Ext CSA Trials

Size: px
Start display at page:

Download "Ext CSA Trials"

Transcription

1 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Et CS Trials )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-a sin Find the value of lim. 0 5 )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-b The polnomial P() is given b P() = 3 + a + b for some real numbers a and b. is a zero of P(). When P() is divided b ( + ) the remainder is 5. i. Write down two equations in a and b. Hence find the values of a and b. 3)! Yearl\Yr-3U\cat-3u.05 Qn3) 3U05-c i. Find the eact values of the gradients of the tangents to the curve = e at the points where = 0 and =. Find the acute angle between these tangents correct to the nearest degree. )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-d D B O E C In the diagram, B and C are points on a circle with centre O. D is a point on B such that DOC is a cclic quadrilateral. DO produced meets the circle again at E. i. Cop the diagram Give a reason wh CD = COE. i Show that DOE bisects COB. 5)! Yearl\Yr-3U\cat-3u.05 Qn5) 3U05-a Evaluate log7 correct to two decimal places. 6)! Yearl\Yr-3U\cat-3u.05 Qn6) 3U05-b i. Show that tan. tan tan Evaluate in simplest eact form. π tan tan 6 6 7)! Yearl\Yr-3U\cat-3u.05 Qn7) 3U05-c (, 0) and B(, 6) are two fied points for some real number. The point P(5, ) divides the interval B eternall in the ratio 3 :. i. Show that 3 = 0. Find an values of. 8)! Yearl\Yr-3U\cat-3u.05 Qn8) 3U05-d EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

2 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 = a F 0 M X P(at,at ) P(at, at ) is a point on the parabola = a with focus F. The tangent to the parabola at P cuts the ais at X. M is the midpoint of PF. i. Show that the tangent to the parabola at P has equation t at = 0. Show that MX is parallel to the ais. 9)! Yearl\Yr-3U\cat-3u.05 Qn9) 3U05-3a Consider the function f() = + n. i. Show that the function f() is increasing and the curve = f() is concave down for all values of in the domain of the function. Find the equation of the tangent to the curve = f() at the point on the curve where =. i iv. Find the equation of the inverse function f (). On the same diagram sketch the graph of the curves = f() and = f (). Show clearl the coordinates of an points of intersection of the two curves and an intercepts made on the coordinate aes. 0)! Yearl\Yr-3U\cat-3u.05 Qn0) 3U05-3b d i. Show that ( sin ). d Evaluate d, giving our answer in simplest eact form. 0 )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-a The equation = 0 has eactl one real root. i. Show that < < 3. Starting with an initial approimation =, use one application of Newton s method to find a further approimation for correct to one decimal place. )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-b 3 Use the substitution u = sin sin to evaluate d, giving the answer in simplest eact form. sin 3)! Yearl\Yr-3U\cat-3u.05 Qn3) 3U05-c particle is moving in a horizontal straight line. t time t seconds, the displacement of the particle from a fied point O on the line is metres, its velocit is v ms, and its acceleration a ms is given b a = 8 3. When the particle is m to the right of O, it is observed to be travelling to the right with speed of 6 ms i. Show that v = Find the set of possible values of. )! Yearl\Yr-3U\cat-3u.05 Qn) 3U05-5a EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

3 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 bag contains nine balls labelled,, 3,, 9, but otherwise identical. Three balls are chosen at random from the bag. Find the probabilit that eactl two even numbered balls are chosen: i. if the balls are selected without replacement. if each ball replaced before the net is selected. 5)! Yearl\Yr-3U\cat-3u.05 Qn5) 3U05-7b h closed, right, hollow cone has a height of metre and semi-vertical angle 5. The cone stands with its base on a horizontal surface. Water is poured into the cone through a hole in its ape at a constant rate of 0 m 3 per minute. i. Show that when the depth of water in the cone is h metres (0 < h < ) the volume of water V m 3 3 in the cone is given b V ( h 3h 3h). 3 Find the rate at which the depth of water in the cone is increasing when h = 05. 6)! Yearl\Yr-3U\cat-3u.05 Qn6) U05-5c C B = sin O π The rectangle OBC has vertices O(0, 0), (,0), B(,), and C( 0, ). The curve = sin is shown passing through the points O and B. Show that this curve divides the rectangle OBC into two regions of equal area. 7)! Yearl\Yr-3U\cat-3u.05 Qn7) U05-6a particle is performing Simple Harmonic Motion about a fied point O on a straight line. t time t seconds it has displacement metres from O given b = cost sint. i. Epress in the form Rcos(t + ) for some R > 0 and 0. Find the amplitude and the period of the motion. i Determine whether the particle is initiall moving towards O or awa from O and whether it is initiall speeding up or slowing down. iv. Find the time at which the particle first returns to its starting point. 8)! Yearl\Yr-3U\cat-3u.05 Qn8) U05-6b Use Mathematical Induction to show that for all positive integers n,... n 9)! Yearl\Yr-3U\cat-3u.05 Qn9) U05-7a n n EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

4 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 particle is projected from a point O with velocit V ms at an angle above the horizontal. t time t seconds it has horizontal and vertical displacements metres and metres respectivel from O. The acceleration due to gravit is g ms. i. Write down epressions for and in terms of V, and t. g Show that tan ( tan ). V 0)! Yearl\Yr-3U\cat-3u.05 Qn0) U05-7b particle is projected from O with velocit 60 ms at an angle above the horizontal. T seconds later, another particle is projected from O with a velocit 60 ms at an angle above the horizontal, where <. The two particles collide 0 metres horizontall from O and at a height of 80 metres above O. Taking g = 0 ms and using results from (a): i. Show that tan = and tan =. Find the value of T in simplest eact form. )! Yearl\Yr-3U\cat-3u.05 Qn) U05-7c The real number is a solution of the equation = 0. Use the Binomial Theorem to show that the sum S of the series n (n =,, 3) n n r is given b S Cr. r )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a d Find ( ) tan. d )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-b The polnomial P() is given b P() = 3 + a + for some real number a. The remainder when P() is divided b ( ) is equal to the remainder when P() is divided b ( ). Find the value of a. 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-c i. The line = m makes an angle of 5 with the line =. Show that m = + m. Hence find the equations of the lines = m which make an angle of 5 with line =. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-c D E B C BC is a triangle in which BC = C. D is a point on the minor arc C of the circle passing through, B and C. D is produced to E. i. Cop the diagram. Give reason wh CDE = BC. EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

5 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 i Hence show that DC bisects BDE. 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-a ( 5, 6) and B(, 3) are two points. Find the coordinates of the point P which divides the interval B eternall in the ratio 5:. 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-c The equation = 0 has roots, and. Find the value of. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-b Consider the geometric series sin + sin cos + sin cos + for 0. i. Show that the limiting sum S of the series eists. Show that S = cot. 8)! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-c = P(t, t ) O M P(t, t ) is a point on the parabola =. The tangent to the parabola at P and the line = t intersect at the point M. i. Show that the tangent to the parabola at P has gradient t and equation t t = 0. Find the Cartesian equation of the locus of M as t varies. 9)! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-3a C cm 8 cm B Three circles with centres, B and C touch eternall in pairs with BC = 90, as shown in the diagram. The circles with centres and B have radii 8 cm and cm respectivel. i. If the circle with centre C has radius cm, show that = 0. EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

6 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Hence find the radius of the circle with centre C. 0)! Yearl\Yr-3U\cat-3u.0 Qn0) 3U0-3b u i. Show that. u u Hence find d using the substitution u,u 0. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-3b Use Mathematical Induction to show that 5 n > n + 3 n for all integers n 3. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a 5 Find the term independent of in the binomial epansion of ( ). 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-b fter t ears, t 0, the number N of individuals in a population is given b N = + Be 0 5 t for some constants > 0 and B > 0. The initial population size is 500 individuals and the limiting population size is 00 individuals. i. Find the values of and B. Find the time taken for the population size to fall within 0 of its limiting value, giving the answer correct to the nearest month. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-c Consider the function f() = cos. i. Show that the equation f() = 0 has a root such that 0 < <. Use one application of Newton s Method with an initial approimation of 0 7 to approimate, giving the answer correct to decimal places. 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-5a In a certain street, 0% of the households have at least cars. If households are chosen at random, find the probabilit that i. Eactl 3 of these households have at least cars. t most 3 of these households have at least cars. 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-5b = h EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0 O mould for a container is made b rotating the part of the curve = which lies in the first quadrant through one complete revolution about the ais. fter sealing the base of the container, water is poured through a hole in the top. When the depth of water in the container is h cm, the depth is 0 changing at a rate cms. ( h) i. Show that when the depth is h cm, the surface area S cm of the water is given b S = ( h).

7 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Find the rate at which the surface area of the water is changing when the depth of the water is cm. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-5c Consider the function f() = e. i. Show that the curve = f() is concave up for all values of. Find the coordinates and nature of the stationar point on the curve = f(). i Hence show that e + for all values of. 8)! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-6a Consider the function f() = cos ( ). i. Find the domain of the function. i Sketch the graph of the curve = f() showing clearl the coordinates of the endpoints. The region in the first quadrant bounded b the curve = f() and the coordinate aes is rotated through one complete revolution about the ais. Find the eact value of the volume of the solid of revolution. 9)! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-6b particle moving in a straight line is performing Simple Harmonic Motion. t time t seconds it has displacement metres to the right of a fied point O on the line, where = cos t sin t. i. Show that = + 3cos t. Hence epress the acceleration i ms of the particlein the form n ( b), where the values of the constants n and b are to be determined. Find the set of possible values of and the period of the motion. Find the distance travelled and the time taken (to the nearest tenth of a second) for the particle to first pass through O. 0)! Yearl\Yr-3U\cat-3u.0 Qn0) 3U0-7a particle is moving in a straight line. t time t seconds it has displacement metres to the right of a fied point O on the line and velocit v ms given b v = sin cos. The particle starts metres to the right of O. d i. Show that ln( tan ). d sin cos Hence show that the displacement of the particle is given b = tan (e t ). i Find the limiting position of the particle and sketch the graph of against t. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-7b 0 ms 0 m 0 5ms O O is a vertical building of height 0 metres. particle is projected horizontall from with speed 0 ms. t the same instant, a second particle is projected from O with speed 0 5 ms at an angle above the horizontal. The two particles travel in the same plane of motion. Take g = 0 ms. EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

8 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 i. Write down epressions for the horizontal and vertical displacements relative to O of each particle after time t seconds. Show that if the two particles collide, then the do so after second. i Show that if the two particles collide, when the do so their paths of motion are perpendicular to each other. )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-a n 5( 0 ) 3 Find the value of lim. n n ( 0 ) )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-b (, 5) and B(7, ) are two points. Find the coordinates of the point M(, ) which divides the line B internall in the ratio :. 3)! Yearl\Yr-3U\cat-3u.03 Qn3) 3U03-c Solve the inequalit. )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-d B C N M BC is a triangle inscribed in a circle. M is a point on the tangent to the circle at B and N is a point on C produced so that MN is parallel to B. Cop the diagram. Give a reason wh MBC = BC. Show that MNCB is a cclic quadrilateral. 5)! Yearl\Yr-3U\cat-3u.03 Qn5) 3U03-a 5 d d If ( ), find and. d d 6)! Yearl\Yr-3U\cat-3u.03 Qn6) 3U03-b Find the value of 5 n n C. 7)! Yearl\Yr-3U\cat-3u.03 Qn7) 3U03-c i. Show that (sin cos ) = sin. Hence find the eact value of sin 5 cos 5. 8)! Yearl\Yr-3U\cat-3u.03 Qn8) 3U03-d EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

9 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 = F O T = (t, t ) T(t, t ) is a point on the parabola = with focus F. The tangent to the parabola at T makes an acute angle with the line FT. Show that the tangent to the parabola at T has gradient t. Find tan in simplest form in terms of t. 9)! Yearl\Yr-3U\cat-3u.03 Qn9) 3U03-3a Consider the function f() = e. Show that the function is increasing and its graph is concave down for all values of in its domain. Use one application of Newton s Method with an initial approimation of = 05 to find the value of the intercept on the graph of = f(), giving the answer correct to one decimal place. Sketch the graph of = f() showing clearl the intercepts on the aes and the equations of an asmptotes. On the same diagram sketch the graph of the inverse function = f (). 0)! Yearl\Yr-3U\cat-3u.03 Qn0) 3U03-3b π particle is moving in a straight line. t time t seconds it has displacement metres (where 0 ) from a fied point O on the line, velocit v ms given b v = cos and acceleration a ms. The particle starts at O. Find epressions for a in terms of and for in terms of t. Sketch the graph of against t. Describe the motion of the particle from its initial position to its limiting position. )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-a Consider the function f ( ) cos. Find the domain and the range of the function and sketch the graph of = f(). Use Simpson s Rule with three function values to find an approimation to the area bounded b the curve = f() and the coordinate aes. Use integration to find the eact area bounded b the curve = f() and the coordinate aes. )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-b particle moving in a straight line is performing Simple Harmonic Motion. t time t seconds it has displacement metres from a fied point O on the line, velocit v ms and acceleration ms given b ( ). The particle is at rest at the fied point O. Show that v = + 6. Find the period and amplitude of the motion. Find the distance travelled b the particle in the first minute of its motion, giving the answer correct to the nearest metre. 3)! Yearl\Yr-3U\cat-3u.03 Qn3) 3U03-5a In the epansion of ( + a) 9 the coefficient of 5 is twice the coefficient of 6. Find the value of the constant a. )! Yearl\Yr-3U\cat-3u.03 Qn) 3U03-5b EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

10 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 9 Evaluate d using the substitution u, epressing the answer in the form n for some positive integer n. 5)! Yearl\Yr-3U\cat-3u.03 Qn5) 3U03-5c container with capacit litres is being filled with water. fter t minutes the volume V litres of water in the container is given b V = ( e kt ) for some constant k > 0. dv Show that k( V ). dt In one quarter of the container is filled in the first two minutes find what fraction of the container is filled in the net two minutes. 6)! Yearl\Yr-3U\cat-3u.03 Qn6) 3U03-6a D h m 0 C 0 m 0 B vertical flagpole CD of height h metres stands with its base C on horizontal ground. is a point on the ground due West of C and B is a point on the ground 0 metres due South of. From and B the angles of elevation of the top D of the flagpole are 0 and 0 respectivel. Find the height of the flagpole correct to the nearest metre. 7)! Yearl\Yr-3U\cat-3u.03 Qn7) 3U03-6b die is biased so that in an single throw the probabilit of an odd score is p, where p is a constant such that 0 < p <, p 05. Show that in si throws of the die the probabilit of at most one even score is 6p 5 5p 6. Find the probabilit that in si throws of the die the product of the scores is even. 8)! Yearl\Yr-3U\cat-3u.03 Qn8) 3U03-6c m m 5 cm h cm cm m n open flat topped water through in the shape of a triangular prism is being emptied through a hole in its base at a constant rate of 6 litres per second. Its top measures metres b metres and its triangular end has a vertical height of 5 centimetres. When the water depth is h centimetres the water surface measures centimetres b metres. Show that when the water depth is h centimetres the volume V cm 3 of water in the trough is given V = 600h. Find the rate at which the depth of water is changing when h = 0 cm. 9)! Yearl\Yr-3U\cat-3u.03 Qn9) 3U03-7a EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

11 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 OB = h metres O = h metres B U ms V ms O vertical building stands with its base O on horizontal ground. and B are two points on the building verticall above each other such that is h metres above O and B is h metres above O. particle is projected horizontall with speed U ms from and 0 seconds later a second particle is projected horizontall with speed V ms from B. The two particles hit the ground at the same point and at the same time. Write down epressions for the horizontal and the vertical displacements relative to O of each particle t seconds after the first particle is projected. Find the time of flight of each particle. Show that V = U. 0)! Yearl\Yr-3U\cat-3u.03 Qn0) 3U03-7b i. Use Mathematical Induction to show that ln ( n! ) n for all positive integers n 6. Hence show that n for all positive integers n 6. n! e 03 i Hence show that.... 5!! 3!! 5! 6! 7! 8! 60 e ( e ) )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a d Find n( e ). d )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-b k ( ) Evaluate. k! k 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-c i. cos Show that tan. cos Hence find the value of tan in simplest eact form. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-d M N B D E C EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

12 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 BC is a triangle inscribed in a circle. MN is the tangent to the circle at. D is a point on B and E is a point on C such that DE MN. i. Cop the diagram. Eplain wh MB = CB. i Hence show that BCED is a cclic quadrilateral. 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-a (, 3) and B(, 5) are two points. Find the coordinates of the point P(, ) which divides the interval B internall in the ratio :. 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-b test consists of 7 multiple choice questions all of which are to be attempted. Each question contains alternative answers of which one and onl one is correct. Find the number of was in which the 7 questions can be attempted so that eactl questions are answered correctl. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-c i. Given that = is a zero of the polnomial P() = 3 3 +, epress P() as a product of three linear factors. Hence solve the inequalit )! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-d F O T P(at, at ) l The tangent t at = 0 at the point P(at, at ) on the parabola = a cuts the directri l at T. F is the focus of the parabola. i. Find the coordinates of T. Show that TF is perpendicular to PF. 9)! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-3a N B C BC is a triangle and N is a point on C. BN = CBN = BCN. BC = a, C = b, B = c. BN = CN = d. i. Given that BN CB, show that c = b ac. Hence show that (a + c) = a + b. 0)! Yearl\Yr-3U\cat-3u.0 Qn0) 3U0-3b P() = i. Show that the equation P() = 0 has a root such that 0 < <. Use one application of Newton s method with a starting value of = 0 5 to find an approimation for, giving the answer to decimal places. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-3c EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

13 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Use the substitution = u (u > 0) to find the eact value of d. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a I sin d 0 i. Find the eact value of I. Use Simpson s rule with 3 function values to approimate the value of I. 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-b The numbers,, 3,, 9 are written one on each of 9 cards. 3 of the cards are chosen at random. i. Find the probabilit that the sum of the 3 numbers chosen is equal to 9. If it is known that the first number chosen is, find the probabilit now that the sum of the 3 numbers chosen is equal to 9. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-c spherical map of the earth is being inflated at a constant rate of 5 cm 3 s. Find the rate at which the length of the equator is changing when the radius is 0 cm. 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-5a f ( ), i. Find the equation of the inverse function f (). On the same diagram sketch the graphs of = f() and = f () showing clearl the coordinates of an points of intersection, the intercepts on the coordinate aes, and the equation of an asmptotes. 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-5b particle is moving in a straight line. O is a fied point on the line. t time t seconds, the particle has displacement metres from O, its velocit v ms is given b v = and its acceleration is a ms. Initiall the particle is m to the right of O. i. Find an epression for a in terms of. Find an epression for in terms of t. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-5c a 6 In the binomial epansion of ( ) the term independent of is 0. Find the value of a. 8)! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-6a i. If = tan + tan B B show that tan. B Hence solve the equation tan 3 tan EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW )! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-6b t time t minutes, the temperature T Celsius of a piece of metal is given b T = 0 + e k t for some constant and k. Initiall the temperature of the piece of metal is 00 C, and after minutes its temperature has dropped to 80 C. i. Find the eact values of and k. Find how much longer it will take for the temperature to drop to 60 C, giving the answer correct to the nearest second. 0)! Yearl\Yr-3U\cat-3u.0 Qn0) π.

14 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 3U0-6c particle moving in a horizontal straight line is performing Simple Harmonic Motion. t time t seconds its displacement metres from a fied point O on the line is given b = 3cos t + sin t, where displacements to the right of O are positive. i. Eplain whether the particle is initiall moving to the right or to the left, and whether it is speeding up or slowing down. Find the time taken b the particle to first reach O, giving the answer in seconds correct to decimal places. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-7a function f() is such that f() > 0 for all real numbers and f(a + b) = f(a). f(b) for an real numbers a and b. i. Show that f(0) = and deduce that f ( ). f ( ) Use the method of mathematical induction to show that f(n) = [f()] n for all positive integers n. i Without using mathematical induction again, deduce that f( n) = [f()] n for all positive integers n. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-7b V h O U particle is projected from a point O with speed U ms at an angle of elevation. t the same instant, another particle is projected from a point (h metres directl above O) with speed V ms at an angle of elevation, where <. Both particles move freel under gravit in the same plane of motion and collide T seconds after projection. i. Write down epressions for the horizontal and vertical displacements of each particle at time t seconds referred to aes O and O. h cos Show that T. U sin( ) )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a k Find the value of ( ) k! k )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-b (, 5) and B(, ) are two points. Find the acute angle between the line B and the line + + = 0, giving the answer correct to the nearest minute. 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-c The polnomial P() = 5 + a 3 + b leaves a remainder of 5 when it is divided b ( ), when a and b are numerical constants. i. Show that P() is odd. Hence find the remainder when P() is divided b ( + ). )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-d EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

15 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 M N B D BCD is a cclic quadrilateral. The tangent at to the circle through, B, C and D is parallel to BD. i. Cop the diagram. Give a reason wh CB = MB. i Give a reason wh CD = BD. iv. Hence show that C bisects BCD. 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-a d Find e d 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-b (, ) and B(, ) are two points. The point P(, 6) divides the interval B eternall in the ratio 5:3. Find the coordinates of B. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-c Find the number of was in which the letters of the word EXTENSION can be arranged in a straight line so that no two vowels are net to each other. 8)! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-d C = EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0 O P(t, t ) T P(t, t ) is a variable point which moves on the parabola =. The tangent to the parabola at P cuts the ais at T. M is the midpoint of PT. i. Show that the tangent PT has equation t t = 0. 3t t Show that M has coordinates (, ). i Hence find the Cartesian equation of the locus of M as P moves on the parabola. 9)! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-3a i. B epanding cos( + ), show that cos 3 = cos 3 3cos. 3 Hence show that if cos then cos )! Yearl\Yr-3U\cat-3u.0 Qn0) 3U0-3b The function f () is given b f ( ) 6 for 6. i. Find the inverse function f () and find its domain. On the same diagram, sketch the graphs of = f () and = f (), showing clearl the i intercepts on the coordinate aes. Draw in the line =. Show that the coordinates of an points of intersection of the graphs = f () and = f () satisf the equation 6 = 0. Hence find an points of intersection of the two graphs. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-a

16 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Use Mathematical Induction to show that 5 n + ( n ) is a multiple of 3 for all positive integers n. )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-b O P Two concentric circles with centres O have radii cm and cm. The points P and Q lie on the larger circle and POQ =, where 0. i. If the area cm of the shaded region is 6 the area of the larger circle, show that satisfies the equation 8sin = 0. Show that this equation has a solution =, where 05 < < 06. i Taking 06 as a first approimation for, use one application of Newton s Method to find a second approimation, giving the answer correct to two decimal places. 3)! Yearl\Yr-3U\cat-3u.0 Qn3) 3U0-5a 9 Evaluate d using the substitution u =, u > 0. Give the answer in simplest eact form. ( ) )! Yearl\Yr-3U\cat-3u.0 Qn) 3U0-5b t an point on the curve = f (), the gradient function is given b d sin. Find the value of d 3 f ( ) f ( ). 5)! Yearl\Yr-3U\cat-3u.0 Qn5) 3U0-5c particle is performing Simple Harmonic Motion in a straight line. t time t seconds it has velocit πt v metres per second, and displacement metres from a fied point O on the line, where 5cos. i. Find the period of the motion. π Find an epression for v in terms of t, and hence show that v ( 5 ). i Find the speed of the particle when it is metres to the right of O. 6)! Yearl\Yr-3U\cat-3u.0 Qn6) 3U0-6a P Q T person on horizontal ground is looking at an aeroplane through a telescope T. The aeroplane is approaching at a speed of 80 ms at a constant altitude of 00 metres above the telescope. When the horizontal distance of the aeroplane from the telescope is metres, the angle of elevation of the aeroplane is radians. 00 i. Show that tan. EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

17 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 Show that dθ dt i Find the rate at which is changing when, giving the answer in degrees per second correct to the nearest degree. 7)! Yearl\Yr-3U\cat-3u.0 Qn7) 3U0-6b particle moves in a straight line. t time t seconds its displacement is metres from a fied point O on the line, its acceleration is a ms, and its velocit is v ms 3 where v is given b v. i. Find an epression for a in terms of. Show that t 6 d, and hence show that = 6 60e t. i Sketch the graph of against t and describe the limiting behaviour of the particle. 8)! Yearl\Yr-3U\cat-3u.0 Qn8) 3U0-7a Four fair dice are rolled. n die showing 6 is left alone, while the remaining dice are rolled again. i. Find the probabilit (correct to decimal places) that after the first roll of the dice, eactl one of the four dice is showing 6. Find the probabilit (correct to decimal places) that after the second roll of the dice eactl two of the four dice are showing 6. 9)! Yearl\Yr-3U\cat-3u.0 Qn9) 3U0-7b particle is projected from a point O with speed 50 ms at an angle of elevation, and moves freel under gravit, where g = 0 ms. i. Write down epressions for the horizontal and vertical displacements of the particle at time t seconds referred to aes O and O. Hence show that the equation of the path of the projectile, given as a quadratic equation in tan, is tan 500 tan + ( + 500) = 0. i Hence show that there are two values of, 0, for which the projectile passes through a given point (X, Y) provided that 500Y < X. iv. If the projectile passes through the point (X, X) whose coordinates satisf this inequalit, and the two values of are and, find epressions in terms of X for tan + tan and tan tan, 3 and hence show that. )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-a Solve the inequalit. )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-b Find the number of was in which girls and 3 bos can be seated in a row so that no two girls are net to each other. 3)! Yearl\Yr-3U\cat-3u.00 Qn3) 3U00-c The curves = ln and = e intersect at the point P(e, ). i. Find the gradients m and m of the tangents to each of the curves at P. e If is the acute angle between the tangents to the curves at P, show that tan. e )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-d EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

18 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 In the diagram, BCD is a cclic quadrilateral. The bisector of BC cuts the circle at E, and meets D produced at F. B D F E i. Cop the diagram showing the above information. Give a reason wh CDE = CBE. i Show that DE bisects CDF. 5)! Yearl\Yr-3U\cat-3u.00 Qn5) 3U00-a (, 5) and B(5, ) are two points. Find the coordinates of the point P which divides the interval B internall in the ratio :. 6)! Yearl\Yr-3U\cat-3u.00 Qn6) 3U00-b The equation 3 5 = 0 has roots, and. Find the value of. 7)! Yearl\Yr-3U\cat-3u.00 Qn7) 3U00-c Consider the series tan + tan 3 + tan 5 +, where 0. i. Show that the limiting sum S of the series eists. Show that s tan. 8)! Yearl\Yr-3U\cat-3u.00 Qn8) 3U00-d P(at, at ), t > 0 is a point on the parabola = a. The normal to the parabola at P cuts the ais at X and the ais at Y. i. Show that the normal at P has equation + t at at 3 = 0. Find the coordinates of the points X and Y. i Find the value of t such that P is the midpoint of XY. 9)! Yearl\Yr-3U\cat-3u.00 Qn9) 3U00-3a 3 Consider the function f ( ). i. Show that the function f() is an increasing function for all values of in its domain. Sketch the graph = f() showing clearl the coordinates of the points of intersection with the aes and the equations of the horizontal and vertical asmptotes. i Show that the coordinates of the points of intersection of the line = m and the curve = f() satisf the equation m (m + 3) + = 0. iv. Find the equations of the tangent lines to the curve = f() which pass through the origin. 0)! Yearl\Yr-3U\cat-3u.00 Qn0) 3U00-3b i. The equation 3 k + = 0 has eactl one root between = 0 and =. Show that k >. C EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

19 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 The equation = 0 has a root where 0 < <. Starting with an initial approimation = 0 3, use one application of Newton s method to find a second approimation to the value of, giving the answer correct to two decimal places. )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-a i. Find the domain and range of the function 3cos. Find the equation of the tangent to the curve 3cos at the point on the curve where = 0 )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-b 3 3 i. Evaluate d using the substitution u = +. 0 The region bounded b the curve and the ais between the lines = and = 3 3 is rotated through one complete revolution about the ais. Find the eact volume of the solid formed. 3)! Yearl\Yr-3U\cat-3u.00 Qn3) 3U00-5a n Use the method of Mathematical Induction to show that n! for all positive integers n. )! Yearl\Yr-3U\cat-3u.00 Qn) 3U00-5b 6 metre high vertical street lamp stands on horizontal ground. metre tall man runs awa from the street lamp at a constant speed of 5 ms. When he is metres from the street lamp his shadow has length metres. 6 i. Show that. 6 Find the rate at which his shadow lengthens. 5)! Yearl\Yr-3U\cat-3u.00 Qn5) 3U00-5c In the Binomial epansion of ( + ) n the coefficient of is 6 times the coefficient of. i. Show that n 5n 66 = 0. Find the value of n. 6)! Yearl\Yr-3U\cat-3u.00 Qn6) 3U00-6a fter t ears, t 0, the number N of individuals in a population is given b N = 000 e kt for some constants > 0, k > 0. The initial population size is 00 individuals and the initial rate of increase of the population size is 80 individuals per ear. i. Find the values of and k. Sketch the graph of N against t. 7)! Yearl\Yr-3U\cat-3u.00 Qn7) 3U00-6b EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

20 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 particle is moving with Simple Harmonic Motion in a straight line about a fied point O. t time t seconds, t 0, it has displacement metres from O given b = a cos(t + ) for some constants a > 0, 0 < <. Initiall it is metres to the right of O. fter seconds it is 3 metres to the left of O. i. Show that a cos = and a sin = 3 Find the eact value of a, and the value of correct to two decimal places. 8)! Yearl\Yr-3U\cat-3u.00 Qn8) 3U00-6c Each time that Bill and Vlad pla a set of tennis there is a probabilit of 3 that Bill wins and a probabilit of 3 that Vlad wins the set. i. If Bill and Vlad pla sets of tennis, find the probabilit that Bill wins sets and Vlad wins sets. If Bill and Vlad pla sets of tennis until one of them wins 3 sets, find the probabilit that Bill wins 3 sets and Vlad wins sets. 9)! Yearl\Yr-3U\cat-3u.00 Qn9) 3U00-7a garden sprinkler is positioned at the centre of a large, flat lawn. Water droplets are projected from the sprinkler at a fied speed of 0 ms and at an angle above the horizontal. The acceleration due to gravit is 0 ms. i. Use integration to show that the horizontal displacement metres and the vertical displacement metres of the water droplets after time t seconds are given b = 0t cos and = 0t sin 5t. Show that the horizontal range R of the water droplets is given b R = 0 sin. i If the angle of projection varies between 5 and 5 above the horizontal, find the eact area of that part of the lawn, which can be watered in this wa. 0)! Yearl\Yr-3U\cat-3u.00 Qn0) 3U00-7b particle is moving in a straight line. fter time t seconds it has displacement metres from a fied point O on the line, velocit ms given b and acceleration a ms. Initiall the particle is at O. i. Find an epression for a in terms of. Show that and hence find an epression for in terms of t. i Describe the motion of the particle, eplaining whether it moves to the left or right of O, whether it slows down or speeds up, and where its limiting position is. [[End Of Qns]] EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

21 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 «) 5» «) i) 8 + a + b = 0, a + b = 5 ii) a =, b =» «3) i) t = 0, m = and at =, m = e ii) 5» «) ii) The eterior angle COE of the cclic quadrilateral DOC is equal to the interior opposite angle CD iii) Proof» «5) 8» «6) i) Proof ii) 3» «7) i) Proof ii) = 3 5,» «8) Proof» «9) i) Proof ii) = iii) f () = e = f () = e (, ) = f() O iv) e» «0) i) Proof ii) 3» 8 «) i) Proof ii)» «) ln» «3) i) Proof ii) 0 0» 5 80 «) i) ii) 3» «5) i) Proof ii) 0 3 m per minute» «6) Proof» «7) i) = cos(t + ) ii) mplitude is m, Period is π iii) Initiall the particle is 3 moving towards O and is speeding up iv) seconds» «8) Proof» «9) i) = Vtcos, = Vtsin gt ii) Proof» «0) i) Proof ii) ( 5 )» «) Proof» «) + tan» «) a = 7» «3) i) Proof ii) = 3 and = 3» [nswers] EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0 «) ii) The eterior angle of a cclic quadrilateral is equal to the opposite interior angle ii) Proof» «5) P(5, )» «6)» «7) Proof» «8) i) Proof ii) =» «9) i) Proof ii) cm» «0) i) Proof ii) ln( ) c» «) Proof» «) » «3) i) = 00, B = 00 ii) 7 ears and 5 months» «) i) Proof ii) 0 7» «5) i) ii)» «6) i) Proof ii) 5 cm s» «7) i) Proof ii) (0, ) is a minimum turning point iii) Proof» «8) i) {: 0 } (0, ) = cos - ( ) O (, 0) ii) iii) «9) i) ( ) ii), π seconds iii) m, sec» 3 units 3» «0) i) ii) Proof iii) metres to the right of O, O t» «) i) = 0t, = 0 5t ii) iii) Proof» «) 5» «) (, )» «3) < or 0 < <» «) ii) The angle between the tangent BM and the chord BC is equal to the angle in the alternate segment. ii) Proof»

22 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 «5) d d 3 0( ), 0( ) ( 9 )» d d «6) 0» «7) i) Proof ii)» «8) i) Proof ii) t» «9) i) Proof ii) 06 = f () = f() = iii)» «0) i) a = cos 3 sin, = tan t = tan t ii) (0, 0) t iii) The particle starts at O, moving to the right and slowing down. It π approaches its limiting position of metres to the right of O.» «) π i) Domain :{ : 0 } Range :{ : 0 } π ( 0, ) cos (, 0) π ii) units π iii) units» «) i) Proof ii) Period = seconds, mplitude = metres iii) 53 metres» 3 «3) a» «) 3» «5) i) Proof ii) 6 3» «6) 8 metres» «7) i) Proof ii) p 6» «8) i) Proof ii) 05 cms» «9) i) EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0 ii) Particle = 0 seconds Particle B = 0 seconds iii) Proof» «0) Proof» ( t Ut, h gt, B V t 0), B h g( e «)» ( e ) 5 «)» 8 «3) i) Proof ii)» «) ii) The angle between the tangent MN and chord B is equal to the angle in the alternate segment. iii) Proof» «5) P(8, 3)» «6) 503» «7) i) P() = ( + )( ) ii) or =» «8) i) T( a( t ), a) ii) Proof» t «9) Proof» «0) 06» «) 6» «) i) ii)» «3) i) 8 ii)» «) 05 cms» «5) i) f ( ), 0 = f () = = (, ) = f() ii) «6) i) a = 3 ii) =» t «7) a =» «8) i) Proof ii)» 6 «9) i) 80, k ln ii) 5 minutes 3 38 seconds» «0) i) The particle is initiall 3 m to the right of O, moving to the right and slowing down. ii) 095 seconds»»

23 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 «) Proof» «) i) For particle projected from O: 0 Utcos, 0 Ut sin gt. For particle projected from : Vt cos, h Vt sin gt ii) Proof» «) 9» «) 85» «3) i) Proof ii) 5» «) ii) The angle between the tangent M and the chord B through the point of contact is equal to the angle CB in the alternate segment iii) The angles subtended in the same segment at B and C b the arc D are equal iv) Proof» «5) ( ) e» «6) B(5, 0)» «7) 60» «8) i) ii) Proof iii) = 9» «9) Proof» «0) i) f () = 6, Domaine: { : 0} = f () = 6 = f() «9) i) = 50t cos, = 50t sin 5t ii) iii) Proof iv) tan tan,tan tan, X X Proof» «) 0 or» «)» «3) i) m,m ii) Proof» e e «) ii) ngles in the same segment standing on arc CE are equal iii) Proof» «5) P(3, )» «6) 5» «7) Proof» «8) i) Proof ii) X(at + at 3, 0), Y(0, a + at ) iii)» «9) i) Proof ii) iii) (3, 3)» «) Proof» «) i) ii) Proof iii) 056» «3) ln» «)» «5) i) seconds ii) v 5( sin t), Proof 3 iii) ms» «6) i) ii) Proof iii) s» 0 «7) i) a ii) Proof 3 6 = 6 60e t O 3 ii) iii) Proof iv) = and = 9» «0) i) Proof ii) 0 35» «) i) Domain is, Range is 0 3 ii) = 0» 3 3 «) i) 8 ii) units 8 «3) Proof» «) i) Proof ii) 5 m/s» «5) i) Proof ii) n =» «6) i) = 800, k = 0 N 000» iii) O The particle is moving to the right and slowing down as it approaches its limiting position 8 metres to the right of O.» «8) i) 039 ii) 07» 00 ii) O «7) i) Proof ii) a = 5, = 0 6» 8 6 «8) i) ii)» 7 8 t» EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

24 GPCC039 C:\M_Bank\Tests\Yr-U\3U csa trials /09/0 «9) i) ii) Proof iii) 00 m» 3 t e «0) i) a ii) iii) Initiall the t e particle is at O, moving at 0 5 m/s to the right and slowing down. The particle continues to move to the right for < and continues to slow down. The limiting position of the particle is m to the right of O.» EDUDT SOFTWRE PTY LTD:995-0 BORD OF STUDIES NSW 98-0 CSS NSW 98-0

Methods of Integration

Methods of Integration U96-b)! Use the substitution u = - to evaluate U95-b)! 4 Methods of Integration d. Evaluate 9 d using the substitution u = + 9. UNIT MATHEMATICS (HSC) METHODS OF INTEGRATION CSSA «8» U94-b)! Use the substitution

More information

Mathematics Extension 1

Mathematics Extension 1 013 HIGHER SCHL CERTIFICATE EXAMINATIN Mathematics Etension 1 General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators

More information

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks) Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.

More information

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.

The region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed. Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.

More information

2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW

2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.

More information

2000 Solutions Euclid Contest

2000 Solutions Euclid Contest Canadian Mathematics Competition n activit of The Centre for Education in Mathematics and Computing, Universit of Waterloo, Waterloo, Ontario 000 s Euclid Contest (Grade) for The CENTRE for EDUCTION in

More information

Mathematics Extension 1

Mathematics Extension 1 Teacher Student Number 008 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension 1 General Instructions o Reading Time- 5 minutes o Working Time hours o Write using a blue or black pen o Approved

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.

More information

MATHEMATICS 4 UNIT (ADDITIONAL) HIGHER SCHOOL CERTIFICATE EXAMINATION. Time allowed Three hours (Plus 5 minutes reading time)

MATHEMATICS 4 UNIT (ADDITIONAL) HIGHER SCHOOL CERTIFICATE EXAMINATION. Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 997 MATHEMATICS UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.

More information

PreCalculus Final Exam Review Revised Spring 2014

PreCalculus Final Exam Review Revised Spring 2014 PreCalculus Final Eam Review Revised Spring 0. f() is a function that generates the ordered pairs (0,0), (,) and (,-). a. If f () is an odd function, what are the coordinates of two other points found

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS / UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value.

More information

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000 Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For

More information

Mathematics Extension 1

Mathematics Extension 1 009 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table

More information

Mathematics Extension 2

Mathematics Extension 2 0 HIGHER SCHL CERTIFICATE EXAMINATIN Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators

More information

Mathematics Extension 1 Time allowed: 2 hours (plus 5 minutes reading time)

Mathematics Extension 1 Time allowed: 2 hours (plus 5 minutes reading time) Name: Teacher: Class: FORT STREET HIGH SCHOOL 014 HIGHER SCHOOL CERTIFICATE COURSE ASSESSMENT TASK 3: TRIAL HSC Mathematics Etension 1 Time allowed: hours (plus 5 minutes reading time) Syllabus Assessment

More information

Name: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination 3 Secondary 4

Name: Index Number: Class: CATHOLIC HIGH SCHOOL Preliminary Examination 3 Secondary 4 Name: Inde Number: Class: CATHOLIC HIGH SCHOOL Preliminary Eamination 3 Secondary 4 ADDITIONAL MATHEMATICS 4047/1 READ THESE INSTRUCTIONS FIRST Write your name, register number and class on all the work

More information

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000

Mathematics. Mathematics 2. hsn.uk.net. Higher HSN22000 hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still

More information

MATHEMATICS HSC Course Assessment Task 3 (Trial Examination) June 21, QUESTION Total MARKS

MATHEMATICS HSC Course Assessment Task 3 (Trial Examination) June 21, QUESTION Total MARKS MATHEMATICS 0 HSC Course Assessment Task (Trial Eamination) June, 0 General instructions Working time hours. (plus 5 minutes reading time) Write using blue or black pen. Where diagrams are to be sketched,

More information

Name DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!!

Name DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!! FINAL EXAM REVIEW 0 PRECALCULUS Name DIRECTIONS: PLEASE COMPLET E ON A SEPARATE SHEET OF PAPER. USE THE ANSWER KEY PROVIDED TO CORRECT YOUR WORK. THIS WILL BE COLLECTED!!! State the domain of the rational

More information

Total marks 70. Section I. 10 marks. Section II. 60 marks

Total marks 70. Section I. 10 marks. Section II. 60 marks THE KING S SCHOOL 03 Higher School Certificate Trial Eamination Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval................... 0 Section. Rolle s Theorem and the Mean Value Theorem...... 0 Section. Increasing and Decreasing Functions and

More information

Mathematics Extension 1

Mathematics Extension 1 NORTH SYDNEY GIRLS HIGH SCHOOL 05 TRIAL HSC EXAMINATION Mathematics Etension General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen Black pen is preferred Board approved

More information

Catholic Schools Trial Examinations 2007 Mathematics. as a single fraction in its simplest form. 2

Catholic Schools Trial Examinations 2007 Mathematics. as a single fraction in its simplest form. 2 0 Catholic Trial HSC Eaminations Mathematics Page Catholic Schools Trial Eaminations 0 Mathematics a The radius of Uranus is approimately 5 559 000m. Write the number in scientific notation, correct to

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

More information

IB Practice - Calculus - Differentiation Applications (V2 Legacy)

IB Practice - Calculus - Differentiation Applications (V2 Legacy) IB Math High Level Year - Calc Practice: Differentiation Applications IB Practice - Calculus - Differentiation Applications (V Legacy). A particle moves along a straight line. When it is a distance s from

More information

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.

1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3. Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient

More information

y A(0, 3) NOT TO SCALE O B(2, 0)

y A(0, 3) NOT TO SCALE O B(2, 0) GPCC9 U 5 CSA Trials )! Yearl\Yr-U\cat-u.5 Qn) U5-a Write down the value of 6. )! Yearl\Yr-U\cat-u.5 Qn) U5-b If, find the value of f when u = 5 and v = 7 5. f u v )! Yearl\Yr-U\cat-u.5 Qn) U5-c Solve

More information

STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE MATHEMATICS. Time allowed Three hours (Plus 5 minutes reading time)

STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE MATHEMATICS. Time allowed Three hours (Plus 5 minutes reading time) STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE 00 MATHEMATICS Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 000 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 000 MATHEMATICS UNIT (ADDITIONAL) AND /4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions

More information

Mathematics Extension 1

Mathematics Extension 1 NSW Education Standards Authority 08 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black pen Calculators approved

More information

1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks)

1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks) 1. Find the area enclosed by the curve y = arctan, the -ais and the line = 3. (Total 6 marks). Show that the points (0, 0) and ( π, π) on the curve e ( + y) = cos (y) have a common tangent. 3. Consider

More information

Mathematics Extension 1

Mathematics Extension 1 Name: St George Girls High School Trial Higher School Certificate Examination 2014 Mathematics Extension 1 General Instructions Reading time 5 minutes Working time 2 hours Write using blue or black pen.

More information

Q.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or

Q.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. Section. Rolle s Theorem and the Mean Value Theorem. 7 Section. Increasing and Decreasing Functions and the First Derivative

More information

Instructions for Section 2

Instructions for Section 2 200 MATHMETH(CAS) EXAM 2 0 SECTION 2 Instructions for Section 2 Answer all questions in the spaces provided. In all questions where a numerical answer is required an eact value must be given unless otherwise

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL

More information

Parabolas. Example. y = ax 2 + bx + c where a = 1, b = 0, c = 0. = x 2 + 6x [expanding] \ y = x 2 + 6x + 11 and so is of the form

Parabolas. Example. y = ax 2 + bx + c where a = 1, b = 0, c = 0. = x 2 + 6x [expanding] \ y = x 2 + 6x + 11 and so is of the form Parabolas NCEA evel Achievement Standard 9157 (Mathematics and Statistics.) Appl graphical methods in solving problems Methods include: graphs at curriculum level 7, their features and their equations

More information

Rancho Bernardo High School/Math Department Honors Pre-Calculus Exit Exam

Rancho Bernardo High School/Math Department Honors Pre-Calculus Exit Exam Rancho Bernardo High School/Math Department Honors Pre-Calculus Eit Eam You are about to take an eam that will test our knowledge of the RBHS Honors Pre-Calculus curriculum. You must demonstrate genuine

More information

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS

More information

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1 Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares

More information

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives

UNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.

More information

1. For each of the following, state the domain and range and whether the given relation defines a function. b)

1. For each of the following, state the domain and range and whether the given relation defines a function. b) Eam Review Unit 0:. For each of the following, state the domain and range and whether the given relation defines a function. (,),(,),(,),(5,) a) { }. For each of the following, sketch the relation and

More information

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity.

(i) find the points where f(x) is discontinuous, and classify each point of discontinuity. Math Final Eam - Practice Problems. A function f is graphed below. f() 5 4 8 7 5 4 4 5 7 8 4 5 (a) Find f(0), f( ), f(), and f(4) Find the domain and range of f (c) Find the intervals where f () is positive

More information

3 UNIT MATHEMATICS (PRELIMINARY) OTHER INEQUALITIES CSSA. Other Inequalities. » 3U89-1c)! Solve the inequality: «x < -1 or 0 < x < 1

3 UNIT MATHEMATICS (PRELIMINARY) OTHER INEQUALITIES CSSA. Other Inequalities. » 3U89-1c)! Solve the inequality: «x < -1 or 0 < x < 1 Other Inequalities 3 UNIT MTHEMTIS (PRELIMINRY) OTHER INEQULITIES SS 3U93-3a)! Solve the inequality 2 3 4 1 «< -7 or > 4» 3U92-2c)! Solve the inequality 2 1 0. «-1 < < 0 or > 1» 3U90-1c)! Solve the following

More information

and hence show that the only stationary point on the curve is the point for which x = 0. [4]

and hence show that the only stationary point on the curve is the point for which x = 0. [4] C3 Differentiation June 00 qu Find in each of the following cases: = 3 e, [] = ln(3 + ), [] (iii) = + [] Jan 00 qu5 The equation of a curve is = ( +) 8 Find an epression for and hence show that the onl

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

Mathematics Extension 2

Mathematics Extension 2 Student Number ABBOTSLEIGH AUGUST 007 YEAR ASSESSMENT 4 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes. Working time 3 hours. Write using blue

More information

Differentiation and applications

Differentiation and applications FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5

More information

k + ln x 2, para x 0, k, en el punto en que x = 2, tiene la 1. La normal a la curva y = x

k + ln x 2, para x 0, k, en el punto en que x = 2, tiene la 1. La normal a la curva y = x 1 1. La normal a la curva = k + ln, para 0, k, en el punto en que =, tiene la ecuación 3 + = b,donde b. halle eactamente el valor de k. Answer:...... (Total 6 puntos).. Sea f la function,tal que, f ( )

More information

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level

Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level file://c:\users\buba\kaz\ouba\c_rev_a_.html Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = 5 + 0 0 + 5 5 Compare ( + ) n with ( ) n. Replace n by 5 and

More information

Review of Essential Skills and Knowledge

Review of Essential Skills and Knowledge Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope

More information

Mathematics Extension 2

Mathematics Extension 2 0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators

More information

Calderglen High School Mathematics Department. Higher Mathematics Home Exercise Programme

Calderglen High School Mathematics Department. Higher Mathematics Home Exercise Programme alderglen High School Mathematics Department Higher Mathematics Home Eercise Programme R A Burton June 00 Home Eercise The Laws of Indices Rule : Rule 4 : ( ) Rule 7 : n p m p q = = = ( n p ( p+ q) ) m

More information

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least

Problems to practice for FINAL. 1. Below is the graph of a function ( ) At which of the marked values ( and ) is: (a) ( ) greatest = (b) ( ) least Problems to practice for FINAL. Below is the graph of a function () At which of the marked values ( and ) is: (a) () greatest = (b) () least = (c) () the greatest = (d) () the least = (e) () = = (f) ()

More information

Add Math (4047) Paper 2

Add Math (4047) Paper 2 1. Solve the simultaneous equations 5 and 1. [5]. (i) Sketch the graph of, showing the coordinates of the points where our graph meets the coordinate aes. [] Solve the equation 10, giving our answer correct

More information

(1,3) and is parallel to the line with equation 2x y 4.

(1,3) and is parallel to the line with equation 2x y 4. Question 1 A straight line passes through the point The equation of the straight line is (1,3) and is parallel to the line with equation 4. A. B. 5 5 1 E. 4 Question The equation 1 36 0 has A. no real

More information

Higher School Certificate

Higher School Certificate Higher School Certificate Mathematics HSC Stle Questions (Section ) FREE SAMPLE J.P.Kinn-Lewis Higher School Certificate Mathematics HSC Stle Questions (Section ) J.P.Kinn-Lewis First published b John

More information

Summary, Review, and Test

Summary, Review, and Test 944 Chapter 9 Conic Sections and Analtic Geometr 45. Use the polar equation for planetar orbits, to find the polar equation of the orbit for Mercur and Earth. Mercur: e = 0.056 and a = 36.0 * 10 6 miles

More information

NATIONAL QUALIFICATIONS

NATIONAL QUALIFICATIONS H Mathematics Higher Paper Practice Paper A Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion

More information

MATH TOURNAMENT 2012 PROBLEMS SOLUTIONS

MATH TOURNAMENT 2012 PROBLEMS SOLUTIONS MATH TOURNAMENT 0 PROBLEMS SOLUTIONS. Consider the eperiment of throwing two 6 sided fair dice, where, the faces are numbered from to 6. What is the probability of the event that the sum of the values

More information

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3 [STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)

More information

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.

Math 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint. Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the

More information

MATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)

MATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1) MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the

More information

Paper 2H GCSE/A2H GCSE MATHEMATICS. Practice Set A (AQA Version) Calculator Time allowed: 1 hour 30 minutes

Paper 2H GCSE/A2H GCSE MATHEMATICS. Practice Set A (AQA Version) Calculator Time allowed: 1 hour 30 minutes Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks Paper 2H GCSE MATHEMATICS CM Practice Set A (AQA Version) Calculator Time allowed: 1 hour 30 minutes

More information

Solutions to O Level Add Math paper

Solutions to O Level Add Math paper Solutions to O Level Add Math paper 4. Bab food is heated in a microwave to a temperature of C. It subsequentl cools in such a wa that its temperature, T C, t minutes after removal from the microwave,

More information

5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)

5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0) C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre

More information

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y. BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n + n ( 9 )( ) + + 6 + 9ab

More information

Possible C4 questions from past papers P1 P3

Possible C4 questions from past papers P1 P3 Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.

More information

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions

y=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)

More information

St Peter the Apostle High. Mathematics Dept.

St Peter the Apostle High. Mathematics Dept. St Peter the postle High Mathematics Dept. Higher Prelim Revision Paper I - Non~calculator Time allowed - hour 0 minutes FORMULE LIST Circle: The equation g f c 0 represents a circle centre ( g, f ) and

More information

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level UNIVERSITY F CMBRIDGE INTERNTINL EXMINTINS General Certificate of Education dvanced Subsidiary Level and dvanced Level *336370434* MTHEMTICS 9709/11 Paper 1 Pure Mathematics 1 (P1) ctober/november 013

More information

AP Calculus (BC) Summer Assignment (169 points)

AP Calculus (BC) Summer Assignment (169 points) AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion

More information

ZETA MATHS. Higher Mathematics Revision Checklist

ZETA MATHS. Higher Mathematics Revision Checklist ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions

More information

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x

UNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x 5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able

More information

Created by T. Madas. Candidates may use any calculator allowed by the regulations of this examination.

Created by T. Madas. Candidates may use any calculator allowed by the regulations of this examination. IYGB GCE Mathematics SYN Advanced Level Snoptic Paper C Difficult Rating: 3.895 Time: 3 hours Candidates ma use an calculator allowed b the regulations of this eamination. Information for Candidates This

More information

Math Review. Name:

Math Review. Name: Math 30-1 Name: Review 1. Given the graph of : Sketch the graph of the given transformation on the same grid Describe how the transformed graph relates to the graph of Write the equation of the image of

More information

I K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists

I K J are two points on the graph given by y = 2 sin x + cos 2x. Prove that there exists LEVEL I. A circular metal plate epands under heating so that its radius increase by %. Find the approimate increase in the area of the plate, if the radius of the plate before heating is 0cm.. The length

More information

CBSE Board Class X Summative Assessment II Mathematics

CBSE Board Class X Summative Assessment II Mathematics CBSE Board Class X Summative Assessment II Mathematics Board Question Paper 2014 Set 2 Time: 3 hrs Max. Marks: 90 Note: Please check that this question paper contains 15 printed pages. Code number given

More information

ADVANCED MATHS TEST - I PRELIMS

ADVANCED MATHS TEST - I PRELIMS Model Papers Code : 1101 Advanced Math Test I & II ADVANCED MATHS TEST - I PRELIMS Max. Marks : 75 Duration : 75 Mins. General Instructions : 1. Please find the Answer Sheets (OMR) with in the envelop

More information

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3

2. Find the value of y for which the line through A and B has the given slope m: A(-2, 3), B(4, y), 2 3 . Find an equation for the line that contains the points (, -) and (6, 9).. Find the value of y for which the line through A and B has the given slope m: A(-, ), B(4, y), m.. Find an equation for the line

More information

NATIONAL QUALIFICATIONS

NATIONAL QUALIFICATIONS Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -

More information

Mathematics 2005 HIGHER SCHOOL CERTIFICATE EXAMINATION

Mathematics 2005 HIGHER SCHOOL CERTIFICATE EXAMINATION 005 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Board-approved calculators may be used A table of standard

More information

2, find c in terms of k. x

2, find c in terms of k. x 1. (a) Work out (i) 8 0.. (ii) 5 2 1 (iii) 27 3. 1 (iv) 252.. (4) (b) Given that x = 2 k and 4 c 2, find c in terms of k. x c =. (1) (Total 5 marks) 2. Solve the equation 7 1 4 x 2 x 1 (Total 7 marks)

More information

2014 HSC Mathematics Extension 1 Marking Guidelines

2014 HSC Mathematics Extension 1 Marking Guidelines 04 HSC Mathematics Etension Marking Guidelines Section I Multiple-choice Answer Key Question Answer D A 3 C 4 D 5 B 6 B 7 A 8 D 9 C 0 C BOSTES 04 HSC Mathematics Etension Marking Guidelines Section II

More information

Circle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle

Circle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle PSf Circle Paper 1 Section A Each correct answer in this section is worth two marks. 1. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0).

More information

Mathematics. Caringbah High School. Trial HSC Examination. Total Marks 100. General Instructions

Mathematics. Caringbah High School. Trial HSC Examination. Total Marks 100. General Instructions Caringbah High School 014 Trial HSC Examination Mathematics General Instructions Total Marks 100 Reading time 5 minutes Working time 3 hours Write using a blue or black pen. Black pen is preferred. Board

More information

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I

APPENDIXES. B Coordinate Geometry and Lines C. D Trigonometry E F. G The Logarithm Defined as an Integral H Complex Numbers I APPENDIXES A Numbers, Inequalities, and Absolute Values B Coordinate Geometr and Lines C Graphs of Second-Degree Equations D Trigonometr E F Sigma Notation Proofs of Theorems G The Logarithm Defined as

More information

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y. BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n + n ( 9 ) ( ) + + 6 + 9ab

More information

Differentiation. Each correct answer in this section is worth two marks. 1. Differentiate 2 3 x with respect to x. A. 6 x

Differentiation. Each correct answer in this section is worth two marks. 1. Differentiate 2 3 x with respect to x. A. 6 x Differentiation Paper 1 Section A Each correct answer in this section is worth two marks. 1. Differentiate 2 3 with respect to. A. 6 B. 3 2 3 4 C. 4 3 3 2 D. 2 3 3 2 Ke utcome Grade Facilit Disc. Calculator

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Page ( ) Water seeps out of a conical filter at the constant rate of 5 cc / sec. When the height of water level in the cone is 5 cm, find the rate at which the height

More information

I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.

I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3. 0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.

More information

Basic Mathematics - XII (Mgmt.) SET 1

Basic Mathematics - XII (Mgmt.) SET 1 Basic Mathematics - XII (Mgmt.) SET Grade: XII Subject: Basic Mathematics F.M.:00 Time: hrs. P.M.: 40 Model Candidates are required to give their answers in their own words as far as practicable. The figures

More information

Practice Assessment Task SET 3

Practice Assessment Task SET 3 PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,

More information

2018 Year 10/10A Mathematics v1 & v2 exam structure

2018 Year 10/10A Mathematics v1 & v2 exam structure 018 Year 10/10A Mathematics v1 & v eam structure Section A Multiple choice questions Section B Short answer questions Section C Etended response Mathematics 10 0 questions (0 marks) 10 questions (50 marks)

More information

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y.

BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH06 Review Sheet y 6 2x + 5 y. BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH06 Review Sheet. Perform the indicated operations and simplif: n n 0 n +n ( 9 )( ) + + 6 + 9ab a+b

More information

MATH 2 - PROBLEM SETS

MATH 2 - PROBLEM SETS MATH - PROBLEM SETS Problem Set 1: 1. Simplify and write without negative eponents or radicals: a. c d p 5 y cd b. 5p 1 y. Joe is standing at the top of a 100-foot tall building. Mike eits the building

More information

, correct to 4 significant figures?

, correct to 4 significant figures? Section I 10 marks Attempt Questions 1-10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1-10. 1 What is the basic numeral for (A) 0.00045378 (B) 0.0004538 (C)

More information

Mathematics Extension 1

Mathematics Extension 1 Northern Beaches Secondary College Manly Selective Campus 04 HSC Trial Examination Mathematics Extension General Instructions Total marks 70 Reading time 5 minutes. Working time hours. Write using blue

More information