Question The function is defined by for . i. Find the set of values of x for which f(x) ≤ 3. ii. Given that the line y=mx+c is a tangent to the curve y = f(x), show that The function g is defined by for x ≥ k, where k is a constant. iii. Express in the form , where a […]
Question The diagram shows the part of the curve for , and the minimum point M. i. Find expressions for , and . ii. Find the coordinates of M and determine the coordinates and nature of the stationary point on the part of the curve for which . iii. Find the […]
Question A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost. i. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day. a. How many […]
Question Three points have coordinates A(0,7), B(8,3) and C(3k,k). Find the value of the constant k for which i. C lies on the line that passes through A and B, ii. C lies on the perpendicular bisector of AB. Solution i. If point C lies on line AB, then coordinates of point C must satisfy equation of line AB. […]
Question i. Prove the identity . ii. Hence solve, for , the equation Solution i. We are given the identity; We have the trigonometric identity; It can be rearranged as; Therefore; We have ; ii. We are required to solve the equation; We have found in (i) that; Therefore; Using calculator we can find the value of […]
Question The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle at P and angle radians. The line OT meets the circle at Q. i. Express the perimeter of the shaded region PQT in terms of r and . ii. In the case where and , find the area of […]
Question In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle radians and angle radians. Denoting the lengths of BM and MC by x, i. find AM in terms of x, ii. show that .Solution i. We are required to find AM. Consider right angled triangle AMC. […]
Question Find the term independent of in the expansion of i. ii. Solution i. 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 […]
Question Relative to an origin , the position vectors of points A and B are given by and The point C is such that . Find the unit vector in the direction of . Solution A unit vector in the direction of is; Therefore for the given case; Therefore, we need to find . We are given that; A vector in […]
Question A curve is such that . Given that the curve passes through the point , find the equation of the curve. Solution We are given that curve passes through the point and we are required to find the equation of the curve. We can find equation of the curve from its derivative through integration; For the given case; Therefore; Rule […]
Question Functions and are defined by , ,, Solve the equation Solution First we find left hand side of required equation . We are given; It can be written as; Therefore; Now we find right hand side of required equation . We are given; These can be written as; Therefore; Now for substitute ; Finally, equating left […]