Question

7  A curve is defined by the parametric equations
                      x = 3t − 2 sin t,            y = 5t + 4 cos t,

    where \(0\leqslant t\leqslant 2\pi \). At each of the points P and Q on the curve, the gradient of the curve is \(\frac{5}{2}\).

    (a) Show that the values of t at P and Q satisfy the equation 10 cos t − 8 sin t = 5.                                                                                        [3] 

    (b) Express 10 cost − 8 sin t in the form \(R\cos (t+\alpha ), where \ R> 0> and \ 0< \alpha < \frac{1}{2}\pi \). Give the exact
           value of R and the value of \(\alpha\)  correct to 3 significant figures.                                                                                                      [3]

    (c) Hence find the values of t at the points P and Q.                                                                                                                                              [4]

Answer/Explanation

Ans

7 (a) Carry out division at least as far as  3x2+ kx

         Obtain quotient 3x2 – 4 – 4

        Confirm remainder is 9 AG 

7 (b) Integrate to obtain at least  k1 x3 and k2 ln(3x + 2) terms

          Obtain x3 – 2x2 – 4x + 3 ln(3x + 2)
          (FT from quotient in part (a)) 

          Apply limits correctly 
          Apply appropriate logarithm properties correctly
          Obtain 125 ln64

7 (c) State or imply \(9x^{3}-6x^{2}-20x-8=(3x+2)(3x^{2}-4x-4) \)

        (FT from quotient in part (a))
       Attempt to solve cubic eqn to find positive value of x (or of 3ey )
       Use logarithms to solve equation of form 3ey = k where k > 0 M1
       Obtain \(\frac{1}{3}ln2\) or exact equivalent 

Question

  It is given that 3 sin 2θ = cos θ where θ  is an angle such that 0o < θ < 90o.
      (a) Find the exact value of sin θ.                                                                                   [2]

      (b) Find the exact value of sec θ .                                                                                  [2]

      (c) Find the exact value of cos 2θ .                                                                                [2] 

Answer/Explanation

Ans

(a) Express left-hand side in terms of sinθ and cosθ

         Obtain 2 cos θ  – 2 sin θ 
         Attempt to express a cos θ + b sin θ in Rcos (θ+β) form 

         Confirm \(R=\sqrt{8}AG\)

         Carry out necessary trigonometry and confirm \(\frac{1}{4}\pi AG\)

 (b) Carry out correct process to find θ from \(\cos \left ( \theta +\frac{1}{4}\pi \right )=\frac{1}{\sqrt{8}}\)

          Obtain 0.424

 (c) Express integrand as \(\sqrt{8}\cos \left ( \frac{1}{2}x+\frac{1}{4}\pi \right )\ or\ as \ 2\cos \frac{1}{2}x-2\sin \frac{1}{2}x\)

          Integrate to obtain \(k\sin (\frac{1}{2}x+\frac{1}{4}\pi )\ or \ 4\sin \frac{1}{2}x+4\cos \frac{1}{2}x\)

          Obtain correct \(2\sqrt{8}\sin \left ( \frac{1}{2}x+\frac{1}{4}\pi \right )\ or 4\sin \frac{1}{2}x+4\cos \frac{1}{2}x\)

Question

Solve the equation \(2 sin ( \theta + 30^o)+5 cos \theta = 2 sin \theta\) for \(0^o<\theta<90^o\).

Answer/Explanation

Ans:

Express first term as \(2 sin \theta cos 30+2cos \theta sin 30\)
Divide by \(cos\theta\) to produce linear equation in \(tan \theta\)
Obtain \(tan \theta = \frac{6}{2\sqrt{3}}\) or 22.39…
Obtain 87.4

Question

(a) Express \(5\sqrt{3}cos x + 5sin x\) in the form \(R cos(x- \alpha )\), where R>0 and \(0<\alpha<\frac{1}{2}\pi\).
(b) As x varies, find the least possible value of
\(4+5\sqrt{3}cos x+5sinx\),
and determine the corresponding value of x where \(-\pi<x<\pi\).
(c) Find \(\int \frac{1}{(5\sqrt{3}cos3\theta+5sin3\theta)^2}d\theta\)

Answer/Explanation

Ans:

  1. State R=10
    Use appropriate trigonometry to find \(\alpha\)
    Obtain \(\alpha=\frac{1}{6}\pi\)
  2. State -6
    Attempt to find x from their \(cos(x-\alpha)=-1\)
    Obtain \(x-\frac{1}{6}\pi=-\pi\) and hence \(-\frac{5}{6}\pi\)
  3. State integrand of form \(k_1sec^2(3\theta-\frac{1}{6}\pi)\)
    Integrate to obtain form \(k_2tan(3\theta-\frac{1}{6}\pi)\)
    Obtain \(\frac{1}{300}tan(3\theta-\frac{1}{6}\pi)+c\)

Question

Solve the equation \(sec^2\theta cot\theta =8\) for \(0<\theta<\pi\).

Answer/Explanation

Ans:

State \(\frac{1}{cos^2\theta} \times \frac{cos \theta}{sin \theta} =8\)
Attempt use of \(sin 2\theta\) identity to obtain \(sin 2\theta = k\)
Obtain \(sin 2\theta = \frac{1}{4}\)
Use correct process to find two values of \(\theta\) between 0 and \(\pi\).
Obtain 0.126 and 1.44

Alternative method for question 2

State \(\frac{1+tan^2 \theta}{tan \theta} =8\)
Attempt solution of 3-term quadratic equation to find values of \(tan\theta\)
Obtain \(tan \theta = \frac{8± \sqrt{60}}{2}\)
Solve \(tan \theta = ….\) to find two values of \(\theta\) between 0 and \(\pi\).
Obtain 0.126 and 1.44

Question

 A curve is defined by the parametric equations
                           x = 3t − 2 sin t,                    y = 5t + 4 cos t,
    where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, the gradient of the curve is \(\frac{5}{2}\)

    (a) Show that the values of t at P and Q satisfy the equation 10 cost − 8 sin t = 5.                                                                                       [3]

    (b) Express 10 cost − 8 sin t in the form \(R \cos(t+\alpha ), \ where \ R> 0\ and \ 0< \alpha \frac{1}{2}\pi\). Give the exact
           value of R and the value of \(\alpha \) correct to 3 significant figures.                                                                                                    [3]

    (c) Hence find the values of t at the points P and Q.                                                                                                                                            [4]

Answer/Explanation

Ans

 (a) Obtain \(\frac{dx}{dt}=3-2\cos t \ and \ \frac{dy}{dt}=5-4\sin t \)

          Equate expression for \(\frac{dy}{dx}\ to \ \frac{5}{2}\)

          Obtain \(10\cos t-8\sin t=5\)

7(b) State \(R=\sqrt{164}\) or exact equivalent 

          Use appropriate trigonometry to find \(\alpha \)

          Obtain 0.675 with no errors seen

(c) Carry out correct method to find one value of t 
         Obtain 0.495 
         Carry out correct method to find second value of t 
         Obtain 4.44 

Question

 It is given that 3 sin 2 θ = cos θ  where θ is an angle such that 0o < 1 < 90o.
     (a) Find the exact value of sin θ .                                                                                          [2]

     (b) Find the exact value of sec θ.                                                                                          [2]

     (c) Find the exact value of cos 2 θ .                                                                                      [2]

Answer/Explanation

Ans

(a) Use sin 2 θ = 2sin θ cos  θ

          Obtain \(\sin \theta =\frac{1}{6}\)

 (b) Use correct identity or identities to find value of secθ

          Obtain \(\frac{6}{\sqrt{35}}\) or exact equivalent

 (c) Use correct identity or identities to find value of cos2θ

          Obtain \(\frac{17}{18}\) or exact equivalent

Question

(a) Show that (sec x + cos x)2 can be expressed as sec2x + a + b cos 2x,                     where  a and b are constants
         to be determined.                                                                                               [2] 

Answer/Explanation

Ans

3(a) Expand to obtain integrand of form sec2 x + k1 + k2 cos 2x

        Obtain correct sec2  \(x+\frac{5}{2}+\frac{1}{2}\cos 2x\)

3(b) Integrate to obtain at least terms of form k3 tan x and k4 sin 2x

        Obtain correct tan \(x+\frac{5}{2}x+\frac{1}{4}2x\)

        Apply limits correctly to integral involving at least two terms

        Obtain \(\frac{5}{4}+\frac{5}{8}\pi \ or \ exact equivalent\)

Question

  By first expanding sin(θ + 30), solve the equation sin(θ + 30) cosec θ = 2 for 0o < θ < 360.    [6]

Answer/Explanation

Ans

2  Express sin ( θ + 30 ) as sin θ cos 30 + cos θ sin 30

    Use  cosecθ=\(\frac{1}{\sin \theta }\)

    Correctly obtain a linear equation in tanθ or cotθ

    Obtain tanθ = \(\frac{1}{4-\sqrt{3}}, \frac{4+\sqrt{3}}{13}, \frac{1}           {2.26795} \ or \ 0.440…\)

    Obtain 23.8 
    Obtain 203.8

Question

(a)By first expanding cos (2θ + θ), show that cos 3θ = 4 cos3θ – 3 cos θ.

(b)Find the exact value of 2 cos3  \(\left ( \frac{5}{18}\pi \right ) – \frac{3}{2} cos \left ( \frac{5}{18} \pi \right )\) .

(c)Find \(\int (12cos^{3}x – 4 cos^{3} 3x)dx\) .

Answer/Explanation

(a)State cos 2θ cos θ – sin 2 θ sin θ 

Attempt correct relevant identities to express in terms of cos θ only

Confirm 4 cos3 θ -3 cos  θ with sufficient detail

(b)Use identity with θ = \(\frac{5}{18}\pi \)

Obtain \(\frac{1}{2} cos \frac{5}{6}\pi and hence – \frac{1}{4}\sqrt{3}\)

(c)Express integrand in form k1 (cos 3x + 3 cos x) + k2 (cos 9x + 3 cos 3x)

Obtain correct integrand 9 cos x –  cos9 x

Integrate to obtain form k3 sin x + k4  sin 9x

Obtain correct 9 sin x – \(\frac{1}{9}\)  sin 9x

Question

The polynomial p (x) is defined by

             p(x) = 4x3 + 16x2 + 9x -15.

(a)Find the quotient when p(x) is divided by (2x+3), and show that the remainder is −6.

(b) Find \(\int \frac{p(x)}{2x+3}dx\) .

(c)Factorise p (x) +6 completely and hence solve the equation

                    p (cosec 2θ) + 6 = 0

for 00 < θ < 1350.

Answer/Explanation

Ans:

(a)Carry out division at least as far as 2x2 + kx

Obtain quotient 2x2 + 5x – 3

Confirm remainder is -6.

(b)Integrate to obtain at least k1 x 3 and  k2 ln (2x 3) + terms

Obtain \(\frac{2}{3}x^{3} + \frac{5}{2}x^{2} – 3x – 3 In (2x+3)\)

(c)State or imply p(x) + 6 = (2x + 3) (2x2 + 5x -3)

Conclude (2x + 3) (2x – 1) (x + 3)

State or imply  sin 2θ = \(-\frac{2}{3}\) or sin 2θ = \(-\frac{1}{3}\) or both

Carry out correct process to find θ in at least one case

Obtain 99.7 and 110.9

Question

(a) Showing all necessary working, solve the equation \(\sec \Theta \csc \Theta =7\)  for \(0^{\circ}< \Theta < 90^{\circ}\).

(b) Showing all necessary working, solve the equation

\(\sin (\beta +20^{\circ})+\sin (\beta -20^{\circ})=6\cos \beta\)  for \(0^{\circ}< \Theta < 90^{\circ}\).

Answer/Explanation

6(a) Express equation as \(\frac{1}{cos\alpha+sin\alpha }=7\)

Attempt use of identity for sin 2α or attempt to obtain a quadratic equation in terms of any one of the following: \(sin^{2}\alpha,cos^{2}\alpha ,cot^{2}\alpha or tan^{2}\alpha \) 

Obtain  sin 2α =\( \frac{2}{7} \)or a correct 3 term quadratic equation, equated to zero in any one of the following: \(sin^{2}\alpha,cos^{2}\alpha ,cot^{2}\alpha or tan^{2}\alpha\) 

Attempt correct process to find at least one correct value of α  Obtain 8.3 and 81.7 and no others between 0 and 90

6(b) Simplify left-hand side to obtain 2sin cos20 β ° 
Attempt to form equation where tan β is only variable, tan β≠3

Obtain  \(tan\beta =\frac{3}{cos20\AA }\) 

Obtain β= 72.6 and no others between 0 and 90

Question

(i) Show that \(2 cosec 21 cot\Theta =cosec^{2}\)

(ii) Hence show that cosec^{2}15Å tan 15Å = 4.

(iii) Solve the equation \(2cosec\Theta cot\frac{\Theta }{2}+cosec\frac{\Theta }{2}=12\) for \(−360Å <\Theta < 360Å\) Show all necessary
working.

Answer/Explanation

(i)State or  imply \( \csc 2\theta =\frac{1}{2\sin \theta \cos \theta }\) 

Attempt to express left-hand side in terms of  \(\sin \theta \) and \( \cos \theta \) only

Simplify to confirm \( \csc ^{2}\theta \)

(ii) Use identity to express left-hand side in terms of \(\sin 30\) or \(\csc 30\)

Obtain\( \frac{2}{\sin 30}\) or \(2\csc 30\) and confirm 4

(iii) Solve quadratic equation of the form\( k\csc ^{2}\frac{\phi }{2}+\csc \frac{\phi }{2}-12=0\) or \(12\sin ^{2}\frac{\phi }{2}-\sin \frac{\phi }{2}-k=0\) correctly for\csc \frac{1}{2}\phi  \)or \(\sin \frac{1}{2}\phi \) to find two values of \(\csc \frac{1}{2}\phi  \)\) or \(\sin \frac{1}{2}\phi \)

Obtain \(\sin \frac{1}{2}\phi =-\frac{1}{4},\frac{1}{3}\)

Use correct process to find at least one correct value of \(\phi \) from \(\sin \frac{1}{2}\phi =\pm \frac{1}{4},\pm \frac{1}{3}\)

Obtain any two of -331.0,-29.0,38.9,321.1.

Question

The diagram shows the curve with equation y = sin 2x + 3 cos 2x for 0 ≤ x ≤ 0. At the points P and Q on the curve, the gradient of the curve is 3.

(i) Find an expression for \(\frac{dy}{dx}\)              

(ii) By first expressing \(\frac{dy}{dx}\)  in the form R cos (2x + \alpha), where \(R>0 and 0<\alpha <\frac{1}{2}\pi\)  R , find the x-coordinates of P and Q, giving your answers correct to 4 significant figures. [8]

Answer/Explanation

7(i) State expression of form \(k_{1}cos2x+k_{2}sin2x\)

7(ii) State\( R=\sqrt{40} or 6.324…\)

Use appropriate trigonometry to find α

Obtain 1.249…

Equate their \(Rcos(2x+\alpha ) to  3  and find  cos^{-1}(3\div R)\)

Carry out correct process to find one value of α

Obtain 1.979

Carry out correct process to find second value of α within the range

Obtain 3.055

Question

Solve the equation \(sec^{2}\Theta \)= 3 cosec 1 for 0Å < 1 < 180Å.

Answer/Explanation

State\( \frac{1}{cos^{2}\theta }=\frac{3}{sin\Theta }\) or \(1+tan^{2}\Theta =\frac{3}{sin\Theta }\)

Produce quadratic equation in sinθ

Solve 3-term quadratic equation to find value between –1 and 1 for sinθ

Obtain  \(sin\Theta =\frac{1}{6}(-1+\sqrt{37})\) and hence 57.9
Obtain 122.1 and no others between 0 and 180

Question

(a) Showing all necessary working, solve the equation \(sec\alpha cosec\alpha   = 7\)  for \(0^{\circ} < \alpha < 90^{\circ}\)

(b) Showing all necessary working, solve the equation

    \(sin(\beta +20)+ sin(\beta – 20) = 6 cos\beta  \)      for    \( 0^{\circ}< \beta< 90^{\circ}\)

Answer/Explanation

(i) Substitute x=−2 and equate to zero

Obtain
or equivalent and hence \(a=7\)

Attempt either division by \(x+2\) and reach partial quotient \(x^{2}+kx \)  where  k is numeric or use of identity or inspection or synthetic  division

Obtain quotient \(x^{2}+5x+4\)

Conclude with  \( (x+1)(x+2)(x+4) \)

Question 

(a) Showing all necessary working, solve the equation \(sec\alpha cosec\alpha   = 7\)  for \(0^{\circ} < \alpha < 90^{\circ}\)

(b) Showing all necessary working, solve the equation

    \(sin(\beta +20)+ sin(\beta – 20) = 6 cos\beta  \)      for    \( 0^{\circ}< \beta< 90^{\circ}\)

Answer/Explanation

(i) Substitute x=−2 and equate to zero

Obtain
or equivalent and hence \(a=7\)

Attempt either division by \(x+2\) and reach partial quotient \(x^{2}+kx \)  where  k is numeric or use of identity or inspection or synthetic  division

Obtain quotient \(x^{2}+5x+4\)

Conclude with  \( (x+1)(x+2)(x+4) \)

Question

The parametric equations of a curve are

      x = 2t − sin 2t, y = 5t + cos 2t, for\( 0 ≤ t \frac{1}{2}\pi \) .

At the point P on the curve, the gradient of the curve is 2.

(i) Show that the value of the parameter at P satisfies the equation 2 sin 2t − 4 cos 2t = 1.                        

(ii) By first expressing 2 sin 2t − 4 cos 2t in the form\( R sin2t − \Theta\), where\( R > 0 and 0 < ! <\Theta\)

Answer/Explanation

(i) Obtain expression for \(\frac{dy}{dx}\) with numerator quadratic, denominator linear

Obtain\(\frac{3t^{2}-6t}{2t+4}\)

Identify t = 3 at P 
Obtain \( \frac{9}{10}\)or equivalent

(ii) Equate first derivative to zero and obtain non-zero value of t 
Obtain t = 2 
Substitute to obtain (12,- 4)

(iii) Equate expression for gradient to m and rearrange to confirm\( 3t^{2}-(2m+6)-4m\)=0

Attempt solution of quadratic inequality or equation resulting from
discriminant

Obtain critical values -\(\sqrt{72}-9,m\geqslant \sqrt{72-9}\)

Conclude
\(m\leqslant \sqrt{72-9},m\geqslant \sqrt{72}-9\)

Question

Solve the equation \(sec^{2}\Theta +tan^{2}\Theta =5tan\Theta  4 for 0Å < 1 < 180Å. Show all necessary working.       [4]

Use identity \(sec^{2}\Theta =1+tan^{2}\Theta \)

Answer/Explanation

Attempt solution of quadratic equation to find two values of tanθ

Obtain \( tan\Theta =-\frac{1}{2},3\)

Obtain 71.6 and 153.4 and no others between 0 and 180

Question

(i) Given that\( tan2\Theta cot\Theta = 8\), show that \(tan^{2}\Theta =\frac{3}{4}\)

(ii) Hence solve the equation tan 21 cot 1 = 8 for 0Å < 1 < 180Å

Answer/Explanation

.

(i) Use identity \(cot\Theta =\frac{1}{tan\Theta }\)

Attempt use of identity for \(tan2\Theta\) 

Confirm given \(tan^{2}\Theta =\frac{3}{4}\)

(ii) Obtain 40.9 B1
Obtain 139.1

Question

Solve the equation \(5 tan 2\Theta = 4cot\Theta for 0Å < 1 < 180Å.\)

Use cot\Theta =1\div tan\Theta

Form equation involving tanθ only and with no denominators involving θ

Answer/Explanation

Obtain\( tan^{2}\Theta =\frac{2}{7}\)

Obtain 28.1 
Obtain 151.9 
Allow other valid methods

Question

(i) Show that sin 2x cot x Ξ 2 cos2x. [2]

   (ii) Using the identity in part (i),

         (a) find the least possible value of
                 3 sin 2x cot x + 5 cos 2x + 8
                 as x varies, [4]

         (b) find the exact value of \(\int_{\frac{1}{8}\pi }^{\frac{1}{6}\pi}\) cosec 4x tan 2x dx. [5]

Answer/Explanation

Ans:

8 (i) State cos 2sin x cos x . \(\frac{cos x}{sin x}\)
          Simplify to confirm 2cos2x

    (ii) (a) Use cos2x= 2cos2 x-1
                    Express in terms of cos x 
                    Obtain 16cos2 x + 3 or equivalent 
                    State 3, following their expression of form a cos2 x + b

           (b) Obtain integrand as \(\frac{1}{2}sec^{2}2x\)
                    Integrate to obtain form k tan 2x
                    Obtain correct \(\frac{1}{2}tan2x\)
                    Apply limits correctly dep 
                    Obtain \(\frac{1}{4}\sqrt{3}-\frac{1}{4}\) or exact equivalent 

Question

 (i) Express 8 sin θ + 15 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. Give the value of α correct to 2 decimal places. [3]

   (ii) Hence solve the equation

8 sin θ + 15 cos θ = 6

for 0° ≤ 1 ≤ 360°. [4]

Answer/Explanation

Ans:

3 (i) State or imply R = 17 
          Use appropriate formula to find α 
          Obtain 61.93 A1 

   (ii) Attempt to find at least one value of θ + α 
          Obtain one correct value of θ (97.4 or 318.7) 
          Carry out correct method to find second answer 
          Obtain second correct value and no others between 0 and 360

Question

 (i) Prove that 2 cosec 2θ tan θ ≡ sec2 θ. [3]

    (ii) Hence

          (a) solve the equation 2 cosec 2θ tan θ = 5 for 0 < θ < π, [3]

          (b) find the exact value of  \(\int_{0}^{\frac{1}{6}\pi }\) 2 cosec 4x tan 2x dx. [4]

Answer/Explanation

Ans:

6 (i) State or imply \(cosec2\theta=\frac{1}{sin2\theta }\)
          Express left-hand side in terms of sinθ and cosθ 
          Obtain given answer sec2 θ correctly

    (ii) (a) State or imply \(cos\theta =\frac{1}{\sqrt{5}}\) or tan θ = 2 at least
                   Obtain 1.11 or awrt 1.11, allow 0.353π
                   Obtain 2.03 or awrt 2.03 , allow 0.648π and no other values between 0 and π 

           (b) State integrand as sec2 2x
                   Integrate to obtain expression of form k tan mx
                   Obtain correct \(\frac{1}{2}\) tan 2x 
                   Obtain \(\frac{1}{2}\sqrt{3}\) or exact equivalent 

Question

(i) Express \(3\cos \Theta +\sin \Theta \)   in the form \(R\cos (\Theta -\alpha ) \), where R > 0 and \(0^{\circ}< \alpha < 90^{\circ}\), giving the exact value of R and the value of correct to 2 decimal places. 
(ii) Hence solve the equation

3 cos 2x + sin 2x = 2,

giving all solutions in the interval \(0^{\circ}\leq x\leq 360^{\circ}\).

Answer/Explanation

(i) State \(R=\sqrt{10}\)
Use trig formula to find α 
Obtain α = 18.43 with no errors seen 
(ii) Carry out evaluation of \(\cos ^{-1}\left ( \frac{2}{R} \right )\approx 50.77^{\circ}\)
Carry out correct method for one correct answer 
Obtain one correct answer e.g. 34.6° 
Carry out correct method for a further answer 
Obtain remaining 3 answers 163.8°, 214.6°, 343.8° and no others in the range

Question

(i) Express 5 sin 2θ + 2 cos 2θ in the form Rsin(2θ + α), where R > 0 and 0◦ < α < 90◦, giving the exact value of R and the value of α correct to 2 decimal places. 
Hence
(ii) solve the equation

5 sin 2θ + 2 cos 2θ = 4,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦

(iii) determine the least value of \(\frac{1}{(10\sin 2\Theta +4\cos 2\Theta )^{2}}\) as θ varies.

Answer/Explanation

(i) State \(R=\sqrt{29}\)
Use trig formula to find α 
Obtain \(\alpha =21.80^{\circ}\) with no errors seen

(ii) Carry out evaluation of \(\sin ^{-1}(\frac{4}{R})\approx 49.97^{\circ}\)
Carry out correct method for one correct answer 
Obtain one correct answer e.g. \(13.1^{\circ}\)

Carry out correct method for a further answer 
Obtain remaining 3 answers \(55.1^{\circ}\),\( 193.1^{\circ}\),\(235.1^{\circ}\)and no others in the range 
(iii) Greatest value of \(10\sin 2\Theta +4\cos 2\Theta =2\sqrt{29}\)

\(\frac{1}{116}\)

Question

(i) Show that \(12\sin ^{2}x\cos ^{2}x=\frac{3}{2}(1-\cos 4x)\)
(ii) Hence show that

\(\int _{\frac{1}{4}\pi }^{\frac{1}{3}\pi }12\sin ^{2}x\cos ^{2}xdx=\frac{\pi }{8}+\frac{3\sqrt{3}}{16}\).

Answer/Explanation

(i) Either
Use \(\sin 2x=2\sin x\cos x\) to convert integrand to \(k\sin ^{2}2x\)

Use \(\cos 4x=1-2\sin ^{2}2x\)

State correct expression \(\frac{1}{2}-\frac{1}{2}\cos 4x\) or equivalent.

Or
Use \(\cos ^{2}x=\frac{1-\cos 2x}{2}\) and/or \(x=\frac{1-\cos 2x}{2}\)  to obtain an equation in cos 2x only

Use \(\cos ^{2}2x=\frac{1+\cos 4x}{2}\).

State correct expression \(\frac{1}{2}-\frac{1}{2}\cos 4x\) or equivalent.

(ii) State correct integral \(\frac{3}{2}x-\frac{3}{8}\sin 4x\)
Attempt to substitute limits, using exact values 
Obtain given answer correctly

Question

Solve the equation \(2\cos 2\Theta =4\cos \Theta -3\),for \(0^{\circ}\leq \Theta \leq 180^{\circ}\).

Answer/Explanation

Make relevant use of the cos 2θ formula 
Obtain a correct quadratic in cos θ 
Solve a quadratic in cos θ 
Obtain answer θ = 60 and no others in the range 
(Ignore answers outside the given range)

Question

(i) Show that \((2\sin x+\cos x)^{2}\) can be written in the form \(\frac{5}{2}+2\sin 2x-\frac{3}{2}\cos 2x\).
(ii) Hence find the exact value of  \(\int ^{\frac{1}{2}\pi }_{0}(2\sin x+\cos x)^{2}dx.\)

Answer/Explanation

(i)Expand  to obtain \(4\sin ^{2}x+4\sin x\cos x+\cos ^{2}x\).

Use \(2\sin x\cos x=\sin 2x\).

Attempt to express \(\sin ^{2}x\) or \(\cos ^{2}x\)(or both) in terms of \(\cos 2x\)

Obtain correct \(\frac{1}{2}k(1-\cos 2x)\) for their or equivalent \(k\sin ^{2}x\) or equivalent 

\(\frac{5}{2}+2\sin 2x-\frac{3}{2}\cos 2x\).

(ii)Integrate to obtain form \(px+q\cos 2x+r\sin 2x\)

Obtain \(\frac{5}{2}x-\cos 2x-\frac{3}{4} \sin 2x.\)

Substitute limits in integral of form  \(px+q\cos 2x+r\sin 2x\) and attempt simplification

Obtain \(\frac{5}{8}\pi +\frac{1}{4}\) or exact equivalent 

Question

(i) Given that \(35+\sec ^{2}\Theta =12\tan \Theta \), find the value of \( \tan \Theta\).
(ii) Hence, showing the use of an appropriate formula in each case, find the exact value of
(a)\(\tan (\Theta -45^{\circ})\)

(b)\(\tan 2\Theta \).

Answer/Explanation

(i) Use \(\sec ^{2}\Theta=1+\tan ^{2}\Theta \)
Attempt solution of quadratic equation in \(\tan \Theta \).
Obtain \(\tan ^{2}\Theta -12\tan \Theta +36=0\) or equivalent and hence  \(\tan \Theta =6\)

(ii) (a) Attempt use of \(\tan (A-B)\) formula 
Obtain \(\frac{5}{7}\)  following their value of \(\tan \Theta \)
(b) Attempt use of \(\tan 2\Theta \) formula 
Obtain \(-\frac{12}{35}\)

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