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    Harkness Test 9b

    Uploaded by

    Lia C
    This document contains 8 mathematics problems: 1) Find values of A and B given an integral equation and find the exact value of another integral in logarithmic form. 2) Prove an identity about cosecant and cotangent and solve an equation involving them. 3) Find the exact value of the tangent of an angle between certain values. 4) Find inverse and composite functions involving several functions and solve an equation between two of them. 5) Calculate the temperature of a cup of coffee at a later time, given the initial temperature and cooling rate. 6) Evaluate two integrals involving trigonometric and polynomial functions. 7) Find the first three non-

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    Harkness Test 9b

    Uploaded by

    Lia C
    46 views1 page
    This document contains 8 mathematics problems: 1) Find values of A and B given an integral equation and find the exact value of another integral in logarithmic form. 2) Prove an identity about cosecant and cotangent and solve an equation involving them. 3) Find the exact value of the tangent of an angle between certain values. 4) Find inverse and composite functions involving several functions and solve an equation between two of them. 5) Calculate the temperature of a cup of coffee at a later time, given the initial temperature and cooling rate. 6) Evaluate two integrals involving trigonometric and polynomial functions. 7) Find the first three non-

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    This document contains 8 mathematics problems: 1) Find values of A and B given an integral equation and find the exact value of another integral in logarithmic form. 2) Prove an identity about cosecant and cotangent and solve an equation involving them. 3) Find the exact value of the tangent of an angle between certain values. 4) Find inverse and composite functions involving several functions and solve an equation between two of them. 5) Calculate the temperature of a cup of coffee at a later time, given the initial temperature and cooling rate. 6) Evaluate two integrals involving trigonometric and polynomial functions. 7) Find the first three non-

    Copyright:

    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    46 views1 page

    Harkness Test 9b

    Uploaded by

    Lia C
    This document contains 8 mathematics problems: 1) Find values of A and B given an integral equation and find the exact value of another integral in logarithmic form. 2) Prove an identity about cosecant and cotangent and solve an equation involving them. 3) Find the exact value of the tangent of an angle between certain values. 4) Find inverse and composite functions involving several functions and solve an equation between two of them. 5) Calculate the temperature of a cup of coffee at a later time, given the initial temperature and cooling rate. 6) Evaluate two integrals involving trigonometric and polynomial functions. 7) Find the first three non-

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    © All Rights Reserved
    Download as DOCX, PDF, TXT or read online from Scribd
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    Harkness Test 9

    5 x−16 A B
    1. Given that ≡ +
    x −6 x+ 8 x−2 x−4
    2

    a. Find the values of A and B.


    7 5 x−16
    ∫5 x 2−6 x +8 dx
    b. Hence, find the exact value of giving your answer in the form

    ln ()
    a
    b where a , b ∈ Z .
    [7]
    2. Prove cosec θ−cot θ ≡cosec θ+cot θ
    4 4 2 2

    4 4 1
    Hence solve cosec θ−cot θ= for 0 ≤ θ ≤2 π
    tan θ
    [6]
    −π θ θ −1
    3. Given that ≤θ ≤ 0, find the exact value of tan when tan =
    4 4 2 2
    [6]

    x 3
    4. The functions f and g are defined by f ( x )= , x ∈ R , x ≠ 2 and g ( x )= , x ∈ R , x ≠ 0
    x−2 x
    a. Find an expression for f −1 ( x ) .
    b. Write down the range of f −1 ( x ) and find its domain
    c. Find an expression for gf (x ) and hence find gf (1.5)
    d. Solve g ( x )=f ( x ) +4
    [7]
    5. At 8.20 on a Wednesday morning Mr Thomas makes a cup of coffee. The coffee is initially
    90 C and 8 mins later is ideal for drinking at 58 C. However, Mr Thomas gets caught up
    with a fiendishly challenging differential equation and at the end of the lesson at 9.20 he
    gets to drink his coffee. What temperature will it be then? You can assume the ambient
    temperature is 20 C. Give your answer to 2 decimal places.
    [5]
    6. Find the following

    ∫ sec2 ( x3 ) dx
    b. ∫
    3
    a.
    x ( x 2−4 ) dx
    [4]
    7. Find the first three non-zero terms of the Maclaurin expansion of f ( x )=ln ⁡(3+ 4 x )
    [5]

    8. A graph is drawn of ( x 2−2 ) ( x+4 y )=12


    a+ bx−x 3
    y=
    a. Show that 4( x 2 −2) where a and b are to be found
    b. Hence find all three asymptotes of the curve
    dy dy
    3 x 2−2+4 x 2 +8 xy−8 =0
    c. Show that dx dx
    d. Hence find the gradient when x=2 leaving your answer as a simplified fraction.
    [10]

    Total [50]

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