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- A mixture of short answer and multiple-choice questions
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- Question 1 of 10
##### 1. Question

Find \( x \) if \({25^3} = {5^x}\).

\( x = \)

CorrectIncorrect##### Hint

\({\left( {{a^x}} \right)^y} = {a^{xy}}\)

- Question 2 of 10
##### 2. Question

Find \(a \) and \(b \) , if $\dfrac{1}{{\sqrt 8 – 2}} + \dfrac{1}{{2\sqrt 8 – 2}} = \dfrac{{a + b\sqrt 2 }}{{14}}$.

\(a = \) , \(b = \)

CorrectIncorrect##### Hint

\(\dfrac{1}{{\sqrt a + \sqrt b }} = \dfrac{1}{{\sqrt a + \sqrt b }} \times \dfrac{{\sqrt a – \sqrt b }}{{\sqrt a – \sqrt b }} = \dfrac{{\sqrt a – \sqrt b }}{{a – b}}\)

- Question 3 of 10
##### 3. Question

The perimeter of a rectangular playing field is 170 m and the length of the diagonals of the field is 65 m. Calculate the dimensions of the field.

shorter side =

longer side =

CorrectIncorrect - Question 4 of 10
##### 4. Question

Given \(f\left( x \right) = 2x – 3\), \(g\left( x \right) = – x + k\) and \(\left( {g \circ f} \right)\left( x \right) = \left( {f \circ g} \right)\left( x \right)\), find \(k\).

CorrectIncorrect##### Hint

\(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\)

- Question 5 of 10
##### 5. Question

Find the angle of inclination, in radian correcting to two significant figures, of a line having a gradient of 1.2.

CorrectIncorrect - Question 6 of 10
##### 6. Question

Solve \(\cos 2x – \cos x = 0\) for \(0^\circ \le x \le 360^\circ \). Write answers in ascending order.

\(x = \) \(^\circ\), \(^\circ\), \(^\circ\), \(^\circ\)

CorrectIncorrect##### Hint

\(\cos 2x = 2{\cos ^2}x – 1\)

- Question 7 of 10
##### 7. Question

Find \(k\), correcting to two significant figures, such a vector \(k\left( {\begin{array}{*{20}{c}}

4\\

{ – 3}\\

1

\end{array}} \right)\) is in the same direction as \(\left( {\begin{array}{*{20}{c}}

4\\

{ – 3}\\

1

\end{array}} \right)\) and with length 3 units.CorrectIncorrect - Question 8 of 10
##### 8. Question

Find for the values of \(x\) for which \(\dfrac{{dy}}{{dx}} = {x^3} – 3x\) is concave upwards.

CorrectIncorrect##### Hint

\(\begin{align} \displaystyle

y” &> 0 \to {\rm{concave upwards}}\\

y” &< 0 \to {\rm{concave downwards}} \end{align}\) - Question 9 of 10
##### 9. Question

Find \(\int {\left( {\sin \dfrac{x}{2} + \cos \dfrac{x}{3} + 4{x^3} + 1} \right)dx} \).

CorrectIncorrect##### Hint

\(\int {\sin \left( {ax + b} \right)dx} = – \dfrac{1}{a}\cos \left( {ax + b} \right) + C\\

\int {\cos \left( {ax + b} \right)dx} = \dfrac{1}{a}\sin \left( {ax + b} \right) + C\) - Question 10 of 10
##### 10. Question

Find the area of the region bounded by \(y = {x^3} – 2{x^2}\) and \(y = {x^2} – 2x\), correcting to one decimal place.

CorrectIncorrect