Power of Maths Paper 1 OL

Power of Maths: Paper 1 – Answers

(g) x 2 – 10x + 2 (h) 8x ___

( j) 3x 2 − 2x + 1 __

(c) −3x 2 + 4x ___

3 − 2 __

5 (i) 15x 2 − 6x ___

3 3. (a) –x (b) 4 + 2x ___

5 + 4 __

3

4. (a) 2x + 1 (b) 16x – 2 = 2(8x – 1) (c) 6x2 – 2x – 1 (d) 3t 2 – 1 (e) 18v – 12 = 6(3v – 2) (f ) 1 5. (a) 1 (b) 19 (c) 2 (d) 3x2 – 8x + 5; 6x – 8 (e) 51; –38 (f ) 60; – 46 6. (a) –2 (b) m = –1; x + y + 3 = 0 (c) m = 10; 10x – y + 2 = 0 7. 3

5 − 1 (d) – 4(1 + 3x 2) (e) 1 − 2x ______

3 − 3x 2 5

0·6 N/m per sec 8. (a) 1 (b) ± 2 (c) –1, 2 9. (a) 9∙2 °C (b) 28∙1 °C (c) Midnight (d) –1∙67°/hr (e) Midday (Noon) 10. 36π m3/m 11. (a) –20 (b) –10 (c) a = –9, c = 14

_ 5 N/m per sec =

Chapter 21 Exercise 3 1. (a) 3 (b) 1 (c) 10 (d) 6 (e) − 1

Exercise 6

1. (a) L Min (2, –2) (i) 0∙6, 3∙4 (ii) 0·6 < x < 3·4, x ∈ R (iii) 0 ≤ x ≤ 2, x ∈ R (b) x = 2 (c) 2x – y – 7 = 0 2. (a) L Min

( 1 __

2 , − 25 ___

4 ) (b) (i) –2 ≤ x ≤ 3, x ∈ R

(ii) No points of intersection of y = –8 and the curve (iii) No, based on answer to (ii) 3. b = –2, c = –3

4. (a) f ′(x) = –5 – 4x; L Max

( − 5 __

4 , 49 ___

8

) (b) –3, 1 _

2

(d) –2·2, 0·7 5. (a) (0, 0) L Max; (4, –32) L Min (b) (0, 0) (6, 0) (d) 0 < x < 4, x ∈ R 6. (a) 3x2 – 6x; (0, 0) L Max; (2, – 4) L Min (c) (i) x = –0·7, 1, 2·7 (ii) 0 < x < 2, x ∈ R 7. (a) f (–1) = –3; f (3) = 1 (b) 3x2 – 6x (c) (0, 1) L Max; (2, –3) L Min (e) (i) 0∙7 < x < 2∙9, x ∈ R (ii) 0 < x < 2, x ∈ R 8. (a) x –1·5 –1 023 3·5 –9 0 3 –9 0 13·5

y

(b) 15x – y + 4 = 0 (c) 5x – y – 10 = 0 (d) 77x – y + 86 = 0 (e) 12x – y + 20 = 0 3. (a) (–1, 1) (b) (2, – 4) (c) (3, 1) (d) (1, – 6) 4. (a) (1, –1) (b) (–1, – 6) (c) (–1, 0), (2, –9) (d) (–2, –3), (1, 0) 5. (a) (2, 13) (b) (–3, 13),

(2, –82) (c) ( −2, − 9),

( − 2 __

3 , − 239 ____

(e) (x1, y1) = (0, 12), (x2, y2) = (4, – 20) Exercise 4

4. x < 0, x ∈ R 5. x > 4, x ∈ R 6. (a) Increasing (b) Increasing (c) Decreasing; accelerating at t = 1, 2 and decelerating at t = 3 7. (a) Increasing (b) Increasing (c) Decreasing (d) Decreasing 8. (a) Increasing (b) Increasing (c) Decreasing

Exercise 5

1. (a) (2, 1) L Min (b) (3, 22) L Max (c) (–1, 2) L Max; (3, –30) L Min (d) (1, 6) L Min; (2, 7) L Max (e) (–2, 3) L Min; (0, 7) L Max 2. 15

3. (

− b ___

2a , 4ac − b2 _______

4a ) 4. (a) (i) (–1, 0) (ii) (3, 5)

(b) (i) 5 (ii) 0 5. (–1, 0) L Max; (2, –54) L Min (a) A (–2, –22); B (–1, 0); C (0, –14); D (2, –54);

E (3, –32) (b) L Max (c) L Min (d) –1 < x < 2, x ∈ R (e) 0 (f ) –54 6. (a) 2x + p, – 6 (b) 8 – 4x (i) (0, 3) (ii) 11 (iii) x < 1, x ∈ R (c) k = 3; (1, 3) (d) k = 12; (–2, 23) L Max; (2, –9) L Min (e) k = 4; (–2, 32) L Max; (2, 0) L Min (f ) a = 2, b = –24, c = 3 (g) k = –9; (–1, 6) L Max; (3, –26) L Min 7. 93∙75 cm; 10∙38 a.m. 8. (a) (i) 40 m (ii) 400 m (b) (i) 40 (ii) – 40

_ 4 2. (a) x + y – 4 = 0

(b) 6x 2 – 10x – 4 (c)

( − 1 __

3 , 100 ____

(f ) x = –1∙1, 0∙4, 3∙1 9. 1 __

27 ) L Max; (2, –9) L Min 2 × 2x (a) Curve does not

cross the x-axis (b) 2·6 10. (a) (0, 0) L Min (c) – 0·8, 2 27 ) (d) (3, –15)

Exercise 7 1. (a) Increasing at €14 per year (b) Increasing at €4 per year (c) Decreasing at €10 per year 2. (a) Increasing at 2 eagles per year (b) Increasing at 8 eagles per year 3. (a) Increasing at €55 000 per year (b) Increasing at €31 000 per year (c) Increasing at €40 000 per year 4. (a) Increasing at €2400 per day (b) Increasing at €1200 per day 5. (a) 16·33 l (b) –13·06 l/min (c) –13·06 l/min 6. (a) 3 m/s (b) 4 m/s2 (c) 20 s, 860 m 7. (a) 77 + 26t – 3t 2 (b) 26 – 6t (c) 11 s (d) 132 m/s (e) 8 m/s2 8. (a) 160 m/s (b) –10 m/s2 (c) 2100 m

Exercise 8

1. (a) 10 – x (b) 10x – x2 (c) 25 m2 2. (a) (i) 20 – 2x (ii) 4000x – 400x2 (b) 5 cm 3. (a) 64 – x (b) 264 + 62x – x2 (c) 1225 cm2 4. (a) 27 – x (b) 27x2 – x3 (c) 2916 cm3 5. (a) 200 – π h (b) 400h – 2πh2 (c) 100 m

6. (a) − 1 _

(b) dP ___

dx = −x 2 + 44x − 160, d 2P ____

3 x3 + 22x 2 − 160x − 1200 7. (a) –2 + 7t – 3t 2 (b) 7 – 6t (c)

dx 2 = −2x + 44 (c) 40 54 ) L Min;

( 1 __

3 , 523 ____

(2, 12) L Max (d) €12 8. (a) (i) €(150x + 30x2) (ii) €(930x – 100x2) (iii) €(780x – 130x2) (b) 3 9. –36 10. 8·75, 17·5 11. (a) (i) 3t 2 – 12t + 9

351

Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Page 6 | Page 7 | Page 8 | Page 9 | Page 10 | Page 11 | Page 12 | Page 13 | Page 14 | Page 15 | Page 16 | Page 17 | Page 18 | Page 19 | Page 20 | Page 21 | Page 22 | Page 23 | Page 24 | Page 25 | Page 26 | Page 27 | Page 28 | Page 29 | Page 30 | Page 31 | Page 32 | Page 33 | Page 34 | Page 35 | Page 36 | Page 37 | Page 38 | Page 39 | Page 40 | Page 41 | Page 42 | Page 43 | Page 44 | Page 45 | Page 46 | Page 47 | Page 48 | Page 49 | Page 50 | Page 51 | Page 52 | Page 53 | Page 54 | Page 55 | Page 56 | Page 57 | Page 58 | Page 59 | Page 60 | Page 61 | Page 62 | Page 63 | Page 64 | Page 65 | Page 66 | Page 67 | Page 68 | Page 69 | Page 70 | Page 71 | Page 72 | Page 73 | Page 74 | Page 75 | Page 76 | Page 77 | Page 78 | Page 79 | Page 80 | Page 81 | Page 82 | Page 83 | Page 84 | Page 85 | Page 86 | Page 87 | Page 88 | Page 89 | Page 90 | Page 91 | Page 92 | Page 93 | Page 94 | Page 95 | Page 96 | Page 97 | Page 98 | Page 99 | Page 100 | Page 101 | Page 102 | Page 103 | Page 104 | Page 105 | Page 106 | Page 107 | Page 108 | Page 109 | Page 110 | Page 111 | Page 112 | Page 113 | Page 114 | Page 115 | Page 116 | Page 117 | Page 118 | Page 119 | Page 120 | Page 121 | Page 122 | Page 123 | Page 124 | Page 125 | Page 126 | Page 127 | Page 128 | Page 129 | Page 130 | Page 131 | Page 132 | Page 133 | Page 134 | Page 135 | Page 136 | Page 137 | Page 138 | Page 139 | Page 140 | Page 141 | Page 142 | Page 143 | Page 144 | Page 145 | Page 146 | Page 147 | Page 148 | Page 149 | Page 150 | Page 151 | Page 152 | Page 153 | Page 154 | Page 155 | Page 156 | Page 157 | Page 158 | Page 159 | Page 160 | Page 161 | Page 162 | Page 163 | Page 164 | Page 165 | Page 166 | Page 167 | Page 168 | Page 169 | Page 170 | Page 171 | Page 172 | Page 173 | Page 174 | Page 175 | Page 176 | Page 177 | Page 178 | Page 179 | Page 180 | Page 181 | Page 182 | Page 183 | Page 184 | Page 185 | Page 186 | Page 187 | Page 188 | Page 189 | Page 190 | Page 191 | Page 192 | Page 193 | Page 194 | Page 195 | Page 196 | Page 197 | Page 198 | Page 199 | Page 200 | Page 201 | Page 202 | Page 203 | Page 204 | Page 205 | Page 206 | Page 207 | Page 208 | Page 209 | Page 210 | Page 211 | Page 212 | Page 213 | Page 214 | Page 215 | Page 216 | Page 217 | Page 218 | Page 219 | Page 220 | Page 221 | Page 222 | Page 223 | Page 224 | Page 225 | Page 226 | Page 227 | Page 228 | Page 229 | Page 230 | Page 231 | Page 232 | Page 233 | Page 234 | Page 235 | Page 236 | Page 237 | Page 238 | Page 239 | Page 240 | Page 241 | Page 242 | Page 243 | Page 244 | Page 245 | Page 246 | Page 247 | Page 248 | Page 249 | Page 250 | Page 251 | Page 252 | Page 253 | Page 254 | Page 255 | Page 256 | Page 257 | Page 258 | Page 259 | Page 260 | Page 261 | Page 262 | Page 263 | Page 264 | Page 265 | Page 266 | Page 267 | Page 268 | Page 269 | Page 270 | Page 271 | Page 272 | Page 273 | Page 274 | Page 275 | Page 276 | Page 277 | Page 278 | Page 279 | Page 280 | Page 281 | Page 282 | Page 283 | Page 284 | Page 285 | Page 286 | Page 287 | Page 288 | Page 289 | Page 290 | Page 291 | Page 292 | Page 293 | Page 294 | Page 295 | Page 296 | Page 297 | Page 298 | Page 299 | Page 300 | Page 301 | Page 302 | Page 303 | Page 304 | Page 305 | Page 306 | Page 307 | Page 308 | Page 309 | Page 310 | Page 311 | Page 312 | Page 313 | Page 314 | Page 315 | Page 316 | Page 317 | Page 318 | Page 319 | Page 320 | Page 321 | Page 322 | Page 323 | Page 324 | Page 325 | Page 326 | Page 327 | Page 328 | Page 329 | Page 330 | Page 331 | Page 332 | Page 333 | Page 334 | Page 335 | Page 336 | Page 337 | Page 338 | Page 339 | Page 340 | Page 341 | Page 342 | Page 343 | Page 344 | Page 345 | Page 346 | Page 347 | Page 348 | Page 349 | Page 350 | Page 351 | Page 352 | Page 353 | Page 354 | Page 355 | Page 356 | Page 357 | Page 358 | Page 359 | Page 360 | Page 361 | Page 362 | Page 363 | Page 364 | Page 365 | Page 366

orderForm.title