h2 math (pure math consultation)

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www.teachmejcmath-sg.webs.com H2 Consultation Exercise Note: Students should take 1 hour to complete the assignment. Topic: Graphing Technique 1 – Rational function 1. Consider the curve C: ) 1 )( 2 ( 2 - + - = x x x y i) State the equations of the asymptotes of C. ii) State the coordinates of any turning points and axial intercepts of C. iii) Sketch the curve C. Find the range of values of k for which ) 1 )( 2 ( 2 - + - = x x x k has at least 1 real root and k is a positive integer. 2. The curve has equation λ λ λ - + - = x x x y 3 2 2 . i) Find the conditions for λ such that C has no real roots. ii) Give sketches of C for a) 0 < λ < 3 b) λ < 0 Hence deduce the number of real roots for the equation ) 1 ( ) 3 2 ( 2 + = - + x k x x x where k is positive. 3. The curve C has equation 1 5 2 + + = x x y λ where λ is a non-zero constant. i) Obtain the equations of asymptotes of C ii) Show that C has exactly two points at which 0 = dx dy iii) Draw a sketch of C to illustrate the case λ > 0, and a separate sketch to illustrate the case λ < 0. iv) Find the set of values of λ for which the line y = 4x and C have at least one point in common. [Source: FM P1 Nov 99] 4. The curve C has equation 2 ax bx c y x d + + = + . It is known that C, that has asymptotes 3 x = and 1 y x = - , cuts the y-axis at –2. (i) Find the values of a, b, c and d. (ii) Sketch the curve C, stating clearly the turning points, asymptotes and any intercepts with the axes. Hence deduce the range of values of k such that the equation

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H2 Mathematics : Pure Math

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  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise Note: Students should take 1 hour to complete the assignment.

    Topic: Graphing Technique 1 Rational function

    1. Consider the curve C: )1)(2(2

    +

    =

    xx

    xy

    i) State the equations of the asymptotes of C. ii) State the coordinates of any turning points and axial intercepts of C. iii) Sketch the curve C.

    Find the range of values of k for which )1)(2(2

    +

    =

    xx

    xk has at least 1 real root and k is

    a positive integer.

    2. The curve has equation

    +=

    x

    xxy 322

    .

    i) Find the conditions for such that C has no real roots. ii) Give sketches of C for

    a) 0 < < 3 b) < 0

    Hence deduce the number of real roots for the equation )1()32( 2 +=+ xkxxx where k is positive.

    3. The curve C has equation 152

    +

    +=

    x

    xy where is a non-zero constant. i) Obtain the equations of asymptotes of C ii) Show that C has exactly two points at which 0=

    dxdy

    iii) Draw a sketch of C to illustrate the case > 0, and a separate sketch to illustrate the case < 0.

    iv) Find the set of values of for which the line y = 4x and C have at least one point in common.

    [Source: FM P1 Nov 99]

    4. The curve C has equation 2ax bx cyx d+ +

    =

    +. It is known that C, that has

    asymptotes 3x = and 1y x= , cuts the y-axis at 2.

    (i) Find the values of a, b, c and d. (ii) Sketch the curve C, stating clearly the turning points, asymptotes and any

    intercepts with the axes. Hence deduce the range of values of k such that the equation

  • www.teachmejcmath-sg.webs.com

    2 ( ) 0ax b k x c kd+ + = has two distinct real roots.

    (iii) Sketch the graph of 2

    ax b x cy

    x d+ +

    =

    +

    stating clearly the turning points, asymptotes and any intercepts with the axes . [Source: SRJC Prelim 2007]

  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise Note: Students should aim to complete this assignment within an hour.

    Topic: Graphing Technique 2 Transformation

    1. The graph y = f(x) is shown below:

    On separate diagrams, sketch the following graphs, showing clearly all relevant coordinates: a) f (1 2 )y x= + e) 2 f ( )y x= b) 3f ( )y x= f) 1

    f ( )y x=

    c) ( )f 1y x= + g) f ( )y x= d) 2 f ( )y x+ = Label your graphs clearly.

    2. The transformations A, B, C and D are given as follows: A: A translation of magnitude 3 units in the negative direction of the x-axis. B: A reflection about the y-axis. C: A translation of magnitude 2 units in the direction of the y-axis. D: A scaling parallel to the y-axis by a factor 3.

    A curve undergoes in succession, the transformations A, B, C and D as above and the equation of the resulting curve is ( ).3ln36 xy =

    Determine the equation of the curve before the transformations were effected.

    3. Give a geometrical description of the following two curves G and H: 2 2

    2 2

    : ( 1) ( 1) 4: ( 1) (2 2) 4

    G x yH x y

    + + =

    + + =

    Sketch, on the same diagram, the graphs of G and H. Describe a sequence of transformations that transforms the graph of G onto the graph of H.

    2 0 x

    y

    (1, -1)

    (3, 1)

    2.5

  • www.teachmejcmath-sg.webs.com

    4. The diagram below shows the graph of ( )fy x= . The points A, B and C have coordinates 1( ,0)

    2 , (1, 2) and (3, 1) respectively.

    Sketch on separate diagrams, the graphs of

    (i) f2xy =

    , (ii) f 1

    2xy =

    (iii) ( )f 'y x= , where ( ) df f ( )d

    ' x xx

    = . [5]

    State, in each case, the coordinates of the points corresponding to A, B and C, where appropriate. [Source: RJC Promo 06]

    5. The graphs of y = | h(x) | and y2 = h(x) for x > 3 are as shown below:

    (i) Explain why h(x) < 0 for 1 < x < 2. [1] Hence sketch the graph of y = h(x) for x > 3, showing clearly the asymptote and the coordinates of the stationary point. [1]

    (ii) Sketch the graphs of

    (a) y = h(x) where h is the derivative function of h, [2]

    y

    x

    ( 16

    , 23

    ) )

    y = | h(x) |

    2

    0

    3

    1

    y2 = h(x)

    x

    2

    0

    3

    1

    x

    y ( )fy x=

    C(3,1)

    B(1,2)

    12( ,0)A

    0

  • www.teachmejcmath-sg.webs.com

    (b) y = 1h( )x for x > 3, [3]

    showing clearly all the asymptote(s) and the coordinates of the stationary point(s).

    [Source: RJC Prelim 07]

  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise

    Topic: Functions and Inequalities

    1. The functions f and g are defined by: 1,),1ln(: >+ xxxxf a

    xxxxg ,44: 2a i. By means of a graphical argument, or otherwise, show that 1g does not exist.

    The function h is defined by kxxxxxh

  • www.teachmejcmath-sg.webs.com

    4(a) The function f is defined by 34)( 2 ++= xxxf , x 3. i) Sketch the graph of f and explain why f -1 does not exist. ii) Write down the value of such that (,] is the largest domain of f which f -1

    exists. Using this new domain of f,

    ii) Express f -1 in similar form.

    (b) The functions f and g are defined as follow: 31,)( 2 = xxxf ; 0,)( 9 = xexf x

    i) Explain why gf exists ii) Express gf in a similar manner and find its range in exact form.

    5. Solve the following inequalities without using the calculator

    (a) 1 52

    xx

    + > (b) 2

    25 4 02 3

    x x

    x x

    +

    .

    9(i) By completing the square or otherwise, show that 3+x2+x 2 is always positive for all real values of x.

    (ii) Solve the inequality x

    3< x - 2.

    10 Solve each of the following inequalities without using calculator.

    i) x

    x2

    > ii) 132 +

  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise Note: Students should aim to complete this assignment within 2 hrs.

    Topic: Sequence and Series Part 1

    1. Find the sum of the series .149...11852 22222 +++++

    2. Find =

    +n

    r

    rn0

    2 )31( in terms of n, simplifying your answer.

    3. Show that )54)(1(61)12(

    1++=+

    =

    nnnrrn

    r

    .

    Hence, evaluate =

    +30

    10)12(

    r

    rr .

    4. Express )2(2+rr

    in partial fractions and hence show that

    )2)(1(32

    23

    )2(2

    1 ++

    +=

    +

    =nn

    n

    rr

    n

    r

    Find the value of

    =+1 )2(1

    r rr.

    5. Express nn 3

    1 in partial fractions.

    Hence, by using method of differences, find the sum of the series

    =

    N

    n nn231

    .

    6. Find an expression for the nth term of the series 2 + 22 + 222 + 2222 +

    and deduce that the sum of the first n terms of the series is

    92)110(

    8120 nn

    .

    7. Use induction to prove that 1)!()1()!)(1(...)!3(13)!2(7)!1(3 22 +=++++++ nnnnn

    8. If 11 23 + = nnn UUU for all positive integral values of n and 21 =U , and 62 =U , give a conjecture for nU in terms of n. Prove the validity of your

    conjecture using Mathematical Induction.

  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise Note: Students should aim to complete this assignment within 2 hrs.

    Topic: Sequence and Series Part 2

    1. The rth term of a series is given by 21 3r . i) Show that the series is an arithmetic series. ii) Find the sum of the first n terms of the series. iii) Find the nth term given that the sum of the first n terms is zero.

    Ans: (39 3 ) ;132

    n n

    2. The sum of the first 20 terms of an arithmetic progression is 50 and the sum of the next 20 terms is 50. Find the sum of the first 100 terms of this progression.

    Ans: 750

    3. An arithmetic progression has first term, a and common difference, 10. The sum of the first n terms is 10000. Express a in terms of n, and show that the nth term is

    ( )10000 5 1nn

    + . Given that the nth term is less than 500, show that

    020001012

  • www.teachmejcmath-sg.webs.com

    6. Write down the first four terms in the binomial expansion, in ascending powers of x, of the following expressions. In each case, state the range of values of x for

    which the expansion is valid.

    (a) ( ) 321 + x (b) 2336

    x

    +

    (c) 1 3

    1x

    x

    +

    7. Give the first four terms in the expansion, in descending powers of x, of 21

    2

    +

    xx .

    State the range of values of x for which the expansion is valid. Hence evaluate

    4 5. correct to four decimal places.

  • H2 Consultation Exercise Note: Students should take 2.5 hour to complete the assignment. Answers are provided in [ ].

    Topic: Differentiation

    Techniques

    1. Differentiate with respect to x:

    a) x1sin b) 22ln xk + , where k is a constant

    c) 25lnxe

    x

    d)

    +

    21

    11

    tanx

    e) xln2

    2. i) For the curve yxyx += 2)( , show that 122122

    +

    =

    yxyx

    dxdy

    ii) Find the gradient of the curve at the points where it cuts the x-axis.

    Applications

    1. The vertical cross-section of a water trough is in the shape of an equilateral triangle with one vertex pointing down. The trough is 15m long. When the water in the trough

    is 1m deep, its depth is increasing at a rate of 15

    ms-1. At what rate is water flowing

    into the trough at that instant? [ 32 m /3 s]

    2. Two variables u and v are connected by the relation 1 1 1u v f+ = , where f is a

    constant. Given that u and v both vary with time, t , find an equation connecting ddu

    t,

    ddv

    t, u and v . Given also that u is decreasing at a rate of 2cm per second and

    that f =10cm, calculate the rate of increase of v when u = 50cm. [81

    cm/s]

    3. A beam is to be cut from a cylindrical log so that its cross-section is a rectangle. The log has diameter d and the beam is to have breadth x and depth y . Given that the

    stiffness of such a beam is proportional to 3xy , find, in terms of d , the values of x

    and y for the stiffest beam that can be cut from the log. [ dydx23

    ,

    2== ]

  • 4. The parametric equations of a curve are )ln(sin =x , 3ln(cos )y = , 2

    0 pi

  • www.teachmejcmath-sg.webs.com

    H2 Consultation Exercise Note: Students should take 1 hour to complete the assignment.

    Topic: Differentiation - Maclaurins Series

    Question 1 Given that y =

    x2cos12

    +, show that 02sin)2cos1( =+ xy

    dxdy

    x .

    Hence, by further differentiation of the result and using Maclaurins theorem, expand y in ascending powers of x, up to the term in x .

    Hence, by putting x = pi/6, find an approximate value of32

    , to three decimal places.

    Question 2

    Using standard series expansion or otherwise, find the series expansion of exsinx, up to and including the term in x5. Hence, find the series expansion of excosx.

    Question 3 If f(x) = xecos , prove that 0=)()(cos+)(')(sin+)(" xfxxfxxf . By differentiating the result further and assuming that xecos can be expanded in a series of ascending powers of x, show that, neglecting powers of x above the fourth,

    xecos =

    + 42

    61

    21

    -1 xxe .

    Hence or otherwise, find the Maclaurin expansion of (sinx) xecos , up to the 3x term.

    Question 4

    A curve C is defined by the equation 12 2 = ydxdyy and (0,3) is a point on C.

    i) Find the Maclaurins series of y up to and including the term in x3. ii) If x is sufficiently small for x3 and higher powers of x to be neglected, show

    that 25429

    353 xxye x + .

  • www.teachmejcmath-sg.webs.com

    Observe, Compare, Substitute

    H2 Consultation Exercise Note: Students should take 2 hours to complete the assignment.

    Topic: Integration Techniques

    Question 1 (a) By using the substitution x = 2 sin, find dx

    x

    x

    +24

    1.

    (b) Find dxxx

    x ++

    +

    3212

    2

    (c) (i) Prove that 2

    2

    11

    x

    xx

    dxd

    = , and find 232 )1( x

    dxd

    .

    (ii) Hence find dxx

    x

    2

    3

    1.

    Question 2 (a) Show that 2 4 20x x+ + can be expressed in the form ( )2x A B+ + , where A and

    B are constants to be determined.

    Hence find 21 d

    4 20x

    x x+ +.

    (b) Find 224 e dxx x .

    Hence find 23 2e dxx x .

    (c) By means of the substitution 21

    xu

    = , find 1 d1

    xx x .

    [Source: RJC Promo 2006]

    Question 3

    a) dxx

    e x

    2sin 22cot

    b) dxxx 3cos5cos c) ( ) + dxxx 1ln 2 d) +

    dxxx

    x

    21

    1

    Question 4

    Show that 1

    111

    2

    =

    +

    x

    x

    x

    x

    Hence, or otherwise, using the substitution x = sec , find dxx

    x +

    11

    .

  • www.teachmejcmath-sg.webs.com

    Observe, Compare, Substitute

    Question 5

    i) Show by means of substitution t = cosx that 2cos1

    sin0 2

    3=

    +pi

    pidx

    x

    x.

    ii) Show that .3sec3tan3)(tan 243 += dd

    Hence, find 404tan

    pi

    d .

    Question 6 a) Find i) dx

    x

    x

    +2416

    1 ii) dxx + )4ln( 2

    b) Using the substitution x = 2cos, find dxx 2416 . Hence, find the area of the

    region bounded by part of the curve y = 2416 x between x = 0 and x = 2. Give your answer in exact form.

    Question 7 a) Find ( )( ) dxxex + tancos 12 b) Show that ( )2

    812cos

    4/

    0

    = pipi

    dxxx . Hence, evaluate dxxx4/

    0

    2cospi

    .

    c) By means of substitutionu

    x12

    = , evaluate the integral dxxx

    1

    2/122 15

    1.

  • www.teachmejcmath-sg.webs.com

    Observe, Compare, Substitute

    H2 Consultation Exercise Note: Students should take 3 hours to complete the assignment.

    Topic: Integration Area & Volume and Differential Equations

    Question 1 Sketch the curve 24

    1x

    y+

    = (x0), and the lines y = 41

    and x = 2.

    The region bounded by the curve 241

    xy

    += and the lines x = 2, y =

    41

    is denoted by R.

    a) Show that the area of R is given by dxx

    +

    2

    0 241

    21

    .

    Hence, find the area of R in terms of pi.

    b) Use substitution x = 2 tan to show that =+4

    022

    0 2cos

    81

    41 pi ddx

    x.

    Evaluate this integral and hence find, in terms of pi, the exact volume of the solid generated when the region R is rotated completely about the x axis.

    c) Show that the volume of the solid obtained when the region R is rotated completely

    about the line x = 2 can be expressed in the form

    41

    81

    414)2(ln dyy

    ypipi .

    By using the substitution 2sin41

    =y or otherwise, evaluate the expression exactly,

    leaving your answer in terms of pi.

    Question 2 A curve has equation y = ( ) 2/124 x for -1 x 1. The region R is enclosed by C, the x-axis and the lines x = -1 and x = 1.

    a) Find the exact area of R. b) Show that the volume generated when R is rotated through two right angles about the

    y-axis is )324( pi .

    Question 3 (a) (i) Find the exact value of xxx d12

    1

    32

    + .

    (ii) Use the substitution 22 += xu to find dxxx

    + 2)2(1

    (b) The diagram shows part of the graph1

    12 +

    =

    xy .

  • www.teachmejcmath-sg.webs.com

    Observe, Compare, Substitute

    By considering n+1 rectangles of equal width from x = 0 to x = 3, show that

    = ++

    = dtd

    Initially the temperature of the hot coffee is 85 C. How long does it take for the coffee to cool down from 85 C to 60 C? Give your answer to the nearest minute.

    Question 6 Solve the following ODE

    a) xx e=dxdy

    .y)e+1(

    y

    x 0

  • www.teachmejcmath-sg.webs.com

    Observe, Compare, Substitute

    b) By means of a substitution, 2y=u , solve the differential equation

    dxdy

    y2 - x=xy2 . Express your answer in a form expressing y in terms of x.

    Question 7

    A cylindrical container has a height of 200 cm. The container was initially full of a chemical but there is a leak from a hole in the base. When the leak is noticed, the container is half-full and the level of the chemical is dropping at a rate of 1 cm per minute. It is required to find for how many minutes the container has been leaking.

    To model the situation it is assumed that, when the depth of the chemical remaining is x cm, the rate at which the level is dropping is proportional to x . Set up and solve an appropriate differential equation, and hence show that the container has been leaking for about 80 minutes.

    x

  • www.teachmejcmath-sg.webs.com

    Confidence+Concepts+Consistency+Composure-Complacency

    H2 Consultation Exercise Note: Students should take 1.5 hours to complete the assignment.

    Topic: Complex Numbers

    Question 1

    a) Find the modulus and argument of the complex number ii

    123

    .

    b) The complex number q is given by q =

    i

    i

    e

    e

    1, where 0 < < 2pi. In either

    order, i) find the real part of q.

    ii) show that the imaginary part of q is

    21

    cot21

    .

    c) The complex numbers z and w are such that |z| = 2, arg(z) = pi32

    , |w| = 5,

    arg(w) = pi43

    . Find the exact values of

    i) the real and imaginary part of z.

    ii) the modulus and argument of 2zw

    .

    Question 2 Two complex numbers p and q are given respectively by p = 3 2i and q = 1 + 3i. show on separate diagrams.

    i) |q| < |z| < |p| ii) arg(z) = arg(pq).

    Solve the equation 013 36 =+ zz , giving your answers in the form ire .

    Question 3 a) Sketch on an Argand diagram the locus of the point Z representing the

    complex number z such that | z + i | = 1. Hence, show that

    6)2arg(

    6pipi ++ iz .

  • www.teachmejcmath-sg.webs.com

    Confidence+Concepts+Consistency+Composure-Complacency

    b) i) Find the fourth roots of the complex number )31(8 i+ . Give your answers

    exactly in the form ire . Hence, sketch the roots of )31(8 i+ on an Argand diagram.

    ii) Consider the complex number

    2

    3

    cos isin4 4

    cos isin3 3

    z

    pi pi

    pi pi

    =

    +

    .

    Find the modulus and the exact value of the argument of z.

  • www.teachmejcmath-sg.webs.com

    Confidence+Concepts+Consistency+Composure-Complacency

    H2 Consultation Exercise Note: Students should take 1 hour to complete the assignment.

    Topic: Vectors

    Question 1

    The plane pi1 and the line l1 have equations 81

    12

    .:1 =

    rpi and

    +

    =

    152

    513

    :1 rl

    respectively where is a constant.

    i) Find the position vector of the point N, the foot of perpendicular from B(3,1,5) to pi1. Hence or otherwise, find the exact distance between pi1 and the line l1.

    ii) Find the Cartesian equation of the plane pi2 which contains the line l1 and the line through B and N.

    Question 2 Two straight lines l1 and l2 have equations given by

    +

    =

    513

    148

    20:1 rl and

    +

    =

    034

    12

    23:2 rl

    a) Show that l1 and l2 intersect and find the coordinates of the point of intersection, C.

    b) Given that A is the point on l1 with parameter =2. Obtain the position vector of the foot F of the perpendicular from A to l2.

    c) Show that an equation for the common perpendicular, p, to l1 and l2 through C is given by

    +

    =

    143

    111

    11: trp

    d) D is the point with parameter 3 on p. Obtain the length of projection of vector AD on l2.

  • www.teachmejcmath-sg.webs.com

    Confidence+Concepts+Consistency+Composure-Complacency

    Question 3

    Consider the two lines

    =+

    =+

    00

    :1zyyx

    l and

    +=

    =

    +=

    tz

    ytx

    l1

    121

    :2

    a) Find the equation of the plane 1 which contains l1 and is parallel to l2. b) Find the equation of the plane 2 which contains l2 and is perpendicular

    to 1 .

    Find the coordinates of the foot of perpendicular, Q, from the point P (1, 1, 1) to the plane 1 . Hence determine the shortest distance between l1 and l2.