Question Functions and are defined by for for i. Solve the equation . ii. Find the range of . iii. Find the set of values of x for which . iv. Find the value of the constant for which the equation has two equal roots. Function is defined by for , and it is given that has an inverse. v. State […]
Question The diagram shows part of the curve and the tangent to the curve at . i. Find expressions for and . ii. Find the equation of the tangent to the curve at P in the form . iii. Find, showing all necessary working, the area of the shaded region. Solution i. First we find the expression […]
Question The equation of a curve is such that . Given that the curve has a minimum point at , find the coordinates of the maximum point. Solution To find the coordinates of a stationary point (in this case a maximum point) we need derivative of equation of the curve. We are given the second derivative of the equation of […]
Question The diagram shows a trapezium ABCD in which BA is parallel to CD. The position vectors of A, B and C relative to an origin O are given by and i. Use a scalar product to show that AB is perpendicular to BC. ii. Given that the length of CD is 12 units, find the position […]
Question i. Prove the identity ii. Solve the equation for . Solution i. We have the trigonometric identity; From this we can substitute in the above equation. Since ; ii. We are required to solve the equation for . From (i), we know that; Therefore; Using calculator; We utilize the periodic/symmetry property of to find […]
Question The diagram shows a sector of a circle with radius cm and centre . The chord AB divides the sector into a triangle AOB and a segment AXB. Angle AOB is radians. i. In the case where the areas of the triangle AOB and the segment AXB are equal, find the value of the constant for which . ii. […]
Question The reflex angle is such that , where . i. Find an expression, in terms of , for a. b. ii. Explain why is negative for . Solution i. A Reflex Angle is one which is more than 180° but less than 360°. a. We are given that; We have the trigonometric identity; From this we can […]
Question Find the coordinates of the point at which the perpendicular bisector of the line joining to meets the x-axis. Solution We are required to find the coordinates of the x-intercept of the perpendicular bisector of the line joining to . The point at which curve (or line) intercepts x-axis, the value of . So we can find the value […]
Question Find the coefficient of in the expansion of . Solution It is evident that to get the terms containing in the product of we need; This will result; Hence we need first to find the terms with and i.e. in the expansion of . First rewrite the given expression in standard form. Expression for the general term in the Binomial expansion of is: In the […]
Question The 1st, 2nd and 3rd terms of a geometric progression are the 1st , 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is r, where r ≠ 1. Find i. the value of r, ii. the 4th term of each […]