Perth Academy Mathematics Department Intermediate 2 Unit 3 Revision Pack. Contents:

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1 Perth Academ Mathematics Department Intermediate Unit Revision Pack Contents: Algebraic Operations: Fractions Fractions Formulae Surds Indices Quadratic Functions: Y = a Y = a + b Y = ( + a) + b Turning Points Quadratic Equations: Graphs Factors Formula Trigonometr Answers

2 Algebraic Operations ~ Fractions Q. Epress these fractions in their simplest for a. c. a b c c p q pq a a p p mn mn p. ab bc def e f Q. Simplif b first finding the common factor. a + b + a. c. + + a ab ac p + p + p ab + b b + c ac + + d ad a + a ab b + b b st rs st n n n p. + p +q s c ac + bc + + Q. Simplif the followin b a. b + q. t. w. c c a b a + b b b + b a a p r. u.. + p p + c. c + c c p p p + p + a a a a a a + p p m m + a a s. v.. p. + q q + d d a + c + c c + c b b a + a + b b +

3 Algebraic Operations ~ Fractions Q. Epress each sum as a fraction in its simplest form a. + + c Q. Epress each difference as a fraction in its simplest form a. c. Q. Epress each product as a fraction in its simplest form a. c. Q. Epress as a single fraction a. c.

4 Q. Epress each sum as a fraction in its simplest form a. a a b b p p + + c p m m a a p p a b m n p q c d + q. + r. + s. + t. + a b a b m n p q u. a + a v. + w. b + b. p + p. m + m Q. Epress each difference as a fraction in its simplest form a. a a b b p p c. + p. m m a a p p a b m n p q c d q. r. s. t. a b a b m n p q u. a a v. w. b b. p p. m m Q. Epress each product as a fraction in its simplest form a. a b p q c c c. a a p p m m b c

5 Q. (continued) m m a a p p t s pq ab c q. r. s t pq c a s. m mn n z z ab a cd a u. v. c bc a cd w. st. s t pq a ( + ) z. a p + Q. Epress as a single fraction a. a a ab a p p c. c c t t k m bc c q q z z p p ab b c ac m mn a st t p. n a v v a a pqr pq d de e q. r. s. t. mn m r d f df

6 Algebraic Operations ~ Formulae Q. Change the subject of each formula to. a. = + = c. = + a = b = = = k = a = p + = t = + = = + a = b + = + Q. Make a the subject of each formula. a. b = a d = a c. = a m = a q = a c = a r = s a t = d a z = b a k = h a p = q a g = a Q. Change the subject of the formula to. a. = a + b = m + c c. t = s r p = q + r m = f n a = b + c k = h m d = b + c g = kc h Q. Change the subject of each formula to the letter shown in brackets. a. P = l (l) V = IR (I) c. S = DT (T) A = lb (b) C = πd (d) G = UT (U) v = u + at (t) P = l + b (l) H = + m () Q. Change the subject of each formula to c. a. b = / c = / c c. = / c m = / c k = / c d = / c a = / c + h = / c p = / c + q = / c t = / c + s r = / c q Q. Make the subject of each formula. a. = d = c c. m = s = a = a + + w = k = z b + c a = p = = + z = h = m + k

7 Q. Change the subject of each formula to a. = k = k c. m = k a = s = k b t k c = q = k d p k h = w = r = k ab = k p = k q = + k c = k d = k k g z k Q. Change the subject of each formula to the letter shown in brackets. a. v = u + as (s) v = u + as (u) c. V = πr h (h) V = πr h (r) R = A π (A) L = + a (a) k = ( p + ) (p) z = t () ar = b (b) st = A ( ) (A) R = A ( ) () na = ( n ) (n) t ( n ) d = n d = (n) a ( + b) = + (R) R r r (a)

8 Algebraic Operations ~ Surds Q. Simplif : a. c. p. q. r. s. t. u. v. w.. Q. Epress each of the following in its simplest for a. + c Q. Epress each of the following in its simplest for a. + c Q. Simplif : a. c. a a c c k k a b p q p. k q. r. s. t. u. v. w.. Q. Simplif : a. c. p. q. r. Q. Multipl out the brackets and simplif, where possibl a. ( + ) ( + ) c. ( + ) ( ) ( + )( ) ( + ) ( + ) ( + )( ) ( + ) + ( + )

9 Q. Epress each of the following with a rational denominator and simplif where possibl a. c. p. Q. Epress each of the following in its simplest form with a rational denominator. a. c. Q. Simplif each of the following b multipling the numerator and denominator b the conjugate sur a. c

10 Algebraic Operations ~ Indices Q. Write each of the following in its simplest inde for a. c. c c a a b b p p d d p. q q q. t t r. f f s. k k t. z z u. v. w. a a. b b Q. Write each of the following in its simplest inde for a. c. a a b b p p c c q q p. d d q. a m s r. s. t. a m s d t w u. v. w.. d t w Q. Write each of the following in its simplest inde for a. ( ) ( ) c. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (a ) (m ) (b ) (p ) (k ) p. (z ) Q. Write the following without brackets. a. (b) (a) c. () () (ab) () (wz) (st) (pq ) ( ) (a b ) (a ) ( ) (c ) (ab ) p. (m k) Q. Simplif these epressions. a. a a c. p p b b ( ) (q ) q (c ) c z (z ) k (k + k ) m (m m ) ( + ) a (a a ) ) ( m c c (q ) q p. m c q q. ( ) (a b ) r. (ab) ( p ) s. p p (ab ) t. a b ab

11 Q. Write down the value of a. c. () ½ a k (mn) (ab ) ( ) ( z ) Q. Rewrite the following with positive indices. a. c. a p b q w a p. c q. t r. Q. Rewrite the following with negative indices. a. c. a p q c Q. Simplif the following epressions. a. m m c. p p a a ( ) (c ) (q ) (w ) b b k k d d ( + ) p (p p ) a (a + a ) p. ½ m (m m ) q. v v h h c c r. s. t. v h c Q. Find the value of a. c. p. q. r. Q. Simplif the following epressions, giving our answers with positive indices. a. ( ) ( p ) c. ( a ) ( ) ( q ) ( k ) (g ) ( m ) (c ) ( h ) ( z ) ( b ) d d p. s s q. r. s. t. u. v. w..

12 Q. Write the following in surd for a. c. a b c a c z m k p w p. d Q. Write the following in inde for a. a c. c a b p z m p. z m a c

13 Quadratic Functions ~ = a Q. Write down the equation of the graphs shown below, which have the form = a. (Diagrams are not drawn to scale) a. c. (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,)

14 Quadratic Functions ~ = a + b Q. Write down the equation of the graphs shown below, which have the form = a + a. c. (,) (,.) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,) (,)

15 Quadratic Functions ~ = ( + a) + b Q. Write down the equation of the graphs shown below, which have the form = ( + a) + a. c. (,) (,) (,) d (,) (,) (,) h (,) (,) (,) (,) (,) (,) (,) (,)

16 Quadratic Functions ~ Turning Points Q. For each of the graphs below, write down the turning point i its nature and ii the equation of the ais of smmetr a b c d e f g h Q. For each of the equations below, write down the turning point i its nature and ii the equation of the ais of smmetr a. = ( ) + = ( ) + c. = ( ) + = ( ) = ( ) = ( ) = ( + ) + = ( + ) + = ( + ) + = ( + ) = ( + ½ ) ¾ = ( +.). = ( ) + = ( + ) + = ( + ) p. = ( ) + q. = ( ) r. = ( ) +.

17 Quadratic Equations ~ Graphs Q. Use the sketches below to solve the quadratic equations. a. = + + = c. = + = + = = Q. Cop and complete the tables below. i Make a sketch of the grap ii Write down the roots of the quadratic equation =. a. = c. = + + = + = Q. For each equation, draw a suitable sketch and find the roots. a. = + = c. = + = + + = + = + + = + + = + = + = + = = = + = =

18 Quadratic Equations ~ Factors Q. Solve these quadratic equations, which are alread in factorised for a. ( ) = ( + ) = c. ( ) = b(b ) = a(a + ) = m(m ) = (a )(a ) = ( )( ) = (c )(c ) = (w + )(w + ) = (s + )(s + ) = (z + )(z + ) = ( + )( ) = (t + )(t ) = ( + )( ) = p. (a )(a + ) = q. (p )(p + ) = r. (c )(c + ) = s. (d )(d ) = t. ( + )( + ) = u. (s + )(s ) = Q. Solve these quadratic equations b factorising first. a. + = c c = c. + = p p = z + z = n + n = t + t = = b b = = a + a = + = = b b = m m = p. w w = q. c c = r. = Q. Solve these quadratic equations b factorising first. a. = b = c. = a = z = k = = p = m = t = a = s = w. =. = h. = p. a = q. c = r. = Q. Solve these quadratic equations b factorising first. a. + + = + + = c. a + a + = m + m + = c + c + = z + z + = p p + = b b + = + = w w + = + = k k + = + = m m = t t = p. a + a = q. c c = r. p + p = Q. Solve these quadratic equations b factorising first. a. + + = p + p + = c. t + t + = k + k + = + + = a + a + = w w + = d d + = + = m m + = c c + = + = = q + q = t t = p. m + m = q. v + v = r. s s =

19 Quadratic Equations ~ Formula Q. Solve these equations using the quadratic formula. a. + + = a + a + = c. c + c + = p + p + = + + = d + d + = + = a a + = p p + = b b + = + = + = = a a = p p = p. c + c = q. = r. w + w = Q. Solve these equations using the quadratic formula, giving our answers correct to decimal places. a. + + = b + b + = c. p + p + = c + c + = + + = a + a + = z z + = q q + = w w + = d d + = + = m m + = + = k + k = c + c = p. t + t = q. a + a = r. z + z = Q. Solve these equations using the quadratic formula, giving our answers correct to decimal places. a. + + = b + b + = c. p + p + = c + c + = + + = a + a + = z z + = q q + = w w + = d d + = + = m m + = + = k + k = c + c = p. t + t = q. a + a = r. z + z = Q. Solve these equations using the quadratic formula, giving our answers correct to significant figures. a. + + = c + c + = c. m + m + = + + = p + p + = a + a + = b b + = z z + = q q + = + = c c + = w w + = k + k = d + d = s + s = p. a + a = q. + = r. c + c = s. + + = t. b + b = u. p p = v. m m + = w. + =. c c =

20 Trigonometr ~ Graphs Q. The graphs represent the functions a sin o and a cos o. Write down the equation for eac a. c.. o o o. ¼ o o o ¼... o o o... o o o. ¾. o o o. ¾.

21 Trigonometr ~ Graphs Q. The graphs represent trigonometric functions. Write down the equation for eac a. o o c. o o o o o o o o

22 o o o o p. o o Q. Make sketches of the following functions, <, clearl marking an important points. a. = cos o = sin o c. = tan o = sin o = cos o = sin o = cos o = sin o = cos o = cos o = sin ½ o = cos / o = tan o = sin o = cos o

23 Trigonometr ~ Graphs Q. The graphs represent the functions sin ( ± a) o and cos ( ± a) o. Write down the equation for eac a. c.

24 Q. Write down the equation for each graph shown below. a. o c... Q. Make a neat sketch of these tri functions showing the important values,. a. = sin ( ) o = sin ( + ) o c. = cos ( ) o = cos ( + ) o = tan ( ) o = sin ( ) o = cos ( + ) o = sin ( ) o = tan ( + ) o = cos ( ) o = cos ( ) o = sin ( + ) o

25 Trigonometr ~ Equations Q. Solve the following equations where a. sin o =. cos o =. c. tan o = cos o =. tan o =. sin o =. tan o =. sin o =. cos o =. sin o =. cos o =. tan o =. Q. Solve the following equations where a. sin o =. cos o =. c. tan o =. cos o =. tan o =. sin o =. tan o =. sin o =. cos o =. sin o =. cos o =. tan o =. Q. Solve the following equations where a. sin o = cos o = c. tan o = cos o = tan o = sin o = tan o =. sin o = cos o = sin o = cos o = tan o = Q. Solve the following equations where a. sin o = cos o + = c. tan o = sin o + = cos o = tan o = cos o = sin o = cos o + = tan o = cos o + = sin o + = Q. Solve the following equations where a. cos o + = sin o = c. tan o = + cos o = tan o + = cos o = sin o + = + cos o = sin o. = cos o + = sin o = sin o + =

26 Trigonometr ~ Periodicit & Identities Q. Write down the period of the following graphs a. c. o o o o. o o. o o

27 Q. Write down the period of each of the following trigonometrical functions. a. = sin o = tan o c. = cos o = tan o = cos o = sin o = cos. o = sin. o = tan. o = sin o = tan o = cos o = tan o = cos o = sin o p. = cos o q. = sin o r. = tan o Q. Write down the period of each of the following trigonometrical functions. a. = sin ½ o = tan / o c. = cos ¼ o = tan / o = cos / o = sin / o = cos o = sin o = tan o = sin o = tan o = cos o Q. Simplif a. cos + sin cos c. cos Atan A sin B o o o sin a tan o o cos a sin ( sin ) cos sin cos cos sin tan sin A + cos A cos B sin B + (cos + sin ) sin cos tan a( sin a) Q. Prove that a. cos A + sin A = cos A tan + tan = sin cos c. ( cosb + sin B) + (cos B sin B) = ( + sin )( sin ) = cos sin θ.tanθ = cos θ cosθ

28 ANSWERS Algebraic Operations ~ Fractions Q. a. c. c a a p n df p. q e Q. a. a + b + a + ( + ) c. b + a + b b ( + b) p + q + b s t b c ( + ) r t a + b p n p. a Q. a. b + c. a c + a + p + q a b + d + m + a a p. p + a + q. b c + r. s. t. b + c + u. p c + a b + v. w... p + c + a + b p b a c Algebraic Operations ~ Fractions Q. a. Q. a. c. c.

29 Q. a. Q. a. Q. a. a n + m mn q. b + a ab u. a + a Q. a. a n m mn q. a b ab u. a a Q. a. Q. a. m s. n. s t b a r. + v. b a r. v. p a z t.. aq p c. c. c. q + p pq b + a ab s. w. b + b c. q p pq b a ab s. w. b b c. ab p a b u. c p m b + a + p ab d + c + p. cd n + m q + p t. mn pq + p m +.. p m p m b a p ab d c p. cd n m q p t. mn pq p m.. p m pq m bc p. ( + ) z. b c. s t v. p q. a d c q a r. w. c b c a

30 m k a d s. n d t. f b a p. sv q. am n pq r. c q Algebraic Operations ~ Formulae Q. a. = = + c. = a = + b = = = = k a = p = + t = + = a b = = = Q. a. a = b a = d c. a = m a = q c s r d t a = a = a = a = b z h k q p g a = a = a = a = Q. a. b c t + r p r = = c. = = a m s q m + n a b h k d b = = = = f c m c kc g = h Q. a. P V S A C G l = I = c. T = b = d = U = R D l π T v u P b H m t = l = = a Q. a. c = b c = c. c = c = m c = k c = d c = (a ) c = (h + ) c = (p q) c = ( + ) c = (t s) c = (r + q) Q. a. c a + = = c. = = d m s z b + c = = w a = a = k + = p = = z + m = h k Q. a. k = k = c. k = m k = a b k = c d k = h g t p k = k = s q z k = k = r k = ab w k = p q

31 Q. a. k = k = c + d k = + v u V s = u = v as c. h = a πr V r = A = π r ( ) a = L πh p = k t = b = z ( ar) st R + A A = = ( A t n = t d R r r = r + r n = a = ( na) d + b Algebraic Operations ~ Surds Q. a. c. p. q. r. s. t. u. v. w.. Q. a. c. Q. a. c. Q. a. c. a c k ab pq p. k q. r. s. t. u. v. w.. Q. a. ½ c. ¼ / ½ ½ / ½ / / / p. / q. / r. / Q. a. + c Q. a. Q. a. c. Q. a. ( + ) p. c. c. ( + ) ( + )

32 ( ) ( + ( ) ( ) ( + ) ( + ) ( ) Algebraic Operations ~ Indices Q. a. c. c a b p d p. q q. t r. f s. k t. z u. v. w. a. b Q. a. c. a b p c q p. d q. r. a s. m t. s u. d v. w. t. w Q. a. c. a m b p k p. z Q. a. b a c. a b w z s t p q a b a c a b p. m k Q. a. a c. p b q c z k + k m m + a a m c p. q q. r. a b s. p t. a b Q. a. c. Q. a. c. p b q w a p. c q. Q. a. c. a p q c Q. a. m c. p a c q w b k d + p p a + a p. m m q. v r. h s. c t. Q. a. c. ½ ½ ¼ / p. / q. / r. / Q. a. p c. a q k g m c h / z b t r. d p. s q. r. s. t. / u. v. w. /. / / Q. a. c. a b

33 c p a c w p z d m k Q. a. a c. z c p m a z a b m p. c Quadratic Functions ~ = a Q. a. = = c. = = / = = = = = = ½ = ¼ = / = = = ¾ Quadratic Functions ~ = a + b Q. a. = + = c. = +. = + = + = = + = + = + = = = + = + = Quadratic Functions ~ = ( + a) + b Q. a. = ( ) + = ( ) + c. = ( ) = ( ) = = ( + ) + = ( + ) = ( + ) = ( ) + = ( ) = ( ) + = ( + ) + = ( ) = + Quadratic Functions ~ Turning Points Q. a. (, ) min, = (, ) min, = c. (, ) min, = (, ) min, = (, ) ma, = (, ) ma, = (, ) ma, = (, ) ma, = Q. a. (, ) min, = (, ) min, = c. (, ) min, = (, ) min, = (, ) min, = (, ) min, = (, ) min, = (, ) min, = (, ) min, = (, ) min, = (½, ¾) min, = ½ (.,.) min, =. (, ) ma, = (, ) ma, = (, ) ma, = p. (, ) min, = q. (, ) min, = r. (,.) min, = Quadratic Equations ~ Graphs Q. a.,, c.,,,, Q. a.,, c.,, Q. a.,, c.,,,,,,,,,,,,,

34 Quadratic Equations ~ Factors Q. a.,, c.,,,,,,,,,,,,, p., q., r., s. ½, t., / u. /, / Q. a.,, c.,,,,,,,, /, /, /,,, p., / q., / r., / Q. a.,, c.,,,,,,,,,,.,..,..,. p., q., r., Q. a.,, c.,,,,,,,,,,,,, p., q., r., Q. a. ½,, ½ c., /, / /,, / ½, /, /, /,, ½ /, /,, ¾ ½, ½ p. /, q. ¼, / r. /, Quadratic Equations ~ Formula Q. a., /, ½ c. /, ½,, ½, / ½,, ½ /, /, ½, / ¾, /,,. ½, ½ p., ½ q., / r., / Q. a..,..,. c..,..,..,..,..,..,..,..,..,..,..,..,..,. p..,. q..,. r..,. Q. a..,.,. c..,..,..,..,..,..,..,..,..,..,..,..,..,. p..,. q..,. r..,. Q. a..,..,. c..,..,..,..,..,..,..,..,..,..,..,..,..,. p..,. q..,. r..,. s..,.t..,. u..,. v..,. w..,...,. Trigonometr ~ Graphs Q. a. = sin o = sin o c. =. sin o = cos o = cos o = ¼ cos o =. sin o =. cos o =. sin o = cos o = sin o = cos o =. sin o = ¾ sin o =. cos o

35 Trigonometr ~ Graphs Q. a. = sin o = sin o c. = cos o = cos o = sin o = cos o = sin o = cos o = sin ½ o = cos ½ o = cos / o = cos / o = tan o = tan o = tan ½ o p. = tan o Q. a. c. o o o o o o o o o o o o o o o

36 Trigonometr ~ Graphs Q. a. = sin ( + ) o = sin ( ) o c. = cos ( ) o = cos ( + ) o = sin ( + ) o = cos ( ) o = cos ( + ) o = sin ( ) o = sin ( ) o = cos ( ) o Q. a. = tan ( ) o = sin ( ) o c. = cos ( + ) o = sin ( + ) o = sin ( + ) o =. cos ( ) o = cos ( ) o = sin ( + ) o Q. a. c. d o o o o

37 Trigonometr ~ Equations Q. a. o, o o, o c. o, o o, o o, o o, o o, o o, o o, o o, o o, o o, o Q. a.. o,. o. o,. o c. o, o o, o o, o o, o o, o o, o o, o o, o o, o o, o Q. a. o, o. o,. o c. o, o o, o o, o. o,. o o, o. o,. o. o,. o o, o. o,. o o, o Q. a. o, o c. o, o o, o o, o. o,. o. o,. o. o,. o. o,. o. o,. o. o,. o. o,. o Q. a.. o,. o. o,. o c.. o,. o o, o. o,. o. o,. o o, o o, o. o,. o. o,. o. o,. o o, o Trigonometr ~ Periodicit & Identities Q. a. c. o o o o o o o o Q. a. o o c. o o o o o o o o o o o o o p. o q. o r. o Q. a. o o c. o o o o o o o o o o Q. a. sin c. sin A cos cos tan a sin sin cos sin A cos B sin a Q. Proofs