Victorian Certificate of Eucation 07 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Written examination Friay 0 November 07 Reaing time: 9.00 am to 9.5 am (5 minutes) Writing time: 9.5 am to 0.5 am ( hour) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answere Number of marks 0 0 40 Stuents are permitte to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners an rulers. Stuents are NOT permitte to bring into the examination room: any technology (calculators or software), notes of any kin, blank sheets of paper an/or correction flui/tape. Materials supplie Question an answer book of pages Formula sheet Working space is provie throughout the book. Instructions Write your stuent number in the space provie above on this page. Unless otherwise inicate, the iagrams in this book are not rawn to scale. All written responses must be in English. At the en of the examination You may keep the formula sheet. Stuents are NOT permitte to bring mobile phones an/or any other unauthorise electronic evices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 07
07 SPECMATH EXAM THIS PAGE IS BLANK
3 07 SPECMATH EXAM Instructions Answer all questions in the spaces provie. Unless otherwise specifie, an exact answer is require to a question. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise inicate, the iagrams in this book are not rawn to scale. Take the acceleration ue to gravity to have magnitue g ms, where g = 9.8 Question (3 marks) Fin the equation of the tangent to the curve given by 3xy + y = x at the point (, ). Question (4 marks) Fin 3 a, expressing your answer in the form log e, where a an b are positive integers. x + x b ( ) TURN OVER
07 SPECMATH EXAM 4 Question 3 (3 marks) Let z 3 + az + 6z + a = 0, z C, where a is a real constant. Given that z = i is a solution to the equation, fin all other solutions. Question 4 (3 marks) The volume of soft rink ispense by a machine into bottles varies normally with a mean of 98 ml an a stanar eviation of 3 ml. The soft rink is sol in packs of four bottles. Fin the approximate probability that the mean volume of soft rink per bottle in a ranomly selecte four-bottle pack is less than 95 ml. Give your answer correct to three ecimal places.
5 07 SPECMATH EXAM Question 5 (4 marks) Relative to a fixe origin, the points B, C an D are efine respectively by the position vectors b= i j+ k, c= i j+ k an = a i j, where a is a real constant. Given that the magnitue of angle BCD is π, fin a. 3 TURN OVER
07 SPECMATH EXAM 6 Question 6 (3 marks) Let f ( x) =. arcsin( x) Fin f (x) an state the largest set of values of x for which f (x) is efine. Question 7 (4 marks) 3 3 π The position vector of a particle moving along a curve at time t is given by r() t = cos () t i + sin (), t j 0 t. 4 Fin the length of the path that the particle travels along the curve from t = 0 to t = π 4.
7 07 SPECMATH EXAM CONTINUES OVER PAGE TURN OVER
07 SPECMATH EXAM 8 Question 8 (4 marks) A slope fiel representing the ifferential equation y = + x y is shown below. y O x a. Sketch the solution curve of the ifferential equation corresponing to the conition y( ) = on the slope fiel above an, hence, estimate the positive value of x when y = 0. Give your answer correct to one ecimal place. marks Question 8 continue
9 07 SPECMATH EXAM b. Solve the ifferential equation y x = with the conition y( ) =. Express your answer + y in the form ay 3 + by + cx + = 0, where a, b, c an are integers. marks TURN OVER
07 SPECMATH EXAM 0 Question 9 (5 marks) A particle of mass kg with initial velocity 3i + j ms experiences a constant force for 0 secons. The particle s velocity at the en of the 0-secon perio is 43i 8j ms. a. Fin the magnitue of the constant force in newtons. marks b. Fin the isplacement of the particle from its initial position after 0 secons. 3 marks
07 SPECMATH EXAM Question 0 (7 marks) a. Show that x x x x arccos = arccos, a a a x where a > 0. mark x b. State the maximal omain an the range of f( x) = arccos. marks c. Fin the volume of the soli of revolution generate when the region boune by the graph of y = f (x), an the lines x = an y = 0, is rotate about the x-axis. 4 marks END OF QUESTION AND ANSWER BOOK
Victorian Certificate of Eucation 07 SPECIALIST MATHEMATICS Written examination FORMULA SHEET Instructions This formula sheet is provie for your reference. A question an answer book is provie with this formula sheet. Stuents are NOT permitte to bring mobile phones an/or any other unauthorise electronic evices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 07
SPECMATH EXAM Specialist Mathematics formulas Mensuration area of a trapezium curve surface area of a cyliner ( a+ b) h π rh volume of a cyliner volume of a cone π r h 3 π r h volume of a pyrami 3 Ah volume of a sphere area of a triangle sine rule 4 3 π r3 bcsin( A) a b c = = sin( A) sin ( B) sin( C) cosine rule c = a + b ab cos (C ) Circular functions cos (x) + sin (x) = + tan (x) = sec (x) cot (x) + = cosec (x) sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x y) = sin (x) cos (y) cos (x) sin (y) cos (x + y) = cos (x) cos (y) sin (x) sin (y) tan( x) + tan ( y) tan( x+ y) = tan( x)tan ( y) cos (x y) = cos (x) cos (y) + sin (x) sin (y) tan( x) tan ( y) tan( x y) = + tan( x)tan ( y) cos (x) = cos (x) sin (x) = cos (x) = sin (x) tan( x) sin (x) = sin (x) cos (x) tan( x) = tan ( x)
3 SPECMATH EXAM Circular functions continue Function sin or arcsin cos or arccos tan or arctan Domain [, ] [, ] R Range π π, [0, ] π π, Algebra (complex numbers) z = x+ iy = r( cos( θ) + isin ( θ) )= r cis( θ ) z = x + y = r π < Arg(z) π z z = r r cis (θ + θ ) z z r = cis θ r θ ( ) z n = r n cis (nθ) (e Moivre s theorem) Probability an statistics for ranom variables X an Y E(aX + b) = ae(x) + b E(aX + by ) = ae(x ) + be(y ) var(ax + b) = a var(x ) for inepenent ranom variables X an Y var(ax + by ) = a var(x ) + b var(y ) approximate confience interval for μ x z s x z s, + n n istribution of sample mean X mean variance E( X )= µ var ( X )= σ n TURN OVER
SPECMATH EXAM 4 Calculus x n ( )= nx n n n+ x= x + c, n n + e ax ae ax ax ( )= e a e ax = + c ( log e() x )= x x = loge x + c ( sin( ax) )= acos( ax) sin( ax) = cos( ax) + c a ( cos( ax) )= asin ( ax) cos( ax) = sin ( ax) + c a ( tan( ax) )= asec ( ax) sin ( ( x) )= x cos ( ( x) )= x ( tan ( x) )= + x prouct rule quotient rule chain rule Euler s metho acceleration sec ( ax) = tan ( ax) + c a x = sin ca 0 a x a +, > a x x = cos + ca, > 0 a a a x x = tan c + a + ( ax b n ) an ( ) ( ax b ) n+ + = + + c, n + ( ax + b) = loge ax + b + c a ( uv)= u v + v u v u u v u v = v y y u = u If y = f( x), x 0 = a an y 0 = b, then x n + = x n + h an y n + = y n + h f (x n ) x v a v v = = = = v t t t arc length + f ( x) or x () t y () t t x x ( ) ( ) + ( ) t Vectors in two an three imensions Mechanics r= xi+ yj+ zk r = x + y + z = r i r y z r = = i+ j+ k t t t t r. r = rr cos( θ ) = xx + yy + zz momentum END OF FORMULA SHEET equation of motion p= mv R = ma