2021 SASS AMath Prelims P1 w Ans

/ 1 2 3 Show that p > 0 for the inequality px2 + 1 > 2px - p for all real values of x. [3] px2 + l > 2px - p px2 - 2px + ( l + p) > 0 Since the quadratic function is positive, then D < 0 and coeff of x2 > 0 (-2p) 2 -4p(l + p) < 0 andp > 0 4p2 -4p-4p 2 < 0 ---~ (i) Sketch the graph of y = In (x + 2) for x > -2, indicating clearly the point where the graph cuts the axes and the asymptotes. [2] ( -tr (ii) In order to solve the equation (x + 2)= ex- 3 , a graph with a suitable straight line is drawn on the same set of axes as the graph of y = In (x + 2). Find the equation of this straight line. [2] ( ) x-3 x+2 = e In (x + 2) = x- 3 :. The line to be drawn is y = x - 3. x 2 -4x+2 . . . Express 2 m partial fractions . [4] X -3x+2 x2 -4x+2 = (x2 -3x+2)-x x2 -3x+2 x2 -3x+2 = l+ -x (x-2)(x-1) Let - x = ___A_ + _lL (x-2)(x-1) x-2 x-1 4 2021 Prelims Exam - A Maths 4049/1 Multiplying by (x - 2)(x - 1 ), -x=A(x- l)+B(x-2) When x = 2, A = -2 When x = 1, B = 1 2 :. x -4x+2 = l+-1 ___ 2_ x2-3x+2 x-1 x-2 It is given that 3 2x-2 (2 x+3 ) = 92x-l . (i) Find the exact value of (%) x . 3 2x-2 (2 x+3) = 92x-l 32x -x2x x23 - 92x 32 -9 32xx2x 32 92x - 23 x9 (32x2 r =l 92 8 .. r = ½ (shown) [3] (ii) Hence find the value of x. [2] x= log2 l 9 8 lgl =-8 lgl 9 = 1.3.£ (3 s.f.) 5 (i) Given that sin 0- cos 0= ¾, show that sin 0 cos 0 = 3 7 2 . [3] sin 0- cos 0= l 4 (sin 0- cos 0)2 = -2... 16 sin2 0- 2sin 0cos 0+ cos2 0= f6 6 (iii) 1 - 2 sin 0 cos 0 = (6 2 sin 0cos 0= 1: sin0 cos 0 = l2 (shown) Hence find the value of 3(cot 0 + tan 0). [2] 3(cot 0 +tan 8)=3fc?s0+sin0) \sm0 cos0 = 3 (co~ 2 0+sin 2 0) sm0 cos0 = 3 (~) 32 = 96 7 The polynomial P(x) is of degree three and has a constant term 20. The roots of the equation P(x) = 0 are 1, -2 and k. When P(x) is divided by (x + 1 ), the remainder is 24. (i) Show that the value of k is 5. (4] P(x) = a(x- l)(x + 2)(x - k) P(0) = 20 2ak = 20 ak= 10 ... (1) P(-1) = 24 a(-2)(-1 - k) = 24 a+ ak= 12 .. . (2) Sub (1) into (2), a+ 10 = 12 a=2 k= 5 (shown) (ii) Find the remainder when P(x) is divided by (x-2). [11 P(x) = 2 (x- l)(x + 2)(x- 5) Remainder= P(2) = 2 (4) (-3) =-24

7 (a) Express each of 2r - 6x + 9 and -x' - 6x - 8 (ii) Using your value of k from part (i), find the 10 (a) Solve the equation lg(x-4)+2lg3=1+1g(j) [3] in the form a(x + b)2 + c, where a, band care time taken by the mouse to find the cheese on lg(x - 4) + 2 lg 3 = 1 + lg(f) constants. [3] the sixth attempt. [l] 2x2 - 6x + 9 = 2 (x2 - 3x) + 9 T = 1.6 e -0.605n + 3 lg(x-4)+lg3 2 =lgl O+lg(j) ={(x- ½J -¾}9 When n = 6, T= l.6e - 0· 605(6) + 3 Ig[(x-4)x9 ]==lglO(j) = 3.04 minutes = 2 (x- ½} -f +9 (iii) Will the mouse be able to find the cheese :. 9(x-4) = 5x within 1.5 minutes? Explain your answer . [2] 4x= 36 = 2(x- ½f +f It is not impossible that the mouse can x=9 find the cheese within 1.5 minutes since Solve the equation (iogg1 x )( loL 3) = 6¼ [4] -x2 - 6x - 8 =