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- Question 1 of 10
##### 1. Question

Find \(\dfrac{d}{{dx}}\left( {{x^2}\sqrt {{e^x}} } \right)\).

CorrectIncorrect##### Hint

\(\begin{align} \displaystyle

\dfrac{d}{{dx}}{e^{f\left( x \right)}} &= f’\left( x \right) \times {e^{f\left( x \right)}}\\

{\left( {uv} \right)^\prime } &= u’v + uv’

\end{align}\) - Question 2 of 10
##### 2. Question

Find \(\dfrac{d}{{dx}}\sqrt x {\log _e}\left( {5x} \right)\).

CorrectIncorrect##### Hint

\(\dfrac{d}{{dx}}{\log _e}f\left( x \right) = \dfrac{1}{{f\left( x \right)}} \times f’\left( x \right)\)

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

Find for the values of \(x\) for which \(f\left( x \right) = {x^3} + 6{x^2} + 9x + 7\) is concave downwards.

CorrectIncorrect##### Hint

\(\begin{align} \displaystyle

y” &> 0 \to {\rm{concave upwards}}\\

y” &< 0 \to {\rm{concave downwards}} \end{align}\) - Question 4 of 10
##### 4. Question

Find \(p\), \(q\) and \(r\), if the curve \(f\left( x \right) = \dfrac{p}{{{{\left( {1 – 2x} \right)}^q}}} + r\) passes through \(\left( {1,3} \right)\) and \(f’\left( x \right) = \dfrac{8}{{{{\left( {1 – 2x} \right)}^5}}}\).

\(p = \) , \(q = \) , \(r = \)

CorrectIncorrect - Question 5 of 10
##### 5. 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 6 of 10
##### 6. Question

Integrate \(\int {{{\left( {{e^x} – {e^{ – x}}} \right)}^3}dx} \).

CorrectIncorrect##### Hint

\(\int {{e^{ax + b}}dx} = \dfrac{1}{a}{e^{ax + b}} + C\\

{\left( {a – b} \right)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3}\) - Question 7 of 10
##### 7. Question

Evaluate \(\int_{ – 2}^0 {{e^{\frac{x}{3}}}} dx\), correcting to two significant figures.

CorrectIncorrect - Question 8 of 10
##### 8. Question

Find the area between the \(x\)-axis and \(y = {x^3}\) from \(x = -1\) to \(x = 1\).

Area =

CorrectIncorrect - Question 9 of 10
##### 9. Question

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

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

A particle is initially at the origin, and is stationary. It accelerates according to \(a\left( t \right) = \dfrac{1}{{{{\left( {t + 1} \right)}^2}}}\) m/s

^{2}. Find the best description of the motion of the particle at \(t = 3\).CorrectIncorrect