MAA HL P3 practice questions2

ǣ ǡ ͵ ̷ ʹͲͲͲǤ Ǥ ͷǡʹͲʹͲ Calculus Statistics & prob Geo & trig Functions Number & algebra Question number Î Maximum mark Î 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 31 30 29 24 31 29 29 31 27 26 23 31 26 28 23 26 30 24 23 32 24 24 29 23 27 Sequences & series x澔澔 x 澔 x 澔 x 澔澔澔澔澔澔澔澔澔 x 澔澔澔 x x 澔 x Exponents & logs 澔 x 澔澔 x 澔澔澔 x 澔澔澔澔澔澔 x x x 澔 x 澔澔 x 澔 x Binomial expansion 澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x澔澔澔澔 Counting 澔澔澔澔澔澔澔 x 澔澔澔澔 x 澔 x 澔澔澔澔澔澔澔澔澔澔 Partial fractions x澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔 Complex numbers 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x x 澔澔澔 Proof by induction 澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Proof by contradiction 澔澔澔澔 x 澔澔澔澔 x 澔澔澔澔澔澔 x 澔澔澔澔澔澔澔 x Systems of equations 澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Domain & range 澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔 x 澔澔澔 x 澔澔澔澔澔 Inverse function 澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Curve sketching 澔 x 澔澔澔澔澔澔澔澔澔澔澔 x 澔 x 澔澔澔 x 澔澔 x 澔澔 Quadratic function 澔澔澔澔澔澔 x 澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔 x 澔澔 Modulus function x Transformations 澔澔澔澔澔澔澔澔澔澔 x 澔澔 x 澔澔澔澔澔澔澔澔澔澔澔 Factor & remainder thrms 澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔 x 澔澔澔澔澔澔澔 Sum & product of roots 澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔 x 澔澔澔 Trigonometry x x x 澔 x澔澔澔澔澔澔 x 澔 x x 澔澔 x x x x x 澔 x 澔 Arcs & sectors 澔澔 x x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Triangle trigonometry x 澔 x澔澔澔澔澔澔澔澔澔澔澔 3D solids x Vectors 澔澔澔澔澔澔澔澔澔澔澔 x 澔澔 x 澔澔澔澔澔澔澔澔 x 澔 Descriptive statistics x 澔澔澔澔澔 x澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔 Correlation & regression 澔澔澔澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔 Probability x 澔澔澔澔澔 x 澔澔澔澔 x 澔澔澔 x 澔 x 澔澔澔澔澔澔 Discrete random variables x 澔澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Binomial distribution 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔澔 Cont. random variables x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Normal distribution x 澔澔澔 x澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Limits x x x 澔 x 澔澔澔澔澔澔澔澔澔澔澔 x 澔 x 澔澔 x 澔 x Differentiation x x 澔 x x x x x x 澔澔 x 澔 x 澔 x x 澔澔 x 澔澔 x 澔 x Integration x x 澔澔 x x 澔 x 澔澔 x 澔 x 澔澔澔 x 澔澔澔 x 澔澔澔 Continuous & differentiable x 澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x L'Hopital's rule 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 x 澔澔 x 澔澔 Differential equations 澔 x 澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔澔 Maclaurin series 澔澔澔澔 x 澔澔澔澔澔澔澔澔澔澔澔澔 x 澔澔澔澔澔澔澔 ͳǤ ሾǣ͵ͳሿ Ǥ ͳǡ ǡǤǤ Ǥ ܺ ൌ ᇱ Ǥ ሺሻሺሻ ܺǤ ሺሻ ݂ܺሺ‫ݔ‬ሻ ൌ ʹ‫ݔ‬Ǥ ሺሻ ሺܺሻǤ ሾ͹ሿ ͳ ̶ ̶ܵ ൌ Ǥ ܺ ሺሻሺሻ ܵǤ ሺሻ ܵ݃ሺ‫ݏ‬ሻ ൌ ʹ‫ି ݏ‬ଷ Ǥ ሺሻ ሺܵሻǤ ǡ ǡǤ ǯ Ǥ ܰ ൌ Ǥ ሾͳͲሿ ͳ ͳ ሺ ሻሺሻሺܰ ൌ ͷሻ ൌ ൬ ൰ ൬ ൰Ǥ ͸ ͷ ሺሻܰͳǤ ሺሻሺܰሻ ൌ ͳ ͳ ͳ ͳ ൅ ൅ ൅ ൅ ‫ڮ‬Ǥ ʹ ͵ Ͷ ͷ ͳ ͳ ͳ ሺሻ ሺܰሻͳ ൬ ൰ ൅ ʹ ൬ ൰ ൅ Ͷ ൬ ൰ ൅ ‫ ڮ‬ǡ ǡ ሺܰሻ ൌ λǤ Ͷ ͺ ʹ ሺሻ ሺܰሻ ൌ λ ǫ ǫ ሾͳͳሿ ሺሻ Ǧ ܵ Ǥ ǫ ǡ ǢǡǤ ሾ͵ሿ ʹǤ ሾǣ͵Ͳሿ ݈ Ǥ y l x ሺሻ ǡ ‫ݔ‬ǡ ‫݈ݕ‬Ǥ ሾʹሿ ሺሻǡ ିଵ ǡ ǡ ǡ Ǥ Ǥ ሺ ሻ ͳିଵ Ǥ ሾ͸ሿ ሺሻ Ǥ ሺሻ ሺ ሻሺሻ ǫ Ǥ ǡ Ǥ Ǥ ሾ͹ሿ y x ሺሻሺሻǡ ǡ ‫ ݔ‬௡ ൅ ‫ ݕ‬௡ ൌ ݈ ௡ ͳ ݊ ൌ ǡ ͳǡ ʹǡ ͵Ǥ ʹ ሺሻ‫ ݔ‬௡ ൅ ‫ ݕ‬௡ ൌ ݈ ௡ ǡ݊ ് Ͳǡሺܽǡ ܾሻ ‫ ݕ‬ൌ െܽ௡ିଵ ܾଵି௡ ‫ ݔ‬൅ ܾଵି௡ ݈௡ Ǥ ሺሻ ǡ ‫ݔ‬ǡ ‫݈ݕ‬Ǥ ሾͳͷሿ ͵Ǥ ሾǣʹͻሿ Ǥ ሺሻ Ǧ Ǥ ǡ ǡ Ǥ ሺሻሺሻሺ݇ ൅ ͳሻଷ Ǥ ሺሻ ሺሻሺሻ݇ ൌ ͳ݇ ൌ ݊ǡ ሾͷሿ ௡ ͳ ͳ ͳ ෍ ݇ ଶ ൌ ݊ଷ ൅ ݊ଶ ൅ ݊Ǥሾ͸ሿ ͵ ʹ ͸ ௞ୀଵ ͳ݊Ǥ ǣȀȀ ǤȀ ȀȀ͵ͻǤ ሺ ሻሺሻ ݇ǡ Ǥ ͳ ͳ ͳ ሺሻ ݊ଷ ൅ ݊ଶ ൅ ݊Ǥሾ͸ሿ ʹ ͵ ͸ ‫ܣ‬ǡ ‫ܤ‬ǡ ‫ܥ‬ǡ ‫ ܦ‬Ǥ ሺሻ‫ܦܥܤܣ‬ ξʹ ሺ݊ െ ͳሻଷ Ǥሾ͹ሿ ͳʹ ሺሻ ǡ ǡ ‫ ܦܥܤܣ‬ Ǥ ሾͷሿ ͶǤ ሾǣʹͶሿ Ǥ ‫ ݎ‬ൌ ‫ ܮ‬ൌ ܴ ൌ ߠ ൌ ʹ ‫ܣ‬ଵ ൌ ‫ܣ‬ଶ ൌ ‫ܣ‬ଷ ൌ ‫ܣ‬ଶ Ͷߠ ሺሻሺሻ ൌ ଶ ߠǤ ‫ܣ‬ଷ ߨ ሺሻ ‫ܣ‬ଶ ߨ ߨ ߠ ൌ ߠ ൌ Ǥሾ͸ሿ ‫ܣ‬ଷ ͸ Ͷ ͷߨ ͳ ߨ ߨ ‫ܣ‬ଶ ߠ ൌ ቀ ൅ ቁ ൌ Ǥ ‫ܣ‬ଷ ʹͶ ʹ ͸ Ͷ ሺሻሺሻ ǡ ǡ ͷߨ ‫ܣ‬ଶ ߠ ൌ Ǥ ‫ܣ‬ଷ ʹͶ ሺሻᇱ Ǥ ሺ ሻሺሻ ሾͶሿ ʹ ߠ ߠ ଶ ‫ܣ‬ଵ ൌ൬ ൰ Ǥ ‫ܣ‬ଷ ͳ ൅ ߠ ሺሻ ‫ܣ‬ଵ Ǥሾ͸ሿ ‫ܣ‬ଷ ‫ܣ‬ଵ Ǥ ‫ܣ‬ଷ ሺሻᇱ Ǥ ሾͺሿ ͷǤ ሾǣ͵ͳሿ ௫ Ǥ ሺሻ ௫ Ǥ ሾͳሿ ሺሻ ൐ ʹǤͷǤ ሾʹሿ ሺ ሻሺሻ ʹ௡ିଵ ൏ ݊Ǩ ݊ ‫ א‬Ժǡ݊ ൒ ͵Ǥ ሺሻ ǡ ǡ ൏ ͵Ǥ ሾͻሿ ଵ ௫ିఓ మ ͳ ቁ ି ቀ ሺሻ ଶ ఙ Ǥ ߪξʹߨ ܽ ܼ̱ሺͲǡ ͳሻǡሺͲ ൏ ܼ ൏ ܽሻ ൎ ܽǤሾͶሿ ξʹߨ ஶ ሺሻ݂ሺ‫ݔ‬ሻ ൌ ෍ ௞ୀ଴ ‫ݔ‬௞ ൐ Ͳ‫ א ݔ‬ԹǤሾͳሿ ݇Ǩ ௡ ሺሻǡ ǡ ݃ሺ‫ݔ‬ሻ ൌ ෍ ௞ୀ଴ ‫ݔ‬௞ ൐ Ͳ‫ א ݔ‬Թǡ݊ ‫ א‬Ժା Ǥሾʹሿ ݇Ǩ ଶ௡ ݄ሺ‫ݔ‬ሻ ൌ ෍ ௞ୀ଴ ‫ݔ‬௞ ݊ ‫ א‬Ժା Ǥ ݇Ǩ ሺሻሺሻ ݄ሺ‫ݔ‬ሻ ݄ሺ‫ݔ‬ሻ ǡ ǡ ݄ሺ‫ݔ‬ሻǤ ௫՜ஶ ௫՜ିஶ ሺሻǡ ǡ݄ሺ‫ݔ‬ሻ ൌ ሺሻ݄ሺ‫ݔ‬ሻ ൐ Ͳ‫ א ݔ‬ԹǤ ‫ ݔ‬ଶ௡ Ǥሾ͸ሿ ሺʹ݊ሻǨ ሾ͸ሿ ͸Ǥ ሾǣʹͻሿ Ǥ M D H 2S-D-E E S ǡ ͵ǡͶǡͶǡ Ǥ ‫ܪ‬ǡ ‫ܯ‬ǡ ܵǡ ǡ Ǥ ܲ ൌ ‫ܵܯܪ‬ ‫ ݐ‬ൌ ሺሻܲߙߚǤ ሺሻ ሺሻ‫ݐ‬ǣ ሺሻ ሺሻ ሺሻ ሺ ሻܲ‫ݐ‬Ǥ ሺሻሺሻ Ǥ ሺሻ ܲ ‫ݐ‬Ǥ ሾͶሿ ሾ͸ሿ ሾͷሿ ሾͺሿ ሺሻሺሻ‫ ܫ‬ൌ ‫ ܵܯܪ‬Ǥ ‫ ܫ‬Ǥ ሺሻܲ Ǥ ሾ͸ሿ ͹Ǥ ሾǣʹͻሿ ǡ ݈ ǡ ǡ ǣ Ǥ ǡ ǡ‫ ݕ‬ൌ ܽ‫ ݔ‬ଷ ൅ ܾ‫ ݔ‬ଶ ൅ ܿ‫ ݔ‬൅ ݀Ǥ ‫ ݕ‬ǯ Ǥ ‫ ݔ‬ǯ Ǥ Ǥǯ ͸ Ǥ Ǥǯ ͳͲͲ ିଵǤ Ǥ ǡ െͲǤͳ ିଶ ͲǤͳ ିଶǤ ሺሻ ݀ܿǤ ሾ͵ሿ ሺሻ ‫ ݔ‬ൌ ݈ǡǡ ݈ܽ ଷ ൅ ܾ݈ ଶ ൅ ͸ͲͲͲ ൌ Ͳ ͵݈ܽ ൅ ʹܾ ൌ ͲǤ ሾʹሿ ሺ ሻሺሻ ǯ Ǥ ሺሻ ǯ Ǥ ሺሻǯ Ǥ ሺሻǯ ܽǡ ܾ‫ݔ‬Ǥ ሾͳͳሿ ሺሻܾ ൌ െͷ ൈ ͳͲି଺ Ǥ ሺሻ ݈Ǥ ሾͶሿ ሺሻ ǡ ݈Ǥ ሾͻሿ ͺǤ ሾǣ͵ͳሿ Ǥ ͸Ǧ͸Ǥ ܺ ൌ Ǥ ሺሻሺሻ ሺܺ ൌ ͳሻǤ ͷ ଷ ͳ ሺሻሺܺ ൌ Ͷሻ ൌ ൬ ൰ ൬ ൰Ǥ ͸ ͸ ሺሻ ሺܺ ൏ ͻሻǤ ሾͷሿ ሾͷሿ ͳ ͷ ‫ ݌‬ൌ ǡ‫ ݍ‬ൌ Ǥ ͸ ͸ ሺሻሺሻሺܺሻ‫ݍ݌‬Ǥ ሺሻ ‫ ݍ‬ൌ ͳ െ ‫݌‬ǡǡ ሺܺሻ ൌ ͸Ǥ ሺ ሻሺሻሺܺሻ ൌ ʹ‫݌‬ሺͳ ൅ ʹ‫ ݌‬൅ ͵‫݌‬ଶ ൅ ‫ ڮ‬ሻ ൅ ሺͳ ൅ ‫ ݌‬൅ ‫݌‬ଶ ൅ ‫݌‬ଷ ൅ ‫ ڮ‬ሻ െ ͵͸Ǥ ሺሻ ሺ‫ ݐ‬൅ ‫ ݐ‬ଶ ൅ ‫ ݐ‬ଷ ൅ ‫ ڮ‬ሻǡǡ ሺܺሻǤሾ͹ሿ ‫ݐ‬ Ǥ ܻ ൌ Ǥ ሺሻሺሻ ሺܻ ൌ ʹሻǤ ሺሻ ሺܻ ൌ ͷሻǤ ሺሻ ሺܻሻǤ Ǥ ܼ ൌ Ǥ ሺሻሺሻ ሺܼ ൌ ͸ሻǤ ሺሻ ሺܼ ൌ ͹ሻǤ ሺሻ ሺܼሻǤ ሾͶሿ ሾͳͲሿ ͻǤ ሾǣʹ͹ሿ Ǥ ሺͲǡ Ͳሻሺ‫݌‬ǡ ‫ݍ‬ሻሺ‫݌‬ǡ െ‫ݍ‬ሻ‫ ݔ‬ଶ െ ‫ ݕ‬ଶ ൌ ͳǤ ‫ ݍܣ‬൒ Ͳǡ െ ‫ ݍܣ‬൏ ͲǤ ǣ ‫ ܣ‬ൌ ‫ ܣ ݌‬ൌ ‫ݍ‬ ௤ ሺሻ‫ ܣ‬ൌ ʹ න ൭ඥ‫ ݕ‬ଶ ൅ ͳ െ ଴ ඥ‫ݍ‬ଶ ൅ ͳ ‫ݕ‬൱ ‫ݕ‬Ǥሾ͵ሿ ‫ݍ‬ ሺሻሺሻ݂ሺ‫ݕ‬ሻ ൌ ‫ݕ‬ඥ‫ ݕ‬ଶ ൅ ͳ ൅ ቀ‫ ݕ‬൅ ඥ‫ ݕ‬ଶ ൅ ͳቁ ‫ݕ‬Ǥ ሺሻ ǡ ǡ ‫ ܣ‬ൌ ൫‫ ݍ‬൅ ඥ‫ݍ‬ଶ ൅ ͳ൯Ǥ ሾ͹ሿ ሺ ሻ ‫ ܣ‬ൌ ஺ ൅ ି஺ ஺ െ ି஺ ‫ ܣ‬ൌ Ǥሾ͸ሿ ʹ ʹ ሺሻሺሻ ଶ ‫ ݔ‬െ ଶ ‫ ݔ‬ൌ ͳǤ ሺሻ ‫ ݔ‬ൌ ‫ ݔ ݔ‬ൌ ‫ݔ‬Ǥ ‫ݔ‬ ‫ݔ‬ ሺሻ ‫ ݔ‬ൌ ‫ݔ‬ ǡ ‫ ݔ‬Ǥሾ͹ሿ ‫ݔ‬ ‫ݔ‬ ǡିଵ ǡିଵ ሺ ‫ݔ‬ሻ ൌ ‫ݔ‬Ǥ ͳ ሺሻ න ‫ ݔ‬ൌ ିଵ ‫ ݔ‬൅ ܿǤሾͶሿ ξͳ ൅ ‫ ݔ‬ଶ ͳͲǤ ሾǣʹ͸ሿ Ǥ ሺሻ ‫ ݎ‬Ǥ ‫ݎ‬Ǥ ሺሻξʹǤ ሾʹሿ ሾ͹ሿ ξ͵ξͳǤͷǤ ሺ ሻǡ ǡ ξʹ‫ ݔ‬൅ ‫ ݕ‬ൌ ξ͵‫ݖ‬ ‫ ݔ‬ൌ ‫ ݕ‬ൌ ‫ ݖ‬ൌ ͲǤ ሾͺሿ ሺሻ൫ξʹǡ ξ͵൯ ሺܽǡ ܾሻሺܿǡ ݀ሻ ሾ͸ሿ ܽǡ ܾǡ ܿǡ ݀ ‫ א‬ԺǤ ሺሻǡ ݊ǡ ݊ Ǥ ሾ͵ሿ ͳͳǤ ሾǣʹ͵ሿ Ǥ ݂ሺ‫ݔ‬ሻ ൌ ܽ‫ ݔ‬ଷ ൅ ܾ‫ ݔ‬ଶ ൅ ܿ‫ ݔ‬൅ ݀ ൌ Ͳߙǡ ߚǡ ߛǤ ܵ௡ ൌ ߙ ௡ ൅ ߚ ௡ ൅ ߛ ௡ Ǥ ሺሻ ܽ‫ ݔ‬ଷ ൅ ܾ‫ ݔ‬ଶ ൅ ܿ‫ ݔ‬൅ ݀ǡߙǡ ߚǡ ߛǤ ሺሻሺሻܵଵ ܽ ൅ ܾ ൌ ͲǤ ሺሻሺߙ ൅ ߚ ൅ ߛሻଶ ǡ ǡ ܵଶ ܽ ൅ ܵଵ ܾ ൅ ʹܿ ൌ ͲǤ ሺሻ ߙ ௡ିଷ ݂ሺߙሻǡߚ ௡ିଷ ݂ሺߚሻǡߛ ௡ିଷ ݂ሺߛሻǡǡ ܵ௡ ܽ ൅ ܵ௡ିଵ ܾ ൅ ܵ௡ିଶ ܿ ൅ ܵ௡ିଷ ݀ ൌ ͲǤ ‫ ݔ‬ଷ െ ‫ ݔ‬൅ ݀ ൌ Ͳǡ݀ ് Ͳǡߙǡ ߚǡ ߛǤ ሺ ሻ ǡ ݀ǡǣ ሺሻߙ ൅ ߚ ൅ ߛ ሺሻߙ ଶ ൅ ߚ ଶ ൅ ߛ ଶ ሺሻߙ ଷ ൅ ߚ ଷ ൅ ߛ ଷ ͳ ͳ ͳ ሺሻ ൅ ൅ ߙ ߚ ߛ ሾ͵ሿ ሾ͹ሿ ሺሻ ͳ ͳ ͳ ൅ ൅ ሾͻሿ ߙߚ ߚߛ ߙߛ ሺሻ ǡ ݀ǡ ߙ ൅ ߚǡߚ ൅ ߛǡߙ ൅ ߛǤ ሾͶሿ ͳʹǤ ሾǣ͵ͳሿ Ǥ ‫ݒ‬଴ ߠǤ ݃Ǥ Ǥ ሺሻሺሻ ᇱ ‫ݔ‬ǡ‫ݒ‬଴ ǡߠ‫ݐ‬Ǥ ݃ ሺሻ ᇱ ‫ ݕ‬ൌ െ ‫ ݐ‬ଶ ൅ ‫ݒ‬଴ ሺ ߠሻ‫ݐ‬Ǥ ʹ ሺሻ ᇱ ‫ ݕ‬ൌ െ ݃ ሺͳ ൅ ଶ ߠሻ‫ ݔ‬ଶ ൅ ሺ ߠሻ‫ݔ‬Ǥሾ͹ሿ ʹ‫ݒ‬଴ ଶ ሺ ሻǡ Ǥ ሺ‫ݔ‬ǡ ‫ݕ‬ሻǡ ߠ Ǥ ሺሻሺሻ ᇱ ߠǤ ሺሻ ‫ ݕ‬ൌ ݃ ଶ ‫ݒ‬଴ ଶ െ ‫ ݔ‬Ǥሾͷሿ ʹ݃ ʹ‫ݒ‬଴ ଶ ǲǳ Ǥ ሺ ሻ Ǥ ሺሻǡ ǡ‫ ݔ‬ଶ ൌ െͶ‫ ݕ‬ଶ ൅ ሾ͸ሿ ʹ‫ݒ‬଴ ଶ ‫ݕ‬Ǥሾ͹ሿ ݃ ሺሻ Ǥ Ǥ ሾ͸ሿ ͳ͵Ǥ ሾǣʹ͸ሿ Ǥ ͳͲ ሺ‫ܪ‬ሻ ሺܶሻǤ ሺሻሺሻ Ǥ ሺሻ ͷܶͷ‫ܪ ܪ‬Ǥ ሺሻ ͸ܶͶ‫ܪ ܪ‬Ǥ ሾͳͲሿ ሺሻ ‫ܪ‬Ǥ ݊Ǥ‫ܥ‬௡ ൌ ‫ܪ‬Ǥ ሺሻ ‫ܥ‬ଷ Ǥ ሾͳሿ ‫ܪ‬ǡ ‫ܪ‬ Ǥ ሺ ሻ‫ܥ‬௡ ൌ ʹ‫ܥ‬௡ିଵ ൅ ʹ௡ିସ െ ‫ܥ‬௡ିସ ݊ ൒ ͶǤ ሾͶሿ ሺሻ ‫ܥ‬ଵ଴ Ǥ ሾ͵ሿ ሺሻ ‫Ͳͳܪ‬Ǥ ሾʹሿ ሺሻ ‫Ͳͳܶܪ‬Ǥ ሾ͸ሿ ͳͶǤ ሾǣʹͺሿ ሺͳ ൅ ‫ ݔ‬ଶ ሻିଵǤ ሺሻǡ ‫ ݕ‬ൌ ሺͳ ൅ ‫ ݔ‬ଶ ሻିଵ ǡ‫ ݕ‬ൌ ሺͳ ൅ ‫ ݔ‬ଶ ሻିଶ ‫ ݕ‬ൌ ሺͳ ൅ ‫ ݔ‬ଶ ሻିଷ Ͳ ൏ ‫ ݔ‬൏ ͳǤ ሾ͵ሿ ଵ ሺሻ න ሺͳ ൅ ‫ ݔ‬ଶ ሻିଵ ‫ݔ‬Ǥሾ͵ሿ ଴ ଵ ሺ ሻ‫ ݔ‬ൌ ߠ ǡ ǡ න ሺͳ ൅ ‫ ݔ‬ଶ ሻିଶ ‫ݔ‬Ǥሾͺሿ ଴ ଵ ‫ܫ‬௡ ൌ න ሺͳ ൅ ‫ ݔ‬ଶ ሻି௡ ‫ݔ‬Ǥ ଴ ሺሻሺሻሺͳ ൅ ‫ ݔ‬ଶ ሻି௡ ሺͳ ൅ ‫ ݔ‬ଶ ሻି௡ ሺͳሻǡǡ ‫ܫ‬௡ାଵ ൌ ൬ͳ െ ͳ ʹି௡ିଵ ൰ ‫ܫ‬௡ ൅ ݊ ൒ ͳǤ ݊ ʹ݊ ଵ ሺሻ න ሺͳ ൅ ‫ ݔ‬ଶ ሻିଷ ‫ݔ‬ǤሾͳͲሿ ଴ ଵ ሺሻ න ሺ‫ ݔ‬ଶ െ ʹ‫ ݔ‬൅ ʹሻିଷ ‫ݔ‬ǤሾͶሿ ଴ ͳͷǤ ሾǣʹ͵ሿ Ǥ ሺ‫ݔ‬ଵ ǡ ‫ݕ‬ଵ ሻǡሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻǡሺ‫ݔ‬ଷ ǡ ‫ݕ‬ଷ ሻǡǥǡሺ‫ݔ‬௡ ǡ ‫ݕ‬௡ ሻǡ σሺ‫ ݔ‬െ ‫ݔ‬ҧ ሻሺ‫ ݕ‬െ ‫ݕ‬തሻ Ǥ ‫ݎ‬ൌ ඥሺσሺ‫ ݔ‬െ ‫ݔ‬ҧ ሻଶ ሻሺσሺ‫ ݕ‬െ ‫ݕ‬തሻଶ ሻ ‫ݔ‬ଵ െ ‫ݔ‬ҧ ‫ݕ‬ଵ െ ‫ݕ‬ത ‫ݕ‬ଶ െ ‫ݕ‬ത ‫ݔ‬ଶ െ ‫ݔ‬ҧ ሺሻሺሻ‫ ܝݎ‬ൌ ൮ ‫ ڭ‬൲ ‫ ܞ‬ൌ ൮ ‫ ڭ‬൲Ǥ ‫ݔ‬௡ െ ‫ݔ‬ҧ ‫ݕ‬௡ െ ‫ݕ‬ത ሺሻ െ ͳ ൑ ‫ ݎ‬൑ ͳǤ ሺሻሺሻ ሺͳǡ ͳሻǡሺʹǡ ʹሻǡሺ͵ǡ ͵ሻǡǥǡሺ݊ǡ ݊ሻǡ‫ ݎ‬ൌ ͳǤ ሺሻ ሺͳǡ ݊ሻǡ ሺʹǡ ݊ െ ͳሻǡ ሺ͵ǡ ݊ െ ʹሻǡ ǥ ǡ ሺ݊ǡ ͳሻǡ ‫ ݎ‬ൌ െͳǤ ᇱ ǡ ‫ݔ‬ǡ ǡ ǡͳǡ ʹǡ ͵ǡ Ͷǡ ͷǡ ͸ǡ ͹Ǥ ǡ ‫ݕ‬ǡ ǡ ǡͳǡ ʹǡ ͵ǡ Ͷǡ ͷǡ ͸ǡ ͹Ǥ ሺ ሻ Ǥ ሾͶሿ ሾͶሿ ሾʹሿ ሺሻሺሻ ටቀ෍ሺ‫ ݔ‬െ ‫ݔ‬ҧ ሻଶ ቁ ቀ෍ሺ‫ ݕ‬െ ‫ݕ‬തሻଶ ቁǤ ሺሻͷ͹‫ݎ‬Ǥ ሺሻሺሻሺܽǡ ܾሻሺܿǡ ݀ሻ ሺܽǡ ݀ሻሺܿǡ ܾሻǡ ሾ͸ሿ ෍ሺ‫ ݔ‬െ ‫ݔ‬ҧ ሻሺ‫ ݕ‬െ ‫ݕ‬തሻ ሺܽ െ ܿሻሺܾ െ ݀ሻǤ ሺሻ ‫ ݎ‬ൌ ͲǤ ሾ͹ሿ ͳ͸Ǥ ሾǣʹ͸ሿ ‫ ݕ‬ൌ ‫ݔ‬ Ǥ ‫ݔ‬ ሺሻ ‫ ݕ‬ൌ ‫ݔ‬ ǡ ‫ݔ‬ ሺሻǦ ሺሻ ሺሻሺሻ ሺሻ ሺሻ ሺሻ ‫ ݕ‬ൌ ሾͻሿ ‫ݔ‬ Ǥሾ͵ሿ ‫ݔ‬ ሺ ሻ ‫ ݕ‬ൌ ‫ݔ‬ ǡߨ ୣ ൏ గ Ǥሾ͵ሿ ‫ݔ‬ ሺሻ ܽ௕ ൌ ܾ௔ ܽǡ ܾ ‫ א‬Թܽ ൏ ܾǡ ܾܽǤ ሺሻ ௔ ܾ ൌ ௕ ܽ ܽǡ ܾ ‫ א‬Թܽ ൏ ܾǡܾܽǤ ሾͷሿ ሾ͸ሿ ͳ͹Ǥ ሾǣ͵Ͳሿ Ǥ ͳͲǤ‫ ܪ‬ൌ Ǥ ሺሻሺሻ ሺ‫ ܪ‬ൌ ͳͲሻǡ ሺሻǤ ሺሻ ሺ‫ ܪ‬ൌ Ͳሻǡ ሺሻǤ ሺሻ Ͳ଴ Ǥ ሾ͸ሿ ݊ ሺ݊ǡ ‫ݎ‬ሻ ൌ ቀ ቁ ‫ ݔ‬௥ ሺͳ െ ‫ݔ‬ሻ௡ି௥ ǡ‫Ͳ ് ݎ‬ǡ ݊Ǥ ‫ݎ‬ ሺሻሺሻ ‫ ݑ‬൏ ‫ ݑ‬െ ͳ‫ ݑ‬൐ Ͳǡ‫ͳ ് ݑ‬Ǥ ͳ ௫ିଵ ‫ ݔ‬൐ ͳǤሾͺሿ ሺሻ‫ ݕ‬ൌ ൬ͳ െ ൰ ‫ݔ‬ ͳ ሺ ሻሺ݊ǡ ݊ െ ͳሻ୫ୟ୶ ൌ ሺ݊ǡ ͳሻ୫ୟ୶ ൌ Ǥሾͺሿ ʹ ሺሻሺሻሺ݊ǡ ‫ݎ‬ሻ ሺ݊ െ ͳǡ ‫ ݎ‬െ ͳሻሺ݊ െ ͳǡ ‫ݎ‬ሻǤ ሺሻሺ݊ǡ ‫ݎ‬ሻ ൑ ሺ݊ െ ͳǡ ‫ ݎ‬െ ͳሻሺ݊ǡ ‫ݎ‬ሻ ൑ ሺ݊ െ ͳǡ ‫ݎ‬ሻǤ ͳ ሺሻǡ ͳ ൏ ‫ ݎ‬൏ ݊ െ ͳǡሺ݊ǡ ‫ݎ‬ሻ ൑ Ǥሾ͹ሿ ʹ ሺሻ ሺ݊ǡ ‫ݎ‬ሻ୫ୟ୶Ǥ ሾͳሿ ͳͺǤ ሾǣʹͶሿ ஶ ෍ ௡ୀଵ ͳ Ǥ ݊௣ ஶ ሺሻ ǡ ǡ ෍ ௡ୀଵ ͳ ൌ λǤሾʹሿ ݊ ሺሻሺሻ ͳ െ ‫ݔ‬ ǡ ǡ ‫ݔ‬ ‫ݔ‬ଶ ‫ݔ‬ସ ‫଺ݔ‬ ൅ െ ൅ ‫ ڮ‬ൌ ͲǤ ͵Ǩ ͷǨ ͹Ǩ ሺሻͳ െ ‫ݔ‬ଶ ‫ݔ‬ଶ ‫ݔ‬ଶ ‫ݔ‬ଶ ‫ݔ‬ସ ‫଺ݔ‬ ൅ െ ൅ ‫ ڮ‬ൌ ቆͳ െ ቇ ቆͳ െ ቇ ቆͳ െ ቇ ǥ ሺߨሻଶ ሺʹߨሻଶ ሺ͵ߨሻଶ ͵Ǩ ͷǨ ͹Ǩ ஶ ଶ ሺሻ ‫ ݔ‬ሺሻሺሻǡ ǡ ෍ ௡ୀଵ ஶ ሺሻ ෍ ௡ୀଶ ݊ଶ ͳ Ǥ ݊ଶ ͳ Ǥሾͳ͵ሿ െͳ ஶ ሺ ሻሺሻ ǡ ǡ ͳ ͳ ͳ ൏ ෍ ଷ ൏ ǤሾͶሿ ݊ ͹ʹ ͻͺ ௡ୀ଻ ஶ ሺሻ ෍ ௡ୀଵ ͳ Ǥሾͷሿ ݊ଷ ͳͻǤ ሾǣʹ͵ሿ ሺ ሻǡ ݊ ݉ ǡ ݊ ൐ ݉ǡ Ǥ ሺሻʹ Ǥ ǡ ǡ ͳ Ǥ ሾʹሿ ሺሻሺሻ Ǥǡ ǡ Ǥ ሺሻ Ǥ Ǥ ሾ͹ሿ ሺ ሻ Ǥ Ǥ ሾ͵ሿ ሺሻሺሻ݊ǡ݊ ‫ א‬Ժା Ǥ ௡ ሺሻ ͹ǡ͹͹ǡ͹͹͹ǡ ǥ ǡ͹ ෍ ͳͲ௞ ǡ ǡ ǡ ௞ୀ଴ ା ݊ ‫ א‬Ժ ǡ Ͳ͹݊Ǥ ሾͷሿ ሺሻ ‫ݔ‬ଵ ǡ ‫ݔ‬ଶ ǡ ‫ݔ‬ଷ ǡ ‫ݔ‬ସ ǡ‫ ݔ‬ൌ ߠ ǡǡ ‫ݔ‬௜ െ ‫ݔ‬௝ Ͳ ൑ ൑ ξ͵Ǥሾ͸ሿ ͳ ൅ ‫ݔ‬௜ ‫ݔ‬௝ ʹͲǤ ሾǣ͵ʹሿ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ǡ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫ݔ‬ Ǥ ሺሻ ǡ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ǡ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ Ǥ ‫ݔ‬ ሺሻ݂ሺ‫ݔ‬ሻ ൌ ௫ െ ͳ ሺሻ݂ሺ‫ݔ‬ሻ ൌ െ ሺ‫ ݔ‬൅ ͳሻ ሺሻ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬െ ͳǡ െ ߨ ߨ ൏ ‫ ݔ‬൏ ሾͻሿ ʹ ʹ ሺሻ ݂ሺ‫ݔ‬ሻ ቆ ቇ ǡ‫Ͳ ് ݔ‬Ǥሾͳሿ ‫ݔ ݔ‬ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ‫ ݕͲ ് ݔ‬ൌ ݉‫ ݔ‬൅ ܿǤ ܿ ݂ሺ‫ݔ‬ሻ ሺ ሻሺሻ ቆ ቇ ൌ െ ଶ ‫Ͳ ് ݔ‬Ǥ ‫ݔ‬ ‫ݔ ݔ‬ ሺሻ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫Ͳ ് ݔ‬Ǥሾͷሿ ‫ݔ‬ ሺሻ‫ ݕ‬ൌ ‫ݔ‬ Ͳ ൏ ‫ ݔ‬൏ ͳǤሾ͵ሿ ‫ݔ‬ ଵ ଵ ܽ௫ ൅ ܾ௫ ௫ ܽ௫ ൅ ܾ௫ ௫ ሺሻሺሻ ൬ ൰ ௚ሺ௫ሻ ǡ ǡ ‫ ݕ‬ൌ ൬ ൰ ǡͲ ൏ ܽ ൏ ܾǡ ʹ ʹ ‫Ͳ ് ݔ‬Ǥ ଵ ܽ௫ ൅ ܾ௫ ௫ ൰ ǡͲ ൏ ܽ ൏ ܾǤሾͳʹሿ ሺሻ ‫ ݕ‬ൌ ൬ ʹ ሺሻǡ ǡ ‫ ݕ‬ൌ ‫݂ݔ‬ሺ‫ݔ‬ሻ ‫ݔ‬Ǥ ሾʹሿ ʹͳǤ ሾǣʹͶሿ Ǥ ሺሻǡ ‫ݖ‬ǡᇱ ߨ ǤሾͶሿ ʹ ܲǡ ܳǡ ܴǡ ܵ Ǥᇱܴܲ ٣ ܴܳܵܲ ൌ ܳܵǤ Q R P S ǡ Ǥ B 2(a+b) C 2a 2(a+b+c) A 0 = 2(a+b+c+d) D ሺሻሺሻ‫ ܤ‬ൌ ʹܽ ൅ ܾ ൅ ܾǤ ሺሻ‫ܣ‬ǡ ‫ܥ‬ǡ ‫ܽܦ‬ǡ ܾǡ ܿǡ ݀Ǥ ሺ ሻܴܲ ٣ ܴܳܵܲ ൌ ܳܵǤ ሺሻܴܲܳܵ Ǥ ሾ͸ሿ ሾ͹ሿ ሾ͹ሿ ʹʹǤ ሾǣʹͶሿ Ǥ ǡ Ǥ ሺሻሺሻ ሺ‫ ܣ‬൅ ‫ܤ‬ሻ ൅ ሺ‫ ܣ‬െ ‫ܤ‬ሻǤ ߨ ሺሻʹ ǡ ǡ ͹ Ͳߨ ʹߨ Ͷߨ ͸ߨ ൅ ൅ ൅ Ǥ ͹ ͹ ͹ ͹ ͵ߨ ͷߨ ͹ߨ ߨ ൅ ൅ Ǥሾͺሿ ሺሻ ൅ ͹ ͹ ͹ ͹ ሺሻሺሻ ͹ߠ ൌ ͲǤ ሺሻ ͹ߠ ൌ ͸Ͷ ଺ ߠ െ ͺͲ ସ ߠ ൅ ʹͶ ଶ ߠ െ ͳ ߠ ് ͲǤ ߠ ሺሻ͸Ͷ‫ ଺ ݔ‬െ ͺͲ‫ ݔ‬ସ ൅ ʹͶ‫ ݔ‬ଶ െ ͳ ൌ ͲǤ ʹߨ ͵ߨ ͹ߨ ߨ ሺሻ ቀ ቁ ൬ ൰ ൬ ൰ ǥ ൬ ൰Ǥ ሾͳʹሿ ͹ ͹ ͹ ͹ ሺ ሻሺሻ ቀ ʹߨ ͵ߨ ͳͲͲߨ ߨ ൰ ൬ ൰ ǥ ൬ ൰Ǥ ቁ ൬ ͳͲͲ ͳͲͲ ͳͲͲ ͳͲͲ ሺሻ ʹߨ ͵ߨ ͳͲͲߨ ߨ ൅ ൅ ൅ ‫ ڮ‬൅ ǤሾͶሿ ͳͲͲ ͳͲͲ ͳͲͲ ͳͲͲ ʹ͵Ǥ ሾǣʹͻሿ ǲ ǳ Ǥ ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଵ ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଵǤହ ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଶ ǡ Ǥ ǡ Ǥ ሺሻ ǡ ǡ ௥ ൬ మ ՜ଵ ௥భ ‫ݎ‬ଶ ‫ݎ‬ଵ ଵǤହ ൅ ‫ݎ‬ଶ ൰ ൌ ͳሾͷሿ െ ‫ݎ‬ଵଵǤହ ʹ ሺሻ ௥ ൫ඥ‫ݎ‬ଶ െ ඥ‫ݎ‬ଵ ൯ ൌ Ǥሾ͸ሿ మ ՜ଵ ͵ ௥భ ଶ ʹ ሺ ሻǡ ‫ݎ‬ǡ‫ݎ‬௡ ൎ ൬ ሺ݊ െ ͳሻ ൅ ඥ‫ݎ‬ଵ ൰ Ǥሾ͵ሿ ͵ ʹ ሺሻሺሻܿଵ ൌ ‫ݐ‬ଵଵǤହ ൅ ‫ݐ‬ଵ ଴Ǥହ Ǥ ͵ ሺሻ‫ݐ‬ଵ ൌ ͳͲͲ‫ݎ‬ଵ ൎ ͳͲͲǤʹʹʹ‫ݎ‬ଶ ൎ ͳͳͶǤͲͳͲǤ ሺሻ ሺ ሻǤ ሺሻǡ ‫ݎ‬௡ ݊ ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଵ ǡ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଵǤହ ‫ ݕ‬ൌ ȁ‫ݔ‬ȁଶǡ ‫ݎ‬ଵ Ǥ ሾͳʹሿ ሾ͵ሿ ʹͶǤ ሾǣʹ͵ሿ Ǥ ‫ ݔ‬ଶ ൅ ‫ ݖ‬ଶ ൌ ͳሺͲǡ ͷǡ ͲሻǤ y T x z ݈ଵ Ǥ Ͳ ߠ ሺሻ݈ଵ ‫ ܚ‬ൌ ൭ͷ൱ ൅ ߣ ൭ െͷ ൱Ǥሾʹሿ Ͳ ߠ Ͳ ͳ ᇱ ݈ଶ ǣ‫ ܚ‬ൌ ൭Ͳ൱ ൅ ‫ ݐ‬൭ ͳ ൱Ǥ ͷ െʹ ሺሻሺሻ ݈ଵ Ǥ ሺሻ ݈ଵ ݈ଶ ߠ ൎ ͲǤͷͷ͵ʹͳǤ ሺ ሻ Ǥ ሺሻ Ǥ ሾ͹ሿ ሾͺሿ ሾͶሿ ሾʹሿ ͳͲ ͳ ᇱ ݈ଷ ǣ‫ ܚ‬ൌ ൭ Ͳ ൱ ൅ ‫ ݐ‬൭ͳ൱Ǥ െͳͲ ͳ ሺሻ Ǥ ʹͷǤ ሾǣʹ͹ሿ Ǥ ǡ ͳǡ െʹǡ െͷǡ െͺ ͳǡ െʹǡ Ͷǡ െͺ ‫ݑ‬ଵ ǡ ‫ݑ‬ଶ ‫ݑ‬ସ Ǥ ሺሻ ͳ ͷ ͳ ͳǡ ǡ ǡ െ Ǥሾͳሿ ͺ ͺ Ͷ ݀ǡ ‫ݎ‬ǡ‫ݑ‬ଵ ǡ‫ݑ‬௣ାଵ ‫ݑ‬௤ାଵ Ͳ ൏ ‫ ݌‬൏ ‫ݍ‬Ǥ ሺሻሺሻ ‫ ݕ‬ൌ ‫ݑ‬ଵ ‫ ݎ‬௫ିଵ ‫ ݔ‬൐ Ͳǡ ‫ ݎ‬൐ ͳǡ‫ ݎ‬ൌ ͳͲ ൏ ͳ ൏ ‫ݎ‬Ǥ ሺሻ ‫ݎ‬ǡ Ǥ ሺ ሻ‫݌‬ሺ‫ ݎ‬௤ െ ͳሻ െ ‫ݍ‬ሺ‫ ݎ‬௣ െ ͳሻ ൌ ͲǤ ݂ሺ‫ݎ‬ሻ ൌ ‫݌‬ሺ‫ ݎ‬௤ െ ͳሻ െ ‫ݍ‬ሺ‫ ݎ‬௣ െ ͳሻǡ‫ݍ݌‬Ǥ ሺሻሺሻ݂ሺ‫ݎ‬ሻ െ ͳͲǤ ሺሻ݂ሺ‫ݎ‬ሻ Ǥ ሾͷሿ ሾͶሿ ሾͳͲሿ ‫ݎ‬௤ െ ͳ ‫ݎ‬Ǥ ௤՜ஶ ‫ ݎ‬ଶ െ ͳ ሺሻሺሻ ሺሻ‫ ݎ‬െ ͳǤ ሾ͹ሿ

MAA HL P3 practice questions2

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MAA HL P3 practice questions2

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