Question

Let f(x) \(\frac{6x^{2}+8x+9}{(2-x)(3+2x)^{2}}\)

(i) Express f(x) in partial fractions.

(ii) Hence, showing all necessary working, show that \(\int_{-1}^{0}f(x)dx=1+\frac{1}{2}In (\frac{3}{4}).\)

Answer/Explanation

(i) State or imply the form

Use a correct method to find a constant

Obtain one of A = 1, B = – 1, C = 3

Obtain a second value

Obtain the third value

[Mark the form \(\frac{A}{2-x}+\frac{Dx+E}{(3+2x)^{2}} \), where A = 1, D = – 2 and E = 0, B1M1A1A1A1 as above.]

(ii) Integrate and obtain terms\( -In(2-x)-\frac{1}{2} In (3+2x)-\frac{3}{2(3+2x)^{2}}\)

Substitute correctly in an integral with terms a ln (2 – x),

b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0

Obtain the given answer after full and correct working
[Correct integration of the

A, D, E form gives an extra constant term if integration by

parts is used for the second partial fraction.]

Question

Let \(fx =\frac{x^{2}+x+6}{x^{2}(x+2)}\)

(i) Express fx in partial fractions.

(ii) Hence, showing full working, show that the exact value of \(\int _{1}^{4f(x)dx is \frac{9}{4}}\)<

Answer/Explanation

(i) State or imply the form \(\frac{A}{x}+\frac{B}){x^{2}}+\frac{c}{x+2}\ Use a correct method for finding a constant

Obtain one of A = – 1, B = 3, C = 2

Obtain a second value

Obtain the third value

(ii) Integrate and obtain terms In \( x-\frac{3}{x}+2 In(x+2)\) Substitute limits correctly in an integral with terms a In x\( \frac{b}{x}\) and c x ln 2 ( + ) , where abc
≠ 0

Obtain\( \frac{9}{4}\) following full and exact working

Question

Let f(x )=\(\frac{16-17x}{(2+x)(3-x)^{2}}\)

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in \(x^{2}\)

Answer/Explanation

(i)State or imply the form \(\frac{A}{2+x}+\frac{B}{3-x}+\frac{C}{\left ( 3-x \right )^{2}}\)

Use a correct method to obtain a constant

Obtain one of A=2,B=2,C=-7

Obtain a second value

Obtain the third value

(ii) Use a correct method to find the first two terms of the expansion of \(\left ( 2+x \right )^{-1},\left ( 3-x \right )^{-1}\) or \(\left ( 3-x \right )^{-2}\), or equibvalent ,e.g.\(\left ( 1+\frac{1}{2}x \right )^{-1}\)

Obtain correct Unsimplified expansions up to the term in\( x^{2}\) of each partial fraction.

Obtain final answer \(\frac{8}{9}-\frac{43}{54}x+\frac{7}{108}x^{2}\)

Question

Let f(x) \(\frac{12+12x-4x^{2}}{(2+x)(3-2x)}\).

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in \(x^{2}\)

Answer/Explanation

(i) State or imply the form\(\frac{5}{2}+\frac{1}{2}i\)

Use a correct method for finding a constant 

Obtain one of A = 2, B = – 4 and C = 6 

Obtain a second value 

Obtain the third value

(ii) Use correct method to find the first two terms of the expansion of\((2+x)^{-1} or (3-2x)^{-1}\), or equivalent

Obtain correct unsimplified expansions up to the term in \(x^{2} \)of each partial fraction 

Add the value of A to the sum of the expansions

Obtain final answer \(2+\frac{7}{3}x+\frac{7}{18}x^{2}\)

Question

(i) Express \(\frac{4+12x+x^{2}}{(3-x)(1+2x)^{2}} \)in partial fractions.

(ii) Hence obtain the expansion of \(\frac{4+12x+x^{2}}{(3-x)(1+2x)^{2}}\) in ascending powers of x, up to and including the term in\( x^{2}\)

Answer/Explanation

(i) Either State or imply partial fractions are of form \(\frac{A}{3-x}+\frac{B}{1+2x}+\frac{C}{(1+2x)^{2}}\)

Use any relevant method to obtain a constant

Obtain A = 1

Obtain B=\(\frac{3}{2}\)

Obtain C=\(-\frac{1}{2}\)

Or State or imply partial fractions are of form \(\frac{A}{3-x}+\frac{Dx+E}{(1+2x)^{2}}\)

Use any relevant method to obtain a constant

Obtain A = 1

Obtain D = 3

Obtain E = 1

(ii) Obtain the first two terms of one of the expansion\( (3-x)^{-1},\left ( 1-\frac{1}{3}x \right )^{-1}
(1+2x)^{-1} and (1+2x)^{-2}\)

Obtain correct unsimplified expansion up to the term in \(x^{2}\) of each partial fraction, following in each case the value of A, B, C

Obtain answer \(\frac{4}{3}-\frac{8}{9}x+\frac{1}{27}x^{2}\)

[If A, D, E approach used in part (i), give M1 A1  A1 for the expansions,  for
multiplying out fully and  for final answer]

Question

 Let \(f\left ( x \right ) = \frac{2x^{2}-7x-1}{\left ( x-2 \right )\left ( x^{2}+3 \right )}\).

   (i) Express fx in partial fractions. [5]

   (ii) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2.[5]

Answer/Explanation

Ans:

 (i) State or imply partial fractions are of the form \(\frac{A}{x-2}+\frac{Bx+C}{x^{2}+3}\)
          Use a relevant method to determine a constant 
          Obtain one of the values A = –1, B = 3, C = –1 
          Obtain a second value 
          Obtain the third value 

   (ii) Use correct method to obtain the first two terms of the expansions of \(\left ( x-2 \right )^{-1},\left ( 1-\frac{1}{2}x \right )^{-1},\left ( x^{2}+3\right )^{-1} or \left ( 1+\frac{1}{3}x^{2} \right )^{-1}\)
           Substitute correct unsimplified expansions up to the term in x2 into each partial fraction 
           Multiply out fully by Bx + C, where BC ≠ 0 
           Obtain final answer \(\frac{1}{6}+\frac{5}{4}x+\frac{17}{72}x^{2}\) , or equivalent 
           [Symbolic binomial coefficients, e.g. \(\begin{pmatrix}-1\\1\\\end{pmatrix}\) are not sufficient for the M1. The f.t. is on A, B, C.]
           [In the case of an attempt to expand \(\left ( 2x^{2}-7x-1 \right )\left ( x-2 \right )^{-1}\left ( x^{2}+3 \right )^{-1}\) , give M1A1A1 for the expansions, M1 for multiplying out fully, and A1 for the final answer.]
           [If B or C omitted from the form of partial fractions, give B0M1A0A0A0 in (i); M1A1 A1 in (ii)]

Question

Let \( f(x)=\frac{4x^{2}+9x-8}{(x+2)(2x-1)}\)

(i) Express f(x) in the form \(A+\frac{B}{x+2}+\frac{C}{2x-1}\)

(ii) Hence show that\(\int_{1}^{4}f(x)dx=6+\frac{1}{2}ln\left ( \frac{16}{7} \right ) \)

Answer/Explanation

8(i) Use a relevant method to determine a constant 
Obtain one of the values A = 2, B = 2, C = –1 
Obtain a second value 
Obtain the third value 
8(ii) Integrate and obtain terms \(2x+2ln(x+2)-\frac{1}{2}ln(2x-1)\) (deduct B1 for each error or omission) [The FT is on A, B and C]

Substitute limits correctly in an integral containing terms a x ln( 2) + and ln(2 1) b x − ,
where ab ≠ 0 Use at least one law of logarithms correctly D Obtain the given answer after full and correct working

Question

 (a) Show that \(\int_{2}^{4}4xlnxdx=56ln2-12\). [5]

    (b) Use the substitution u = sin 4x to find the exact value of \(\int_{0}^{\frac{1}{24}\pi }cos^{3}4xdx.\) [5]

Answer/Explanation

Ans:

 (a) Carry out integration by parts and reach \(ax^{2}lnx+b\int \frac{1}{2}x^{2}dx\)
           Obtain \(2x^{2}lnx-\int \frac{1}{x}.2x^{2}dx\)
           Obtain \(2x^{2}lnx-x^{2}\)
           Use limits, having integrated twice 
           Confirm given result 56 ln 2 – 12

   (b) State or imply \(\frac{du}{dx}=4cos4x\)
           Carry out complete substitution except limits
           Obtain \(\int \left ( \frac{1}{4}- \frac{1}{4}u^{2}\right )du\) or equivalent
           Integrate to obtain form k1u + k2u3 with non-zero constants k1, k2
           Use appropriate limits to obtain \(\frac{11}{96}\)

Question

 For each of the following curves, find the gradient at the point where the curve crosses the y-axis:

(i) \(y=\frac{1+x^{2}}{x+e^{2x}};\) [3]

(ii) \(2x^{3}+5xy+y^{3}=8\) [4]

Answer/Explanation

Ans:

 (i) Use correct quotient rule or equivalent 
          Obtain \(\frac{\left ( 1+e^{2x} \right )2x-\left ( 1+x^{2} \right )2e^{2x}}{\left ( 1+e ^{2x}\right )^{2}}\) or equivalent 
          Substitute x = 0 and obtain \(-\frac{1}{2}\) or equivalent 
  (ii) Differentiate y3 and obtain \(3y^{2}\frac{dy}{dx}\)
          Differentiate 5xy and obtain \(5y+5x\frac{dy}{dx}\)
          Obtain \(6x^{2}+5y+5x\frac{dy}{dx}+3y^{2}\frac{dy}{dx}=0\)
          Substitute x = 0, y = 2 to obtain \(-\frac{5}{6}\) or equivalent following correct work 

Question

(i) Express \(\frac{1}{x}(2x + 3)\) in partial fractions.

(ii) The variables x and y satisfy the differential equation

                                                                \(x(2x+3)\frac{dy}{dx}=y\).

and it is given that y = 1 when x = 1. Solve the differential equation and calculate the value of y when x = 9, giving your answer correct to 3 significant figures.

Answer/Explanation

9(i) Carry out a relevant method to obtain A and B such that \(\frac{1}{x(2x+3)}\equiv \frac{A}{x}+\frac{B}{2x+3},\)or

equivalent

Obtain   A =\( \frac{1}{3}\) and   B = \(−\frac{2}{3}\) , or equivalent

9(ii) Separate variables and integrate one side 
Obtain term ln  y
Integrate and obtain terms \(\frac{1}{3}lnx -\frac{1}{3}ln(2x+3,)\)

Use x = 1 and y = 1 to evaluate a constant, or as limits, in a solution containing

Obtain correct solution in any form, e.g \(\frac{1}{3}lnx -\frac{1}{3}ln(2x+3,)\)
Obtain answer y = 1.29 (3s.f. only)

Question

Let f(x)=\(\frac{x(6-x)}{(2+x)(4+x^{2})}\)

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in \(x^{2}\)

Answer/Explanation

(i) State or imply the form  \(\frac{A}{2+x}+\frac{Bx+C}{4+x^{2}}\)

Use a relevant method to determine a constant

Obtain one of the values A = – 2, B = 1, C = 4

Obtain a second value

Obtain the third value

Question

 Let \(f(x)=\frac{8+5x+12x^{2}}{(1-x)(2+3x)^{2}}\)   .
     (a) Express (fx) in partial fractions.

     (b) Hence obtain the expansion of f (x) in ascending powers of x, up to and including the term in x2.         [5]

Answer/Explanation

(a) State or imply the form \(\frac{A}{1-x}+\frac{B}{2+3x}+\frac{C}{(2+3x)^{2}}\)

        Use a correct method for finding a coefficient 
        Obtain one of A = 1, B = –1 , C = 6 
        Obtain a second value 
         Obtain the third value 

 (b) Use a correct method to find the first two terms of the expansion

          of  \((1-x)^{1}, (2+3x)^{-1}, \left ( 1+\frac{3}{2}x \right )^{-1}, (2+3x)^{-2}\ or \ \left ( 1+\frac{3}{2}x \right )^{-2}\)

          Obtain correct un-simplified expansions up to the term in
          of each partial fraction 

          Obtain final answer \(2-\frac{11}{4}x+10x^{2}, or\ equivalent\)

Question

Let \(f(x)=\frac{\cos x}{1+\sin x}\)

     (a) Show that f′x < 0 for all x in the interval \(-\frac{1}{2}\pi < x< \frac{3}{2}\pi\)

     (b) Find \(\int_{\frac{1}{6}\pi }^{\frac{1}{2}\pi }f(x)dx \). Give your answer in a simplified exact form.       

Answer/Explanation

Ans

(a) Use quotient or product rule 
         Obtain derivative in any correct form e.g \(\frac{-\sin x(1+\sin x)-\cos x(\cos x)}{(1+\sin x)^{2}}\)

        Use Pythagoras to simplify the derivative 
        Justify the given statement

(b) State integral of the form a ln (1 + sin x) 
         State correct integral ln (1 + sin x) A1
         Use limits correctly

          Obtain answer \(ln\frac{4}{3}\)

Question

Let \(f(x)=\frac{2+11x-10x^2}{(1+2x)(1-2x)(2+x)\)
(a) Express f(x) in partial fractions.
(b) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in \(x^2\).

Answer/Explanation

Ans:

(a) State or imply the form \(\frac{A}{1+2x}+\frac{B}{1-2x}+\frac{C}{2+x}\)
Use a correct method for finding a constant
Obtain one of A = -2, B = 1 and C = 4
Obtain a second value
Obtain the third value
(b) Use correct method to find the first two terms of the expansion of \((1+2x)^{-1}\).
\((1-2x)^{-1}.(2+x)^{-1}or (1+\frac{1}{2}x)^{-1}\)
Obtain correct unsimplified expansions up to the terms in \(x^2\) of each partial fraction
Obtain final answer \(1+5x-\frac{7}{2}x^2\)

Question

Find \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi}x sec^2xdx\). Give your answer in a simplified exact form.

Answer/Explanation

Ans:

Integrate by parts and reach axtanx + b\(\int \)tan x dx
Obtain x tan x – \(\int \) tan x dx
Complete the integration, obtaining a term \(\pm In cosx\), or equivalent
Obtain integral xtan x + In cos x, or equivalent
Substitute limits correctly, having integrated twice
Use a law of logarithms
Obtain answer \(\frac{5}{18} \sqrt{3 \pi} – \frac{1}{2} In 3\), or exact simplified equivalent

Question

 The variables x and t satisfy the differential equation \(\frac{dx}{dt}=x^{2}(1+2x),\ and\ x=1 \ when\ t=0\)

     Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.                        [11]

Answer/Explanation

Ans

  State a suitable form of partial fractions for  \(\frac{1}{x^{2}(1+2x)}\)

      Use a relevant method to determine a constant

      Obtain one of A = – 2, B = 1 and C = 4 

      Obtain a second value 
      Obtain the third value 
      Separate variables correctly and integrate at least one term 

      Obtain terms \(-2\ln\ x-\frac{1}{x}+ 2ln(l+2x) \ and \ t\)

      Evaluate a constant, or use limits x = 1, t = 0 in a solution containing terms t,
      a ln x and b  ln  (1 + 2x ), where ab≠0

      Obtain a correct expression for t in any form, e.g  \(t=-\frac{1}{x}+2 ln\left ( \frac{1+2x}{3x} \right )+1\)

Question

 (a) Prove that  \(\frac{1-\cos 2\theta }{1+\cos 2\theta }=\tan ^{2}\theta\)                                                                               [2]

    (b) Hence find the exact value of \(\int_{\frac{1}{6}\pi }^{\frac{1}{3}\pi } \frac{1-\cos 2\theta }{1+\cos 2\theta }\)    [4]

Answer/Explanation

Ans

4 (a) Use correct double angle formula or t-substitution twice

          Obtain \( \frac{1-\cos 2\theta }{1+\cos 2\theta }=tan^{2}\theta\)  from correct working

4 (b) Express 2 tanθ in terms of 2 secθ

           Integrate and obtain terms tanθ – θ

          Substitute limits correctly in an integral of the form a  tanθ + bθ  , where ab≠0 

          Obtain answer \(\frac{2}{3}\sqrt{3}-\frac{1}{6}\pi \)

Question

Let \(f(x) = \frac{5a}{(2x-a)(3a-x)}\), where a is a positive constant.
(a) Express f(x) in partial fractions.
(b) Hence show that \(\int_{a}^{2a}f(x)dx=In6\).

Answer/Explanation

Ans:

  1. Carry out a relevant method to determine constants A and B such that
    \(\frac{5a}{(2x-a)(3a-x)}=\frac{A}{2x-a}+\frac{B}{3a-x}\)
    Obtain A=2
    Obtain B=1
  2. Integrate and obtain terms In(2x-a)-In(3a-x)
    Substitute limits correctly in a solution containing terms of the form bIn(2x-a) and cIn(3a-x), where bc≠0
    Obtain the given answer showing full and correct working

Question

(i) Prove the identity tan \(tan2\Theta -tan\Theta sec2\Theta \)

(ii) Hence show that \(\int_{0}^{\frac{1}{6}\pi }tan\Theta sec2\Theta d\Theta =\frac{1}{2}In\frac{3}{2}\)

Answer/Explanation

(i) EITHER: Use tan 2A formula to express LHS in terms of tanθ Express as a single fraction in any correct form Use Pythagoras or cos 2A formula Obtain the given result correctly

OR: Express LHS in terms of sin 2θ, cos 2θ, sin θ and cosθ
Express as a single fraction in any correct form
Use Pythagoras or cos 2A formula or sin(A – B) formula Obtain the given result correctly

(ii) Integrate and obtain a term of the form aln(cos2 ) θ or bln(cos ) θ (or secant equivalents)

Obtain integral \(-\frac{1}{2}\) ln(cos 2 θ ) ln(cos θ )   , or equivalent

Substitute limits correctly (expect to see use of both limits)
Obtain the given answer following full and correct working

Question

Let f(x )= \(\frac{4x^{2}+12}{(x+1)(x-3)^{2}}\)

(i) Express f(x )in partial fractions.

(ii) Hence obtain the expansion of f(x )in ascending powers of x, up to and including the term in

(i) State or imply the form 

Use a correct method to determine a constant M1
Obtain one of the values A = 1, B = 3, C = 12  Obtain a second value 
Obtain a third value

[Mark the form where A =1, D = 3, E = 3, B1 M1 A1 A1 A1 as above.]

(ii) Use correct method to find the first two terms of the expansion of \((x+1)^{-1},(x-3)^{-1},(1-\frac{1}{3}x)^{-1}\) \

\((x-3)^{-2},(1-\frac{1}{3}x)^{-2}\) 

Obtain correct unsimplified expansions up to the term in\( x^{2}\)

Answer/Explanation

of each partial fraction

Obtain final answer\( \frac{4}{3}-\frac{4}{9}x+\frac{4}{3}x^{2}\) , or equivalent

Question

(a)Find the quotient and remainder when 8x3 + 4x2 + 2x + 7 is divided by 4x2 + 1.

(b) Hence find the exact value of \( \int_{0}^{\frac{1}{2}} \frac{8x^{3}+4x^{2}+2x+7}{4x^{2}+1}dx.\) 

Answer/Explanation

Ans:

(a)Commence division and reach quotient of the form 2x ± 1

Obtain (quotient) 2x + 1

Obtain (remainder) 6

(b)

Obtain terms x2+x

Obtain term of the form a tan-1 2x

Obtain term 3tan-1 2x

Use x = 0 and x = \(\frac{1}{2}\) as limits in a solution containing a term of the form a tan-1 2x

Obtain final answer \(\frac{3}{4}\left ( 1+\pi \right )\) , or exact equivalent 

Question

Let f(x) = \(\frac{3x^{3}+6x-8}{x\left ( x^{^{2}}+2 \right )}\) .

    (i) Express f(x) in the form \(A+\frac{B}{x}+\frac{Cx+D}{x^{2}+2}\) .[5]

    (ii) Show that \(\int_{1}^{2}f\left ( x \right )dx = 3 – ln 4\) .[5]

Answer/Explanation

Ans:

(i) State or obtain A = 3 
          Use a relevant method to find a constant 
         Obtain one of B = −4, C = 4 and D = 0 
         Obtain a second value 
         Obtain the third value

   (ii) Integrate and obtain 3x – 4ln x 
         Integrate and obtain term of the form k ln(x2 +2) 
         Obtain term 2ln(x2 +2)
         Substitute limits in an integral of the form axb ln x + c ln(x2 + 2) , where abc ≠ 0 
         Obtain given answer 3 − ln 4 after full and correct working 

Question

(i) By differentiating \(\frac{1}{cosx}\) , show that if y = sec x then\( \frac{dy}{dx}\) = sec x tan x.

(ii) Show that \(\frac{1}{(secx-tanx)}\) ≡ sec x + tan x.

(iii) Deduce that \(\frac{1}{(secx-tanx)^{2}}\) ≡ \(2 sec^{2}\)
x − 1 + 2 sec x tan x.

(iv) Hence show that \(\int_{0}^{\frac{1}{4}\pi }\frac{1}{(secx-tanx)^{2}}dx\)=\(\frac{1}{4}(8\sqrt{2}-\pi )\)

Answer/Explanation

(i) Use correct quotient or chain rule Obtain the given answer correctly having shown sufficient working 
(ii) Use a valid method, e.g. multiply numerator and denominator by sec x + tan x, and a version of Pythagoras to justify the given identity
(iii) Substitute, expand 
(sec x +\( tan x)^{2}\)
and use Pythagoras once 
Obtain given identity 
(iv) Obtain integral 2 tan x – x + 2 sec x 
Use correct limits correctly in an expression of the form a tan x + bx + c sec x, or
equivalent, where abc 0
Obtain the given answer correctly

Question

By first expressing \(\frac{4x^{2}+5x+3}{2x^{2}+5x+2} \) in partial fractions, show that 

\(\int_{0}^{4}\frac{4x^{2}+5x+3}{2x^{2}+5x+2} \)dx =8-9In9

Answer/Explanation

Question

The diagram shows the curve y\(=8sin\frac{1}{2}x-tan\frac{1}{2}x\) for 0 ≤ x < π. The x coordinate of the maximum point is α and the shaded region is enclosed by the curve and the lines x = α and y = 0.

(i) Show that\( α \frac{2}{3}\pi \)

(ii) Find the exact value of the area of the shaded region.

Answer/Explanation

(i) Differentiate to obtain  \(4cos\frac{1}{2}x-\frac{1}{2}sec^{2}\frac{1}{2}x\)

Equate to zero and find value of \(cos\frac{1}{2}x\)

Obtain \( cos\frac{1}{2}x=\frac{1}{2} \) and confirm \(\alpha =\frac{2}{3}\pi\)

(ii) Integrate to obtain \(-16cos\frac{1}{2}x\)

\(+2Incos\frac{1}{2}x \)or equivqlent

Using limits 0 and \(\frac{2}{3}\pi in acos\frac{1}{2}x+bIncos\frac{1}{2}x\)

Obtain\( 8+2In\frac{1}{2}\)  or exact equivalent

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