Question A function is defined by for . i. State the range of . ii. State the exact value of . iii. Sketch the graph of . iv. Obtain an expression, in terms of , for Solution i. We have the function; for This can be written as; We know that range […]
Question The equation of a curve is . i. Find an expression for and determine, with a reason, whether the curve has any stationary points. ii. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line is rotated through 360o about the x-axis. iii. Find the set of values of for which […]
Question The equation of a curve is . i. Show that the equation of the normal to the curve at the point (3, 6) is . ii. Given that the normal meets the coordinate axes at points A and B, find the coordinates of the mid-point of AB. iii. Find the coordinates of the point at which the normal meets […]
Question The diagram shows two circles, C1 and C2, touching at the point T. Circle C1 has centre P and radius 8 cm; circle C2 has centre Q and radius 2 cm. Points R and S lie on C1 and C2 respectively, and RS is a tangent to both circles. i. Show that RS=8 cm. ii. Find angle RPQ in radians […]
Question The diagram shows a metal plate consisting of a rectangle with sides cm and cm and a quarter-circle of radius cm. The perimeter of the plate is 60 cm. i. Express in terms of . ii. Show that the area of the plate, cm2, is given by . Given that can vary, iii. find the value of at […]
Question The diagram shows a pyramid OABC with a horizontal base OAB where OA = 6 cm, OB = 8 cm and angle . The point C is vertically above O and OC = 10 cm. Unit vectors , and are parallel to , and as shown. Use a scalar product to find angle ACB. Solution i. We […]
Question i. Prove the identity ii. Hence solve the equation for . . Solution i. We have the equation; We have the relation , therefore, We have the trigonometric identity; We can write it as; Therefore; Using the formula; ii. To solve the equation for , as demonstrated in (i), we can […]
Question Find Solution Rule for integration of is: Rule for integration of is: Rule for integration of is:
Question Functions and are defined for by Express in the form , where a, b and c are constants. Solution i. We have the functions; For ; We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘4’ from the terms which involve ; Next we complete the square for the terms […]
Question In the expansion of , where is constant, the coefficient of is . Find the coefficient of . Solution Expression for the Binomial expansion of is: In the given case: Hence; We know that coefficient of is . i.e. The coefficient of is ;
Question a) The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75. Find the first term and the common difference. b) The first term of a geometric progression is 16 and the fourth term is . Find the sum to infinity of the progression. Solution a) From […]