Question A function f is defined by for . i. Find the range of f. ii. Sketch the graph of iii. Solve the equation , giving answers in terms of . The function is defined by for , where k is a constant. iv. State the largest value of k for which g has an inverse. v. For this value of k, find […]
Question Relative to an origin O, the position vectors of points A, B and C are given by and i. Use a scalar product to find angle AOB. ii. Find the vector which is in the same direction as and of magnitude 15 units. iii. Find the value of the constant p for which perpendicular to . Solution i. It is evident […]
Question a) A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km. He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km. (i) How far will he travel on May 15th? (ii) On what date will he finish the event? b) […]
Question The equation of a curve is . i. Obtain an expression for ii. Explain why the curve has no stationary points. At the point P on the curve, x = 2. iii. Show that the normal to the curve at P passes through the origin. iv. A point moves along the curve in such a way that its x-coordinate is decreasing […]
Question The diagram shows a metal plate ABCD made from two parts. The part BCD is a semicircle. The part DAB is a segment of a circle with centre O and radius 10 cm. Angle BOD is 1.2 radians. i. Show that the radius of the semicircle is 5.646 cm, correct to 3 decimal places. ii. Find the perimeter of the […]
Question The line , where a and b are positive constants, intersects the x- and y-axes at the points A and B respectively. The mid-point of AB lies on the line and the distance . Find the values of a and b. Solution We need to work through the problem statement very carefully to glean the information scattered […]
Question In the expansion of , the coefficient of x is 7. Find the value of the constant n and hence find the coefficient of . Solution Binomial Theorem states that if is a natural number; First we expand . In the given case: Hence; We will have the given product as; We consider only the terms containing ; We are […]
Question A curve has equation . i. Find the set of values of for which . ii. Find the value of the constant for which the line is a tangent to the curve. Solution i. We are required to find the set of values of x for which . We are given that; Therefore; We solve the following equation to find […]
Question i. Express the equation in the form , where k is a constant. ii. Hence solve the equation for . Solution i. We are given that; We know that , therefore; Comparison with given yields; ii. We are required to solve for . From (i) we know that can be written in the form; Therefore, we solve for . […]
Question A curve is such that . The point (2,5) lies on the curve. Find the equation of the curve. Solution We are given that; We can find equation of the curve from its derivative through integration; Therefore; Rule for integration of is: If a point lies on the curve , we can find out value of . We […]