2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW

Size: px
Start display at page:

Download "2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW"

Transcription

1 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. [5]

2 Given that sin and both cos and tan are negative, find an epression, in q terms of q, for (i) tan, [] (ii) sin. []

3 (i) The graph of y 4 c passes through the point (, ). Find the possible values of the constant c. [] (ii) Solve the inequality 4 7. []

4 4 4 The height, h m, of a stone t seconds after it has been thrown vertically upwards from ground level is given by h at bt, where a and b are constants. t h The table shows eperimental values of the variables t and h, but an error has been made in recording one of the values of h. (i) Epress the given equation in the form suitable for drawing a straight line graph and, using graph paper, draw the graph for the values given. [4] Use the graph to (ii) correct the reading of h for which an error has been made, [] (iii) estimate the value of a and b. [] A second stone is thrown into the air from ground level. The height, h m, is directly proportional to t and h = 60 m when t = 0 s. (iv) Draw a line on your graph to illustrate the motion of the second stone. [] (v) Hence find the time when the two stones meet. []

5 5

6 6 5 Solve the equation log log log. [5]

7 7 6 A cubic polynomial f () is such that the roots of f ( ) 0 are, 5 and - and it gives a remainder of 4 when it is divided by ( ). (i) Epress f () as a cubic polynomial in with integer coefficients. [4] (ii) Hence, solve f ( ) 0. []

8 8 cos sin 7 (i) Prove the identity cosec. [] sin cos (ii) Hence, find all the angles between 0 and 80 which satisfy the equation cos sin tan 70. sin cos []

9 9 8 A curve has the equation 4 y. (i) dy Find an epression for and eplain why the curve has no turning points. [] d (ii) Find the gradient of the curve when y = 0. [] (iii) Given that y is increasing at the rate of 0.8 units per second at the instant when =, find the rate of change of at this instant. []

10 0 9 In the epansion of of is p where p is a positive constant, the term independent (i) Show that p = 4. [4] (ii) With this value of p, find the coefficient of 9 p 9 9 in the epansion of 9. [4]

11 0 A line y 0 cuts a curve at two points A and B. y (i) Find the coordinates of A and B. [4] (ii) Show that the perpendicular bisector of AB cuts the y -ais at 0,. 4 [4]

12 EXAM ANSWERS y 6 k 7 k 7 6 line is tangent t o curve, b 6 4( k k 4 when curve and line 9 7)() meet,, y Coordinate s of point is,. 0 4ac 0 A A [5] (i) tan tan q 4 q 4 A (ii) sin sin cos cos sin sin sin cos q 4 q q 4 q q 4 A [5]

13 (i) 4 c y or 4 c y 4 4 c c 5 or c or (ii) A, A A [5] 4 (i) h at bt h a bt t t h h t B B for table of values 5 0 B: B (0.0,.68) (iv) B for straight line graph (i) 5 E: (.47, 4.) Slope CD = A: (5.89, -9.4) -5 (i) -0 (ii) h 9 t h 54 B (iii) From graph, a = y-intercept = 4 b = gradient = 4 B, A (iv) Refer to line drawn above. B (v) t =.5 s B

14 4 5 Given log log log log log log log [ ] log log [ A] log log log log log log log log log log [ M ] log log log [ A] [ A] A A [5] a a 9 6 (i) Let f ( ) a 5 Since f ( ) 4, 0 0 a 9 4 8a 4 a 0 0 f ( ) (ii) f ( ) ,, 4 A A [6]

15 5 7 (i) cos sin sin cos cos cos sin (sin ) cos cos (sin ) cos cos (sin ) cos cosec A (ii) cos sin tan 70 sin cos cos ec tan 70 sin tan ,..4,66. 7 A [6] 8 (i) 4 y dy 4 d 4 dy Since 0, the curve has no turning point. d A B (ii) when y = 0, gradient of the curve A

16 6 8 (iii) dy 4 7 when =, d 8 dy dy d dt d dt d dt.8 units / s A [8] 9 (i) r k r r th term 9 C 9 r r r r r r C p 9 8 r C p 9 r r 8r r r C p [] 9 8 r 8r 0 r 8 r 6 [ A] p 6 T C p 576 [] p 84 6 p p 4 [] r A AG (ii) T4 C 9 6 For, term in 9 67 [B] term in [] coefficient of 477 [A] B A [8]

17 7 0 [] [] [A] [A] [] [] [] [AG] [8]

18 FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Given that 4 8, find the eact value(s) of. [5]

19 The equation of a circle C is y (i) Find the centre and radius of the circle. [] (ii) The circle C is reflected in the y -ais to obtain the circle D. Write down the equation of the circle D. y [] Circle D P Circle C Y Q X (iii) The two circles intersect at points P and Q, as shown in the diagram above. Find the length of line segment PQ. [4] (iv) Two line segments are then drawn from P to meet each of the circle at X and Y respectively. Find the coordinates of X and Y, given that PX and PY are diameters of the circles C and D respectively. [6]

20

21 4 (i) State the amplitude of the curve y sin. [] (ii) Sketch the graph of y sin for the interval 0. [] (iii) By drawing a suitable line on the same diagram in part (ii), determine how many solutions there are of the equation 0 6sin for the interval 0. []

22 5 4 (i) Solve the equation sin 8 sin sin for [4] (ii) Find all angle(s) between 0 and for which 4sin 6cos 5. []

23 6 5 A particle moves in a straight line so that its velocity, v m/s, is given by 6t t, where t is the time in seconds after passing O. Find an epression in terms of t for (i) its acceleration, [] (ii) its displacement from O. [] Calculate (iii) the value of t at which the particle passes through O again, [] (iv) the minimum velocity of the particle, [] (v) the total distance traveled by the particle in the interval t = 0 and t = 0. [5]

24 7

25 8 6 The diagram shows a rectangle PQRS. PQT is an isosceles triangle with PT = 4 cm and PTQ = radians. TV is parallel to PS, and TV = 7 cm. (i) Show that the perimeter, W cm, of the rectangle PQRS is given by 56sin 8cos 4. [] (ii) Epress W in the form a b T 7 cm sin, where a 0 and [] (iii) Find the value of for which W = 7 cm. [] P S 4 cm V Q R

26 9 7 The equation 5 0 has roots and and the equation 4 p 0 k k has roots and. Find the value of k and of p. [8]

27 0 8 A M P E O D B C The diagram shows two circles with centres O and P intersecting at two points A and B. P is a point on the circumference of the circle with centre O. A straight line through B meets the circles at D and C. The line CP meets AB at E and CP produced meets AD at M. Prove that (i) AP AC = AE CP, [4] (ii) CM is perpendicular to AD. [5]

28

29 9 (i) a Given that b, where a and b are integers, find, without using a calculator, the value of a and of b. [4] (ii) Given that ln y y = a and ln = b, epress ln y in terms of a and b. [5]

30

31 4 0 The diagram shows part of the graph of the curves y sin and y cos for 0. y y cos y sin 0 P (i) Show that the point of intersection of the curves is,0.5. [4] (ii) Given that the curve y cos cuts the -ais at the point P. Show that the -coordinate of P is approimately.9. [] (iii) The line is drawn to divide the area enclosed by the curves and the -ais into regions, A and B. A student claimed that the regions A and B are of the same size by just looking at it. Determine if this claim is true, eplaining your argument clearly. [5]

32 5

33 6 EXAM ANSWERS 4 8 y Let. 8 y y 4 y y y y 7 y y or y 8 or or for sub. y for factorisation A, A [5] [] OR [A] [A] 0y 4 0 y Centre of circle = (, 0) Radius = 0 4 = 8 [B] [] [A] [B]

34 7

35 8 [] [A] [] [A] [4]

36 9 (i) Amplitude = B (ii) y B for ½ cycle of 5 4 (ii) sine curve B for correct position (iii) 0 - (iii) - 0 6sin 5 sin Draw y 5 for getting eqn of line for correct line drawn No. of solutions = A [6]

37 0 4 (i) sin 8 sin sin cos 8 sin 8 sin cos5 sin sin sin cos5 0 sin 0 or cos ,60, ,00,40,660, 780 0,60,0, 80,60,84,, 56 0,,60,84,0,,56, 80 (ii) 4sin 6cos 5 4sin 6 sin 5 4sin 6 6sin 5 sin sin A A [7] 5 (i) (ii) v 6t t a t v 6t t s t c when t 0, s 0, c 0 s t t t B B (iii) s 0 when particle passed O again t t 0 t t 0 t 5 s 4 4 A

38 5 (iv) s m v s t t / a velocity, min for finding t A (v) m m s s m s t t ) 8 7 (4 950 distance Total 950 (00) (0) 0 t to s 7 from t travelled distance t 0 to from t travelled distance 7 0, 0-7 t t 0 - t 6 0, when v A []

39 6 (i) PQ 4sin PS 7 4cos 8sin 8sin 7 4cos W W 56sin 8cos 4 (shown) AG (ii) a tan P 6.6sin 6.6 A 4 (iii) 6.6sin sin , 8.0º 6 A [7] k k k 600 p k 4(.5) 5.5 k k k p B B A A A [8]

40 8 (i). Join PB. PA=PB ( radii of circle with centre P). PAE PBE (base angles, isos ) 4. PBE PCA ( s in the same seg) 5. PAE PCA 6. APE CPA (common angle) 7. APE is similar to CPA (AA) 8. AP AE CP AC 9. AP AC AECP (proven) (ii). APB ADB ( at centre = at circumference). APB PAE PBE 80 ( s sum of ). ADB PAE ADB PAE PAE BCE ( s in the same seg) 6. ADB BCE ADB BCE CMD CMD 90 CM is perpendicular to AD. A A [9] a 9 (i) Given b a a a ( b )( ) b b() (b ) ( b ) Hence by comparing coefficients, we obtain a b () b () Solving, we obtain b = - and a = - A, A

41 4 OR a a b a b [ ] a b a 0 [ A] Hence by comparing coefficients, we obtain a b () a () [A] Solving, we obtain b = - and a = - [A] A, A 9 (ii) ln y = a ln + ln y = a ln + ln y = a () ln y = b ln ln y = b () Solving simultaneously eqns () and (), ln = (a+b), a b y = ln y ln = ln y ln ln y = ln a b [ (a+b) ] a 4b = = A A [9]

42 0 5

43 6 0 The claim is not true as area of region A area of region B. [A] [] [] OR [A] [A] 0y 4 0 y Centre of circle = (, 0) Radius = 0 4 = 8 [B] [] [A] [B]

44 FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the range of values of k if y intersects ky 4 at two distinct points. [5]

45 The roots of the equation 5 0 are and. (a) State the value of and. [] (b) Find the quadratic equation in whose roots are and. [5]

46 A wire is bent to form a rectangle of area 5 cm. If the breadth of the rectangle is cm, find the length of the wire, giving your answer in surds. [4]

47 Epress ( ) in partial fractions. [5]

48 5 5 Solve the simultaneous equations [8] log log y, y.

49 6 6 Solve the equation Hence solve [7]

50 7 7 Write down and simplify, in ascending powers of, the first three terms of the epansion of (a) 5 (b) 5 Hence, or otherwise, obtain the first three terms of the epansion of 5 and use it to estimate the value of.94 5 correct to decimal places. [8]

51 8 8 A circle with centre C passes through points A,7 and B,8 0. (i) Eplain why the perpendicular bisector of AB will pass through C. [] (ii) Given further that the line y passes through the centre of the circle, show that the coordinates of C is,4. [4] (iii) Hence find the equation of the circle. []

52 9 9 The following diagram shows the graph of y asin b c for 0. (a) State the possible values of a and the value of b and of c. [] (b) Copy the graph on your answer sheet. By sketching an additional graph on the same aes and domain, find the number of solutions of a sin b cos c 0. [4] y y asin b c 0 4 4

53 0 0 Answer the whole of this question on a sheet of graph paper. The table below shows the eperimental values of two variables and y. 4 5 y m It is known that and y are related by an equation of the form y where m and n are n unknown constants. Draw a graph of y against y and use the graph to estimate (i) the value of m and of n, (ii) the value of when y [8]

54

55 EXAM ANSWERS y y () 5 ky 4 () Sub () into (): k( ) 4 k k 4 k 4 k 0 For distinct roots, b 4ac 0 ( k) 4k 4()(4 k) 0 6 k 0 k k 4 0 ( k 4)( k ) 0 k 4 or k A (a) 5 0 a, b, c 5 c a b a 5 B B (b) ( ) A A A

56 cm rectangle of Length cm Perimeter wire of Length A A 4 4 ) ( (), into Sub 4 : () () () , When () , When 4 6, When ) ( ) ( 4 6 ) ( 4 6 B A A A B A B A B A B A C C C B A C B A A 5 5 (4) 6 7 From (), () log 4 log log log log From (), () 9 8 () log log y y y y y y y y y y 8

57 4 and (4), into Sub ) ( ) 7)( ( (4), Sub ()into y y y or NA y y y y y y y A A 6 Let ) ( f 7 or or 7 or or 0 4 ) 7( ) 6( ) ( or or ) )( )( ( 0 7) )( ( ) ( , of t coefficien By comparing 7) )( ( ). ( of a factor is ) ( 0 4 ) 7( ) 6( ) ( ) ( f b b b b f f A A 7

58 5 7 ( a) ( b) C C C () C () By substituti ng 0. into (), (0.) (0.) () A A A A 8 8(i) By the property of circle, the perpendicular bisector of chord passes through centre of circle. Therefore, the perpendicular bisector of chord AB passes through C. B 8(ii) Mid point of AB, 8 7 Gradient of AB 0 ( ), 5 Gradient of perpendicu lar bisector of AB 4 When Sub ()into (), Sub into and y 7 y (), 5 y, () () y 7 4 c c 7 The coordinates of C are,4. A

59 6 8(iii) Radius = ( 0) (8 4) 5 units Equation of circle: ( ) ( y 4) 5 ( ) ( y 4) 5 or 6 y 8y 0 A 9(a) Possible values of a or Value of b and c y B0.5,B0.5 B0.5,B0.5 y asin b c 9(b) y cos a sin b cos c 0 a sin b c cos Award marks for correct graph drawn (with labels). Number of solutions = B B

60 7 0 m y n y ny m ny y m y n y m n 8 y y y y (i) (ii) Award point for each of the following: Correct labelling of the graph, -ais and y-ais Appropriate use of scale and plotting of all points Drawing of best fit line gradient n. 7 n n y-intercept m. 0 n From the graph, y. 95 when y m M A A A

61 FEB EXAM 4 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW. Find the coordinates of the points of intersection A and B of the line y 0 and the curve. Hence show that the distance AB is 5 units. [5] y

62 . Given that the epression 5 is eactly divisible by p 4. (a) Find the value of p. [] (b) Hence solve the equation 5 ( )( ). [5]

63 . Given the quadratic equation k k 8 0, where k is a real number, (a) if the equation has equal real roots, find the values of k, [4] (b) if the equation has comple roots, find the range of values of k. []

64 4. (a) Solve the equation (b) (c) 7( ) 6, leaving your answers in two decimal places. [4] Without using a calculator, find the value of p for which Simplify (log 45)(log 5 p). [] 7 7. []

65 9 5. (a) Epress in partial fractions. [5] 4 cos cos cot cos (b) Prove the identity ec [4]

66 6. (a) Sketch the following graphs y cos and y sin on the same aes, for [] (b) Hence, find the number of solutions for the following equation [] sin cos.

67 7. (a) Given that 4 6, find the value of for which tan + cot = 5 cosec. [5] (b) Epress 9sin 7cos in the form Rsin where R is a positive number and α is acute. Hence, or otherwise, solve the equation 9sin7cos for [6]

68 EXAM 4 ANSWERS. + y + = y : y = : y + 4 = y Sub into, ( ) + 4 = ( ) [] = + 4 = 0 [] ( + 4)( ) = 0 = or y = or 4 (, y) = (, ), (, 4) [A] AB ( ) ( 4) 45 5 [B] (a) 5 ( p 4)( ) Coeff of : = () +p() [] p = [A] (b) 5 ( )( ) ( + )( + ) = 0 ( + 4)( + ) ( + )( + ) = 0 ( + )[ + 4 ] = 0 [] ( + )( 5 + ) = 0 ( + )( )( ) = 0 [] =,,.5 [A] (a) k k 8 0 k + (k+ 8) = 0 if equation has equal real roots, b 4ac = 0 [] ( k) 4()(k + 8) = 0 9k k = 0 [] (k 8)(k + 4) = k or k [A] (b) if comple roots, b 4ac < 0 [] 4 8 [A] 4(a) 7( ) 6

69 . = 7(. ) + 6 [] Let y =. y 4y 6 = 0 (y + 4)(y 9) = 0 [] y = or 9 = or = 9 [] (rejected) = lg 9 / lg =.7 ( dec pl) [A] 4(b) (log 45)(log 5 p) log 5 log p [] log 4 log5 log5 log p log log5 log p = log [] p = = 8 [A] 4(c) = [] = 7 4 [] = 9 [A] 9 5(a) 4 0 = 4 0 = ( )( ) 0 A B Let ( )( ) [] [] 0 = A( ) + B( + ) when =, when =, 8 = 4B B = = 4A A = [A] [A] 9 4

70 = [A] cos cos ec cot cos 5(b) cos RHS = sin sin cos = sin cos = sin ( cos ) cos cos = [] [] [] = cos = RHS [A] cos 6(a) Correct Graph M Correct Range and Ais 6(b) From the graph, solutions [B] 7(a) tan + cot = 5 cosec sin cos 5 [] cos sin sin sin + cos = 5cos ( cos ) + cos = 5cos cos + 5cos = 0 [] (cos )(cos + ) = 0 cos = 0.5 or cos = [M] basic =.05 (rejected) Soln in st and 4 th quadrant, =.05, 5. [A] (rejected)

71 7(b) 9sin 7cos = Rsin R [] 7 tan 9 9sin 7cos [] 0 =.40sin sin = 0 sin( 7.87 ) 0.6 [A] [] = 5.5 0, , , , = 5. 0, 0.6 0, 4. 0, , 77. 0, = 7.7 0, , ( dec place) [A]

72 FEB EXAM 5 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW. (a) (i) Epand 9 up to the first terms. [] 4 (ii) Hence, given that h k..., 4 find the values of h and k. [] (b) Evaluate the coefficient of 7 in the binomial epansion of 4. []

73 . (a) If α and β are the roots of the equation h 9 0 and β = α, calculate the values of h. [] (b) Given that α and β are the roots of the equation 5 0, form another equation whose roots are and. [4]

74 . The line y74cuts the circle y y at two points, A and B. Find (a) the coordinates of A and B, [5] (b) the equation of the perpendicular bisector of AB and show that it passes through the centre of the circle. [6]

75 4 4. The diagram shows a prism such that each cross-section is a quadrant of a circle of radius cm, with angle at the centre equal to 90. The cross-sections are OAB and PDC where A, B, C, D lie on the curved surface of the prism and the vertical line OP is the intersection of the vertical plane faces OADP and OBCP. The cross-sections are horizontal and y cm apart. C P D y cm B O cm A (a) Given that the volume of the prism is 0 cm, epress y in terms of. Hence show that the total surface area, A cm, of the prism is given by 404 A. [4] (b) Find the value of for which A has a stationary value, [] (c) Find the stationary value of A, [] (d) Determine if the stationary value of A is a maimum or a minimum. [] (e) Given also that the total surface area, A, is increasing at a constant rate of cm s -, find the rate at which is changing when 4. []

76 5

77 6 5. (a) Solve the following equations for 0 60: (i) cos sin 0, [] (ii) sec 4 tan 6. [4] (b) Prove the identity sin 7 sin 5 tan. [] cos 7 cos5

78 7 6. Differentiate the following with respect to : (a) ( ) 0 [] (b) (c) e [] ln, leaving your answer to the simplest form. [4] 4

79 8 b 7. Variables and y are related by the equation y a, where a and b are constants. The table below shows measured values of and y y (a) On a graph paper, plot y against, using a scale cm to represent unit on the ais and cm to represent unit on the y ais. Draw a straight b line graph to represent the equation y a. [] (b) Use your graph to estimate the value of a and of b. [] (c) On the same diagram, draw the line representing the equation y and b hence find the value for which. [] a

80 9

81 0 8. A particle moves along a straight line so that its displacement, s metres, from a fied point P is given by Find the, where t is the time in seconds after passing P. s t 4t 5t (a) initial velocity and acceleration of the particle, [4] (b) minimum velocity, [] (c) range of values of t for which the velocity is negative. []

82 9. The diagram shows the curve 4 y crosses the -ais at P. y S R P Q (a) Find the coordinates of P. [] (b) Find the equation of normal to the curve at P. [4]

83 0. The solution to this question by accurate scale drawing will not be accepted. The diagram shows a rectangle ABCD. The coordinates of A and D are A(4, ) and D(6, 5) and the equation of AC is y =. y C B D(6, 5) O A(4, ) Find (a) the equation of CD, [] (b) the coordinates of C and B, [] (c) the length of AB, [] (d) the area of the rectangle ABCD. []

84 EXAM 5 ANSWERS (a) (i) h k (ii) = = +. [M] [M] h6 and k [A] (b) 4 General term or r th term of the epansion 4 r Cr = 4 = r 4 8 r Cr For the term in 7 = 8 r r = 7 Coeff of 7, r 7 8 r C [] [] [A] (a) h 9 0 h 9 [] Given β = α α = 9 α = [] h 6 9 [A]

85 4 (b) Sum of new roots [] = = = [] Product of new roots 9 [] = New equation is 9 0 [A] (a) y y y Sub y = 4 7 into eqn : [] [] = 4 or 5 y = 6 or [A] A(4, 6) and B(5, ) [A] (b) Gradient of line AB = Gradient of line = 7 [] Midpoint of AB =,, [] Eqn of bisector of AB is 5 9 y 7

86 5 7 y [] y y Centre of Circle is g =, g = f = 4, f = [M] (, ) Sub (, ) into eqn : LHS = RHS [A] 4(a) volume of prism = 0π cm base area of prism = 4 [] volume of prism = 4 y = 0π 80 y [A] 4 Curved surface area ABCD = y 80 4 = = 40 cm Surface Area of OADP and OBCP = y = 60 cm Surface Area of quadrant OAB and PCD = cm Total Surface Area of Prism, A = = A = y ( 4) cm (shown)

87 6 da 4(b) For stationary value, 0 d da 40 4 = 0 [] d = 4.50 cm 4(c) Stationary value of A = 95. cm [A] [A] 4(d) da 40 4 d d A d 80 4 [] At A = 95. cm, d A d 80 4 >0 minimum point. [A] 4(e) da Given cm s dt da da dt [] d dt d when = 4, da 40 4 d = 5.88 dt [] d d cm/s [A] dt

88 7 5(a)(i) cos sin 0 cos 4sin cos 0 cos ( 4sin ) 0 [] cos 0 or 4sin 0 sin 4 90, ,.4 [A] 5(a)(ii) sec 4 tan 6 tan 4 tan 6 [] tan 4 tan 5 0 tan 5 tan 0 [] tan = 5 or tan = basic = basic = 45 = 5,78.7,5,58.7 [A] 5(b) sin 7 sin 5 tan cos 7 cos5 sin 7 sin 5 LHS = cos 7 cos cos sin = cos cos [M] = sin tan RHS cos [A] d d 6(a) 0 9 = 0 6 [] 60 [A] = 9 d d d d = e e [] d d e e [] 6(b) e = = ( ) e [A]

89 8 6(c) d In d 4 d = [ln ln 4 ] d [] = 4 [M] 4 [A] = 7(a) y a b y a b Y = mx + C y y [] Correct Graph B Correct Points B 7(b) a = gradient = b = y-intercept =.45 [B] [B]

90 9 7(c) Draw the line Y = X [B] b a a = b = a + b Points of intersection of the line Y = X and Y = ax + b (4.6,.8) [A] 8. s t 4t 5t (a) initial velocity = ds initial acceleration = dt t0 = t 8t 5 [] = 5 m/s [A] = t 8 ds dt t0 [] = 8 m/s [A]

91 0 8(b) Minimum velocity t 8 = 0 t = 4 s ds 0 dt [] min velocity = t 8t 5 = m/s [A] 8(c) t 8t 5 < 0 t5 t 0 [] t 5 [A] 9. 4 y (a) when y = 0, = P(, 0) [B] (b) Gradient at point P = = d 4 d d d = d d [] = [] Normal gradient = [] Eqn of normal at point P: y [A] 0(a) Gradient of AD = Gradient of line to AD = [] Eqn of CD, y 5 ( 6) y 5 [A]

92 0(b) y y = --- = = 56 = 4 y = C(4, ) [A] Midpoint of AC and BD is the same. 6 y 5 4 4,, =, y = 7 B(, 7) [] [A] 0(c) Length of AB units = [B] 0(d) Area of ABCD = = 80 [] = 80 units [A]

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

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

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

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2) . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

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

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

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series

More information

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions

Pure Mathematics Year 1 (AS) Unit Test 1: Algebra and Functions Pure Mathematics Year (AS) Unit Test : Algebra and Functions Simplify 6 4, giving your answer in the form p 8 q, where p and q are positive rational numbers. f( x) x ( k 8) x (8k ) a Find the discriminant

More information

Learning Objectives These show clearly the purpose and extent of coverage for each topic.

Learning Objectives These show clearly the purpose and extent of coverage for each topic. Preface This book is prepared for students embarking on the study of Additional Mathematics. Topical Approach Examinable topics for Upper Secondary Mathematics are discussed in detail so students can focus

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

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16) Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +

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

Core Mathematics 2 Coordinate Geometry

Core Mathematics 2 Coordinate Geometry Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Coordinate Geometry 1 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle

More information

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5

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

Add Math (4047/02) Year t years $P

Add Math (4047/02) Year t years $P Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The

More information

Possible C2 questions from past papers P1 P3

Possible C2 questions from past papers P1 P3 Possible C2 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P1 January 2001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.

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

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

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

Pure Core 2. Revision Notes

Pure Core 2. Revision Notes Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic

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

MT - MATHEMATICS (71) GEOMETRY - PRELIM II - PAPER - 6 (E)

MT - MATHEMATICS (71) GEOMETRY - PRELIM II - PAPER - 6 (E) 04 00 Seat No. MT - MTHEMTIS (7) GEOMETRY - PRELIM II - (E) Time : Hours (Pages 3) Max. Marks : 40 Note : ll questions are compulsory. Use of calculator is not allowed. Q.. Solve NY FIVE of the following

More information

( ) 2 + 2x 3! ( x x ) 2

( ) 2 + 2x 3! ( x x ) 2 Review for The Final Math 195 1. Rewrite as a single simplified fraction: 1. Rewrite as a single simplified fraction:. + 1 + + 1! 3. Rewrite as a single simplified fraction:! 4! 4 + 3 3 + + 5! 3 3! 4!

More information

Part (1) Second : Trigonometry. Tan

Part (1) Second : Trigonometry. Tan Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,

More information

Circles - Edexcel Past Exam Questions. (a) the coordinates of A, (b) the radius of C,

Circles - Edexcel Past Exam Questions. (a) the coordinates of A, (b) the radius of C, - Edecel Past Eam Questions 1. The circle C, with centre at the point A, has equation 2 + 2 10 + 9 = 0. Find (a) the coordinates of A, (b) the radius of C, (2) (2) (c) the coordinates of the points at

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

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

MT - GEOMETRY - SEMI PRELIM - I : PAPER - 4

MT - GEOMETRY - SEMI PRELIM - I : PAPER - 4 07 00 MT A.. Attempt ANY FIVE of the following : (i) Slope of the line (m) 4 y intercept of the line (c) 0 By slope intercept form, The equation of the line is y m + c y (4) + (0) y 4 MT - GEOMETRY - SEMI

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

DEPARTMENT OF MATHEMATICS

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 2 - C2 2015-2016 Name: Page C2 workbook contents Algebra Differentiation Integration Coordinate Geometry Logarithms Geometric series Series

More information

Cambridge International Examinations Cambridge Ordinary Level

Cambridge International Examinations Cambridge Ordinary Level Cambridge International Examinations Cambridge Ordinary Level *054681477* ADDITIONAL MATHEMATICS 4037/11 Paper 1 May/June 017 hours Candidates answer on the Question Paper. No Additional Materials are

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

QUESTION BANK ON STRAIGHT LINE AND CIRCLE

QUESTION BANK ON STRAIGHT LINE AND CIRCLE QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,

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

Edexcel New GCE A Level Maths workbook Circle.

Edexcel New GCE A Level Maths workbook Circle. Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint

More information

AP Calculus I and Calculus I Summer 2013 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS?

AP Calculus I and Calculus I Summer 2013 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS? Name: AP Calculus I and Calculus I Summer 0 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS? Calculus is a VERY RIGOROUS course and completing this packet with your best effort will help you

More information

Solutions to O Level Add Math paper

Solutions to O Level Add Math paper Solutions to O Level Add Math paper 04. Find the value of k for which the coefficient of x in the expansion of 6 kx x is 860. [] The question is looking for the x term in the expansion of kx and x 6 r

More information

MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2

MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2 Class Inde Number Name MRIS STELL HIGH SCHOOL PRELIMINRY EXMINTION DDITIONL MTHEMTICS 406/0 8 September 008 Paper hours 0minutes dditional Materials: nswer Paper (6 Sheets RED THESE INSTRUCTIONS FIRST

More information

ARE YOU READY FOR CALCULUS?? Name: Date: Period:

ARE YOU READY FOR CALCULUS?? Name: Date: Period: ARE YOU READY FOR CALCULUS?? Name: Date: Period: Directions: Complete the following problems. **You MUST show all work to receive credit.**(use separate sheets of paper.) Problems with an asterisk (*)

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

6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary

6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary 6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3

More information

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

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. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by

MATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli

More information

Sec 4 Maths SET D PAPER 2

Sec 4 Maths SET D PAPER 2 S4MA Set D Paper Sec 4 Maths Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e Answer all questions. Write your answers and working on the separate Answer Paper provided.

More information

AP Calculus AB/BC ilearnmath.net

AP Calculus AB/BC ilearnmath.net CALCULUS AB AP CHAPTER 1 TEST Don t write on the test materials. Put all answers on a separate sheet of paper. Numbers 1-8: Calculator, 5 minutes. Choose the letter that best completes the statement or

More information

Mathematics Extension 2

Mathematics Extension 2 009 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions o Reading Time- 5 minutes o Working Time hours o Write using a blue or black pen o Approved calculators may be

More information

MATHEMATICS EXTENSION 2

MATHEMATICS EXTENSION 2 Sydney Grammar School Mathematics Department Trial Eaminations 008 FORM VI MATHEMATICS EXTENSION Eamination date Tuesday 5th August 008 Time allowed hours (plus 5 minutes reading time) Instructions All

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

Brief Revision Notes and Strategies

Brief Revision Notes and Strategies Brief Revision Notes and Strategies Straight Line Distance Formula d = ( ) + ( y y ) d is distance between A(, y ) and B(, y ) Mid-point formula +, y + M y M is midpoint of A(, y ) and B(, y ) y y Equation

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

DISCRIMINANT EXAM QUESTIONS

DISCRIMINANT EXAM QUESTIONS DISCRIMINANT EXAM QUESTIONS Question 1 (**) Show by using the discriminant that the graph of the curve with equation y = x 4x + 10, does not cross the x axis. proof Question (**) Show that the quadratic

More information

CAMI Education links: Maths NQF Level 4

CAMI Education links: Maths NQF Level 4 CONTENT 1.1 Work with Comple numbers 1. Solve problems using comple numbers.1 Work with algebraic epressions using the remainder and factor theorems CAMI Education links: MATHEMATICS NQF Level 4 LEARNING

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

Circles, Mixed Exercise 6

Circles, Mixed Exercise 6 Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5

More information

1 / 23

1 / 23 CBSE-XII-07 EXAMINATION CBSE-X-009 EXAMINATION MATHEMATICS Series: HRL Paper & Solution Code: 0/ Time: Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question paper

More information

MATHEMATICS Higher Grade - Paper I (Non~calculator)

MATHEMATICS Higher Grade - Paper I (Non~calculator) Prelim Eamination 005 / 006 (Assessing Units & ) MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefully. Calculators may not be used in this paper.. Full credit

More information

The Fundamental Theorem of Calculus Part 3

The Fundamental Theorem of Calculus Part 3 The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative

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

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS

SURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -

More information

2013 Bored of Studies Trial Examinations. Mathematics SOLUTIONS

2013 Bored of Studies Trial Examinations. Mathematics SOLUTIONS 03 Bored of Studies Trial Examinations Mathematics SOLUTIONS Section I. B 3. B 5. A 7. B 9. C. D 4. B 6. A 8. D 0. C Working/Justification Question We can eliminate (A) and (C), since they are not to 4

More information

x n+1 = ( x n + ) converges, then it converges to α. [2]

x n+1 = ( x n + ) converges, then it converges to α. [2] 1 A Level - Mathematics P 3 ITERATION ( With references and answers) [ Numerical Solution of Equation] Q1. The equation x 3 - x 2 6 = 0 has one real root, denoted by α. i) Find by calculation the pair

More information

Secondary School Mathematics & Science Competition. Mathematics. Date: 1 st May, 2013

Secondary School Mathematics & Science Competition. Mathematics. Date: 1 st May, 2013 Secondary School Mathematics & Science Competition Mathematics Date: 1 st May, 01 Time allowed: 1 hour 15 minutes 1. Write your Name (both in English and Chinese), Name of School, Form, Date, Se, Language,

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

Higher Mathematics Course Notes

Higher Mathematics Course Notes Higher Mathematics Course Notes Equation of a Line (i) Collinearity: (ii) Gradient: If points are collinear then they lie on the same straight line. i.e. to show that A, B and C are collinear, show that

More information

MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D.

MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maximum Marks : 100. [ Q. 1 to 60 carry one mark each ] A. 0 B. 1 C. 2 D. M 68 MULTIPLE CHOICE QUESTIONS SUBJECT : MATHEMATICS Duration : Two Hours Maimum Marks : [ Q. to 6 carry one mark each ]. If sin sin sin y z, then the value of 9 y 9 z 9 9 y 9 z 9 A. B. C. D. is equal

More information

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP

Trans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP Solved Examples Example 1: Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5. Method 1. Consider the equation (x + y 6) (2x + y 4) + λ 1

More information

CBSE MATHEMATICS (SET-2)_2019

CBSE MATHEMATICS (SET-2)_2019 CBSE 09 MATHEMATICS (SET-) (Solutions). OC AB (AB is tangent to the smaller circle) In OBC a b CB CB a b CB a b AB CB (Perpendicular from the centre bisects the chord) AB a b. In PQS PQ 4 (By Pythagoras

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

Special Mathematics Notes

Special Mathematics Notes Special Mathematics Notes Tetbook: Classroom Mathematics Stds 9 & 10 CHAPTER 6 Trigonometr Trigonometr is a stud of measurements of sides of triangles as related to the angles, and the application of this

More information

Sec 4 Maths. SET A PAPER 2 Question

Sec 4 Maths. SET A PAPER 2 Question S4 Maths Set A Paper Question Sec 4 Maths Exam papers with worked solutions SET A PAPER Question Compiled by THE MATHS CAFE 1 P a g e Answer all the questions S4 Maths Set A Paper Question Write in dark

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

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

Algebra y funciones [219 marks]

Algebra y funciones [219 marks] Algebra y funciones [9 marks] Let f() = 3 ln and g() = ln5 3. a. Epress g() in the form f() + lna, where a Z +. attempt to apply rules of logarithms e.g. ln a b = b lna, lnab = lna + lnb correct application

More information

FILL THE ANSWER HERE

FILL THE ANSWER HERE HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP. If A, B & C are matrices of order such that A =, B = 9, C =, then (AC) is equal to - (A) 8 6. The length of the sub-tangent to the curve y = (A) 8 0 0 8 ( ) 5 5

More information

Mathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman

Mathematics syllabus for Grade 11 and 12 For Bilingual Schools in the Sultanate of Oman 03 04 Mathematics syllabus for Grade and For Bilingual Schools in the Sultanate of Oman Prepared By: A Stevens (Qurum Private School) M Katira (Qurum Private School) M Hawthorn (Al Sahwa Schools) In Conjunction

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

Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours

Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mark scheme Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question

More information

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS

WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS Surname Centre Number Candidate Number Other Names 0 WJEC LEVEL 2 CERTIFICATE 9550/01 ADDITIONAL MATHEMATICS A.M. MONDAY, 22 June 2015 2 hours 30 minutes S15-9550-01 For s use ADDITIONAL MATERIALS A calculator

More information

MATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT )

MATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT ) Total No. of Printed Pages 6 X/5/M 0 5 MATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 80 Pass Marks : 4 ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 00

More information

Objective Mathematics

Objective Mathematics . A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four

More information

Newbattle Community High School Higher Mathematics. Key Facts Q&A

Newbattle Community High School Higher Mathematics. Key Facts Q&A Key Facts Q&A Ways of using this booklet: 1) Write the questions on cards with the answers on the back and test yourself. ) Work with a friend who is also doing to take turns reading a random question

More information

International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS PAPER 2 MAY/JUNE SESSION 2002

International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS PAPER 2 MAY/JUNE SESSION 2002 International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL EXAMINATIONS ADDITIONAL MATHEMATICS 0606/2 PAPER 2 MAY/JUNE SESSION 2002 2 hours Additional materials: Answer paper Electronic

More information

1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)

1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given

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

Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.

Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to

More information

+ 2gx + 2fy + c = 0 if S

+ 2gx + 2fy + c = 0 if S CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the

More information

SAMPLE QUESTION PAPER Class-X ( ) Mathematics. Time allowed: 3 Hours Max. Marks: 80

SAMPLE QUESTION PAPER Class-X ( ) Mathematics. Time allowed: 3 Hours Max. Marks: 80 SAMPLE QUESTION PAPER Class-X (017 18) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

1 / 23

1 / 23 CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question

More information

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse

More information

Mathematics Class X Board Paper 2011

Mathematics Class X Board Paper 2011 Mathematics Class X Board Paper Solution Section - A (4 Marks) Soln.. (a). Here, p(x) = x + x kx + For (x-) to be the factor of p(x) = x + x kx + P () = Thus, () + () k() + = 8 + 8 - k + = k = Thus p(x)

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

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y.

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y. PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line 3x 4 k 0 is

More information

*P46958A0244* IAL PAPER JANUARY 2016 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA. 1. f(x) = (3 2x) 4, x 3 2

*P46958A0244* IAL PAPER JANUARY 2016 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA. 1. f(x) = (3 2x) 4, x 3 2 Edexcel "International A level" "C3/4" papers from 016 and 015 IAL PAPER JANUARY 016 Please use extra loose-leaf sheets of paper where you run out of space in this booklet. 1. f(x) = (3 x) 4, x 3 Find

More information

Objective Mathematics

Objective Mathematics Chapter No - ( Area Bounded by Curves ). Normal at (, ) is given by : y y. f ( ) or f ( ). Area d ()() 7 Square units. Area (8)() 6 dy. ( ) d y c or f ( ) c f () c f ( ) As shown in figure, point P is

More information

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436) HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum

More information

CBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80

CBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80 CBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided

More information

Solutionbank C1 Edexcel Modular Mathematics for AS and A-Level

Solutionbank C1 Edexcel Modular Mathematics for AS and A-Level Heinemann Solutionbank: Core Maths C Page of Solutionbank C Exercise A, Question Find the values of x for which f ( x ) = x x is a decreasing function. f ( x ) = x x f ( x ) = x x Find f ( x ) and put

More information