REVISION PAPER 3 1 (a) x > 2/3 1 (b) x < 1 1 (c) x > 1/5 or x < - 1/3 1 (d) Quotient = 3x3 – 9 x2 + 28 x – 91 Remainder = 279 1 (e) a = 6, other quadratic factor is x2 – x + 3. 2 2x 4 7 1 (f) (i) ; (ii) 5 + 2 ½ x – 3 ¾ x2 – 1 x3 2 2 x 1 x 8 1 4 2 1 (g) (i) x 1 x 2 x 1 1 4 8 1 (h) (i) 2 x 1 x 2 ( x 2) 2 2 (a) x = -1, 3 2 (b) x = -3 2 (c) y = 20, -4(reject) 2 (d) x = 6.5, y = 1.5 2 (e) y = 1000 x2.
3 (e) (i) 13 sin(2 x 0.558) ; (ii) x = 1.14, 2.99 rad 4 (a) 6 cot2x 4 (b) cos x e 1+ sinx 1 1 4 (c) x 2 x 4 (d) 2e3x (3x + 1) 2(sin 2 x 1) 4 (e) cos 2 2 x 4 (f) t = 0, dy/dx = ½ ; t = 10, dy/dx = 1 ½ 4 (g) gradient = - 1/ 9
3 2 x 5 e C 2 cos(3 x 2) 5 (b) C 3 1 5 (c) tan(2 x 1) C 2 5 (a)
ln(2 x 1) C 2 sin 2 x 5 (e) x C 2 1 sin 4 x 5 (f) x C 2 8 ln(1 x 2 ) 1 5 (g) ln(2 x 1) C 2 2 x 5 (h) – lncos x + C 1 5 (i) sin 2 x C 2 3 5 (j) ln( x 2 2) C 2 5 (k) xln(2x+1) – x + ln(2x +1) + C 5 (l) x2 ex – 1 – 2x ex – 1 + 2ex – 1 + C 16 5 (m) 105 5 (n) 6 5 (o) 2.62 (2 d.p) 5 (d)
6 (a) (iii) 0.0553 (3 s.f.) 7 (a) θ = 97.1° 7 (b) 7i + 8j + 6k 7 (c) (i) d n = 0. 7 (c) (ii) r = i – j + 2k + λ(i + j – k) 7 (d) (i) r = 7i + j + 7k + λ(i + 4j + k) 7 (d) (ii) OP = 6i – 3j + 6k 7 (d) (iii) 5x – 3y + 7z = 82 dv 400 k h dt dh dh dv 1 (400 100 h ) 0.04 0.01 h dt dv dt 10 2 8 (a) (iii) t = 378.5 s or 379 s
8 (a) (i)
9 (a) (i) 3 + i 9 (a) (ii) i 1 3i 9 (b) 2 9 (c) parallelogram