find the term independent of in the expansion of . . Solution Expression for the general term in the Binomial expansion of is: In the given case: Hence; Since we are looking for the coefficient of the term independent of i.e. , so we can equate Hence, substituting ; Becomes; Hence coefficient of the term independent of i.e. is .