Question Find the coefficient of in the expansion of . Solution First rewrite the given expression in standard form. Expression for the Binomial expansion of is: In the given case: Hence; Hence the coefficient of is .
Question Solve the equation , for Solution i. We have the equation; Dividing both sides of the equation by ; We have the relation , therefore, To solve this equation, we can substitute . Hence, Since given interval is , for interval can be found as follows; Multiplying both sides of the inequality with 2; Since ; […]
Question The diagram shows the line and part of the curve . i. Show that the equation can be written in the form . ii. . Hence find the area of the shaded region. iii. The shaded region is rotated through about the y-axis. Find the exact value of the volume of revolution obtained. Solution i. We are given […]
Question It is given that a curve has equation , where . i. Find the set of values of for which the gradient of the curve is less than 5. ii. Find the values of at the two stationary points on the curve and determine the nature of each stationary point. Solution i. Gradient (slope) of the curve is […]
Question The coordinates of A are (−3, 2) and the coordinates of C are (5, 6). The mid-point of AC is M and the perpendicular bisector of AC cuts the x-axis at B. i. Find the equation of MB and the coordinates of B. ii. Show that AB is perpendicular to BC. iii. Given that ABCD is a square, […]
Question Two vectors and are such that and , where is a constant. i. Find the value of for which is perpendicular to . ii. For the case where , find the angle between the directions of and . Solution i. If and & , then and are perpendicular. Therefore, we need the scalar/dot product […]
Question The diagram shows the curve and the line , where is a constant. The curve and the line intersect at the points A and B. i. For the case where , find the x-coordinates of A and B. ii. Find the value of for which is a tangent to the curve . Solution i. For the case where […]
Question A watermelon is assumed to be spherical in shape while it is growing. Its mass, Mkg, and radius, r cm, are related by the formula , where is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and […]
Question In the diagram, ABC is an equilateral triangle of side 2 cm. The mid-point of BC is Q. An arc of a circle with centre A touches BC at Q, and meets AB at P and AC at R. Find the total area of the shaded regions, giving your answer in terms of and . Solution It […]
Question The function is defined for the domain , where and are constants. i. Express in the form , where and are constants. ii. State the range of in terms of . iii. State the smallest value of for which is one-one. iv. For the value of found in part (iii), find an expression for and state the domain of , giving your answers […]
Question a) The first two terms of an arithmetic progression are 1 and respectively. Show that the sum of the first ten terms can be expressed in the form , where a and b are constants to be found. b) The first two terms of a geometric progression are 1 and respectively, where . i. Find the […]