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    Senior Maths Olympiads

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    mwabamatutu
    The document provides instructions and 10 tasks for a 3-hour senior mathematics olympiad involving 4 candidates from David Kaunda National Technical High School. The tasks cover a range of mathematics topics including algebra, trigonometry, calculus, logarithms and coordinate geometry. Candidates are instructed to show their working and attempt all tasks.

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    © All Rights Reserved
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    Senior Maths Olympiads

    Uploaded by

    mwabamatutu
    26 views5 pages
    The document provides instructions and 10 tasks for a 3-hour senior mathematics olympiad involving 4 candidates from David Kaunda National Technical High School. The tasks cover a range of mathematics topics including algebra, trigonometry, calculus, logarithms and coordinate geometry. Candidates are instructed to show their working and attempt all tasks.

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    The document provides instructions and 10 tasks for a 3-hour senior mathematics olympiad involving 4 candidates from David Kaunda National Technical High School. The tasks cover a range of mathematics topics including algebra, trigonometry, calculus, logarithms and coordinate geometry. Candidates are instructed to show their working and attempt all tasks.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    26 views5 pages

    Senior Maths Olympiads

    Uploaded by

    mwabamatutu
    The document provides instructions and 10 tasks for a 3-hour senior mathematics olympiad involving 4 candidates from David Kaunda National Technical High School. The tasks cover a range of mathematics topics including algebra, trigonometry, calculus, logarithms and coordinate geometry. Candidates are instructed to show their working and attempt all tasks.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 5

    LUSAKA PROVINCE JETS

    SUBJECT: SENIOR MATHEMATICS


    OLYMPAIDS

    DURATION: 3 HOURS

    VENUE: DAVID KAUNDA NATIONAL


    TECHNICAL HIGH SCHOOL

    INSTRUCTIONS TO CANDIDATES
    1. Write your name, age, grade and school
    on all your answer paper
    2. There are 10 tasks on this paper
    3. Attempt all the tasks.

    1
    1. x 2  8x  29  ( x  a)2  b ,
    Where a and b are constants.

    (a) Find the value of a and the value of b. [3]

    (b) Hence, or otherwise, show that the roots of

    x2  8x  29  0
    are c  d 5 , where c and d are integers to be found. [3]

    2.

    A fence from a point A to a point B is in the shape of an arc AB of a


    circle with centre O and radius 45m, as shown in Figure 1. The length
    of the fence is 63m.

    (a) Show that the size of AOB is exactly 1.4 radians. [2]

    The points C and D are on the lines OB and OA respectively, with


    OC = OD = 30m.

    A plot of land ABCD, shown shaded in Figure 1, is enclosed by the


    arc AB and the straight lines BC, CD and DA.

    (b) Calculate, to the nearest m2 , the area of this plot of land. [5]

    2
    3. Solve , for  900 <x<900, giving your answers to 1 decimal place,

    3
    (a) tan(3x  200 )  , [6]
    2

    10
    (b) 2 sin 2 x  cos 2 x  [4]
    9

    4. An arithmetic series has first term a and common difference d.

    (a) Prove that the sum of the first n terms of the series is
    n2a  (n  1)d  .
    1
    [4]
    2

    The rth term of a sequence is (5r – 2).

    (b) Write down the (x – 3)th term of this sequence. [1]


    n
    1
    (c) Show that  (5r  2)  2 n(5n  1) .
    r 1
    [3]

    200
    (d) Hence, or otherwise, find the value of  (5r  2) .
    r 5
    [4]

    5. The region enclosed by the curve with equation y  e2 x  4 , the x- axis,


    the y-axis and the line x = 2 is rotated through 3600 about the x- axis.
    Find , in terms of e and  , the volume of the solid generated. [5]

    d 1
    6. (a) Show that (tan )  [3]
    d cos 2 

    (b) Solve, in degree to one decimal place, the equation 5sin  7 cos ,
    00   < 360 .
    0
    [3]

    3
    7.
    Figure 2

    Figure 2 shows part of the curve C with equation


    3

    y  2 x 2  6 x  10, x  0 .
    The curve C passes through the point A(1, 6) and has a minimum turning
    point at B.

    (a) Show that the x-coordinate of B is 4. [3]

    The finite region R, shown shaded in Figure 2, is bounded by C and the


    straight line AB.

    (b) Find the exact area of R. [7]

    8. Solve
    (a) log q 343  3 [2]

    (b) log 4 (5n  9)  3 [3]

    (c) log m 4  8 log 4 m  6, [6]

    (d) 2 log3 x  3x log3 x  6 x  4. [5]

    4
    1
    1
    9. (a) Expand (1  x) 3 in ascending powers of x, up to and including the
    4
    term in x2, simplifying each term as far as possible. [3]
    1
    1 
    (b) Expand (1  x) 3 in ascending powers of x, up to and including the
    4
    term in x2, simplifying each term as far as possible. [3]

    (c) State the range of values of x for which both of your expansions
    are valid. [1]
    Using your answers to parts (a) and (b),
    1
    4  x 3
    (d) Expand   in ascending powers of x, up to and including the
    4 x
    term in x2, simplifying each term as far as possible. [3]
    1
    4 x
    (e) Hence obtain an estimate, to 3 significant figures, of  
    0.3 3
     dx .
    0 4 x
    [3]

    2x  1 1
    10. A curve has equation y  ,x   .
    4x  2 2
    (a) Write down an equation for the asymptote to the curve which is
    parallel to
    (i) the x-axis,
    (ii) the y-axis. [2]

    (b) Find the coordinates of the points where the curve crosses the
    coordinate axes. [2]

    (c) Sketch the curve, showing clearly the asymptotes and the
    coordinates of the points where the curve crosses the coordinate axes.
    [3]
    The curve intersects the y-axis at the point P.

    (d) Find an equation for the normal to the curve at P. [4]

    The normal at P meets the curve again at Q.


    (e) Find the coordinates of Q. [4]

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