21N.1.SL.TZ0.6

METHOD 1

attempt to write all LHS terms with a common denominator of $x-1$                 (M1)

$2x-3-\frac{6}{x-1}=\frac{2x\left(x-1\right)-3\left(x-1\right)-6}{x-1}$   OR   $\frac{\left(2x-3\right)\left(x-1\right)}{x-1}-\frac{6}{x-1}$

$=\frac{2x^2-2x-3x+3-6}{x-1}$   OR   $\frac{2x^2-5x+3}{x-1}-\frac{6}{x-1}$                 A1

$=\frac{2x^2-5x-3}{x-1}$                 AG

 

METHOD 2

attempt to use algebraic division on RHS                 (M1)

correctly obtains quotient of $2x-3$ and remainder $-6$                 A1

$=2x-3-\frac{6}{x-1}$ as required.                 AG

 

[2 marks]

consider the equation $\frac{2 {\sin }^2 2\theta -5 \sin 2\theta -3}{\sin 2\theta -1}=0$                 (M1)

$\Rightarrow 2 {\sin }^2 2\theta -5 \sin 2\theta -3=0$

EITHER

attempt to factorise in the form $\left(2 \sin 2\theta +a\right)\left(\sin 2\theta +b\right)$                 (M1)

Note: Accept any variable in place of $\sin 2\theta$.

$\left(2 \sin 2\theta +1\right)\left(\sin 2\theta -3\right)=0$

OR

attempt to substitute into quadratic formula                 (M1)

$\sin 2\theta =\frac{5\pm \sqrt{49}}{4}$

THEN

$\sin 2\theta =-\frac{1}{2}$  or  $\sin 2\theta =3$                 (A1)

Note: Award A1 for $\sin 2\theta =-\frac{1}{2}$ only.

one of $\frac{7\mathrm{\pi }}{6}$  OR  $\frac{11\mathrm{\pi }}{6}$   (accept $210$ or $330$)                 (A1)

$\theta =\frac{7\mathrm{\pi }}{12}, \frac{11\text{π}}{12}$  (must be in radians)                 A1

Note: Award A0 if additional answers given.

 

[5 marks]

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