AAMSHSZS CALCULUS AND ALGEBRA Question 1 it) The coordinates of points A, B and C are IE41 1), (it, 5) and {10. -'i') respectively. (i) Find the distance of AC. (3 marks] (it) Find the value of at given the midpoint of BC is (T, -I). (3 marks) {iii} Find the gradient of BC. (2 marks) (iv) Find the equation of the straight line that passes through point B and C. (3 marks) (v) Find the equation of the straight line that passes through the midpoint of BC and it is perpendicular to BC. (4 marks) b) Find the Cartesian equation from the following parametric equations. r + 2 x = - , t l I 3 = . 5 marks y I + I ( :l c) The equation ofa circle with oentre C is 4::2 + 4y1 4x + 24y 63 = 0. Find the coordinates of the centre C and the radius of the circle. (5 marks) [Total: 25 marks] Question 1 a) Given two lnCtiDl'le{x:l =10 ix, x e R and g(x) = 126, x e R. (i) Find 1: f a g)(x}. (3 marks) (ii) EvaluateU o g)(8). (2 marks) (iii) Find f" (x). {3 marks) (in) Find (3 o f")(x). (3 marks} unison: CALCULU AND ALGEBRA WENMA} x2+9 b} Given a function f(x} = , x :5 U. .1' (i) Find the image if the object is I5. (2 marks) (ii) Find the object if the image is ill (4 marks] 2 (iii) Use answer obtained in Question 2 b) (ii), determine whether the function J((th = x + 9 x is one to one. State a reason for your answer. (2 marks) 6} Find the domain and range of the following Jnotions. (i) f(x} = 22" {2 marks) {ii} g(x)=Jx+l (2 malts} (iii) M1) = mg; + 2} {2 marks) ['I'otal: 25 marks] Question 3 3} Find the equation of the tangent line to the cuwe y = x2 +211 8 at the point P where x = 2. (5 marks] b) Find coordinates of the stationary points for the curve 3: 2 at3 -%x2 +12x+1, and determine the nature of each stationary points using second derivative method. (1' marks] c} Differentiate the Following functions with respect to x. (i) y = (x1 + sin $5 (3 marks} (ii) y = L'f' {3 marks] (iii) y=x=(tanx) (3 marl-ts) d} The radius of a circle is increasing at a rate of [.5 cmfhour. Find the rate at which the area of the circle is increasing when the radius of the circle is 4 cm. Leave your answer in term of n: . {Area of circle. .4 2 m1] {4 marks) [Tote]: 25 merits] This question paper consists of 4 questions on 5 printed pages. AAMS1623 CALCULUS AND ALGEBRA Question 4 a) Solve the inequality x2 - 9x+1450. (4 marks) b) Solve the following simultaneous equations. x+2y =7, x2 + y2 - 2x+6y-26 =0. (5 marks) c) Given two matrices -(3 6 2) and B - 2 (i) Find 4A + BT (3 marks) (ii) Find (AB) -1. (5 marks) 2 2 0 d) Find the determinant and inverse of matrix A= 1 3 1 using matrix method. Hence, solve 5 1 2 the following system of linear equations. 2x +2y=6, x+3y+z=8, 5x+ y+2z =17. (8 marks) [Total: 25 marks]