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

Evaluate \(\mathop {\lim }\limits_{x \to \infty } \dfrac{{{x^2} + 3x – 1}}{{{x^2} – 2x – 3}}\).

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

Differentiate \(y = \sqrt[4]{{5x – 1}}\).

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

Find \(a\), \(b\) and \(c\) gave that \(f\left( x \right) = 5x – 4{x^{ – 2}} + 9{x^{ – 3}} + {x^{ – 4}}\) and \(f”’\left( x \right) = 96{x^{ – 5}} – 540{x^{ – 6}} – 120{x^{ – 7}}\).

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

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

Find \(A\) and \(B\), if the equation of the normal for \(y = \dfrac{1}{{{x^2}}}\) at \(x = 2\) is \(Ax + By – 63 = 0\).

\(A = \) , \(B = \)

CorrectIncorrect##### Hint

\(\text{gradient of tangent } \times \text{gradient of normal } = – 1\)

- Question 5 of 10
##### 5. 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 6 of 10
##### 6. 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 7 of 10
##### 7. Question

A particle \(P\) moves in a straight line with a velocity function of \(s\left( t \right) = {t^3} – 6{t^2} + 1\) metres, where \(t \ge 0\), \(t\) in seconds. Find the total distance travelled for the first 5 seconds.

metres

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

Find \(p\) and \(q\), if \(\int {{{\left( {\dfrac{{5x}}{2} + 2} \right)}^3}} dx = \dfrac{1}{p}{\left( {5x + 4} \right)^q} + C\).

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

CorrectIncorrect##### Hint

\(\int {{{\left( {ax + b} \right)}^n}} dx = \dfrac{{{{\left( {ax + b} \right)}^{n + 1}}}}{{a\left( {n + 1} \right)}} + C\)

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

Find \(p\) and \(q\), if \(\int {\dfrac{x}{{{{\left( {{x^2} + 2} \right)}^4}}}} dx = – \dfrac{1}{{p{{\left( {{x^2} + 2} \right)}^q}}} + C\).

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

CorrectIncorrect##### Hint

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

A particle has velocity function \(v\left( t \right) = \cos 3t\) cm/s as it moves in a straight line. The particle is initially 0.5 cm to the right of the origin. Write a formula for the displacement function.

cm

CorrectIncorrect