Question In the diagram, OAB is a sector of a circle with centre O and radius r. The point C on OB is such that angle ACO is a right angle. Angle AOB is radians and is such that AC divides the sector into two regions of equal area. i. Show that It is given […]
Question Points A(2,9) and B(3,0) lie on the curve y=9+6x−3×2, as shown in the diagram. The tangent at A intersects the x-axis at C. Showing all necessary working, i. find the equation of the tangent AC and hence find the x-coordinate of C, ii. find the area of the shaded region ABC. Solution i. To find the equation of […]
Question The function is defined by for x > −1. i. Find . ii. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function is defined by for x < −1 iii. Find the coordinates of the stationary point on the curve . Solution i. We are given that; We can […]
Question The point A has coordinates and the point B has coordinates , where p is a constant. i. For the case where the distance AB is 13 units, find the possible values of p. ii. For the case in which the line with equation 2x+3y=9 is perpendicular to AB, find the value of p. Solution i. […]
Question The diagram shows the graph of , where is defined by for 0 < x ≤ 2. i. Find an expression for and state the domain of . ii. The function is defined by for x ≥ 1. Find an expression for , giving your answer in the form ax + b, where a and b are constants to be […]
Question Relative to an origin O, the position vectors of the points A, B and C are given by and i. Show that angle ABC is . ii. Find the area of triangle ABC, giving your answer correct to 1 decimal place. Solution i. We recognize that angle ABC is between and . Therefore, we need […]
Question i. Express the equation in the form and solve the equation for . ii. Solve the equation for . Solution i. We are given; We know that; Therefore; Now we solve this equation for . Using calculator; We utilize the periodic/symmetry property of […]
Question A curve is such that and the point (4, 7) lies on the curve. Find the equation of the curve. Solution We are required to find the equation of the curve whose derivative is given as below. We can find equation of the curve from its derivative through integration; Rule for integration of is: If […]
Question Express in the form , where a, b and c are constants. Solution We have the expression; We use method of “completing square” to obtain the desired form. We take out factor ‘2’ from the terms which involve ; Next we complete the square for the terms which involve . We have the algebraic formula; For the given case we […]
Question i. Write down the first 4 terms, in ascending powers of , of the expansion of . ii. The coefficient of in the expansion of is −200. Find the possible values of the constant . Solution i. Expression for the Binomial expansion of is: First we rewrite the expression in the standard form; In the given case: […]
Question a) The first term of an arithmetic progression is −2222 and the common difference is 17. Find the value of the first positive term. b) The first term of a geometric progression is and the second term is , where . Find the set of values of for which the progression is convergent. Solution a) From […]