Question A curve is such that . The curve passes through the point . i. Find the equation of the curve. ii. Find . iii. Find the coordinates of the stationary point and determine its nature. Solution i. We are given that curve passes through the point and we are required to find the equation of the curve. We […]
Question A line has equation and a curve has equation . i. For the case where the line is a tangent to the curve, find the value of the constant . ii. For the case where , find the x-coordinates of the points of intersection of the line and the curve. Find […]
Question The diagram shows the function defined for , where i. State the range of . ii. Copy the diagram and on your copy sketch the graph of . iii. Obtain expressions to define the function , giving also the set of values for which each expression is valid. Solution i. To find the range of we can substitute […]
Question i. Prove the identity ii. Hence solve the equation , for . Solution i. We have the equation; We have the identity; Substituting in above equation; We have the identity , therefore, ii. To solve the equation , for , as demonstrated in (i), we can write the given equation as; Now we […]
Question Relative to an origin , the position vectors of the points and are given by and i. Find the value of p for which angle is . ii. For the case where , find the unit vector in the direction of . Solution i. We are given that angle , hence, and are perpendicular and therefore […]
Question The coordinates of points A and B are and respectively, where and are constants. The distance AB is units and the gradient of the line AB is 2. Find the possible values of and of . Solution Expression for slope of a line joining points and ; Therefore, slope of line AB with points and ; Expression to find […]
Question The diagram shows triangle ABC in which AB is perpendicular to BC. The length of AB is 4 cm and angle CAB is radians. The arc DE with centre A and radius 2 cm meets AC at D and AB at E. Find, in terms of, i. the area of the shaded region, ii. the perimeter […]
Question A curve has equation . Find the equation of the tangent to the curve at the point where the line intersects the curve. Solution We are required to find the equation of the tangent to the curve at the point where curve and line intersect. To find the equation of the line either we need coordinates of […]
Question i. Express in the form . ii. Hence, or otherwise, find the set of values of satisfying . Solution i. We have the expression; 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; For the given case we can compare the given […]
Question The diagram shows part of the graph of . State the values of the constants a and b. Solution We know that; We also know that; We can see from the given diagram that at ; Substituting in the above equation yields value of . We can see from the given diagram that at ; Substituting and in the above equation yields […]
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 7.
Question An arithmetic progression has first term and common difference . It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms. i. Find in terms of . ii. Find the 100th term in terms of . Solution From the given information, we can compile following data about […]