Question The diagram shows part of the curve . The point P(2,1) lies on the curve and the normal to the curve at P intersects the x-axis at A and the y-axis at B. i. Show that B is the mid-point of AP. The shaded region is bounded by the curve, the y-axis and the line y = 1. […]
Question The curve y=f(x) has a stationary point at (2,10) and it is given that i. Find f(x). ii. Find the coordinates of the other stationary point. iii. Find the nature of each of the stationary points. Solution i. We are given; We are required to find . We can find equation of the curve from […]
Question The function f is defined, for , by , where a and b are constants. i. In the case where a = 6 and b = −8, find the range of f. ii. In the case where a = 5, the roots of the equation f(x)=0 0 are k and −2k, where k is a constant. Find the […]
Question Relative to an origin O, the position vectors of the points A, B and C are given by and i. In the case where ABC is a straight line, find the values of p and q. ii. In the case where angle BAC is 90o, express q in terms of p. iii. In the case […]
Question Points A, B and C have coordinates A(-3,7), B(5,1) and C(-1,k), where k is a constant. i. Given that AB=BC, calculate the possible values of k. The perpendicular bisector of AB intersects the x-axis at D. ii. Calculate the coordinates of D. Solution i. Expression to find distance between two given points and is: We are […]
Question The diagram shows a metal plate OABC, consisting of a right-angled triangle OAB and a sector OBC of a circle with centre O. Angle AOB = 0.6 radians, OA = 6 cm and OA is perpendicular to OC. i. Show that the length of OB is 7.270 cm, correct to 3 decimal places. ii. Find the perimeter of […]
Question i. Prove the identity . ii. Hence solve the equation for . Solution i. We are given the identity; We have the relation; Therefore; We have the trigonometric identity; It can be rearranged as; Hence; We have the algebraic identity; Therefore, we can write; ii. We are required to solve following equation. From (i), we know that left hand […]
Question Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are identical isosceles triangles. Angle ABE = angle BAE = 30o. The length of AD is 40 cm. The tank is fixed in position with the open top ABCD horizontal. Water is poured into the tank at a […]
Question In the expansion of , where k is a non-zero constant, the coefficients of and are equal. Find the value of k. Solution Expression for the general term in the Binomial expansion of is: We are given the expression for expansion as; First we rewrite the expression in the standard form; In the given case: Hence; Since we […]
Question Functions f and g are defined by , , Solve the equation . Solution First we find . We have; We write it as; To find the inverse of a given function we need to write it in terms of rather than in terms of . Interchanging ‘x’ with ‘y’; Now we find . We have; We write it as; Therefore; […]