Exam-style practice (harder): Maths Methods multiple choice quiz

Let $f(x) = \dfrac{\ln x}{x}$ for $x > 0$.

What is the value of $f'(e)$?

What is $\dfrac{d}{dx}\left[\sin^2(3x)\right]$?

The tangent to the curve $y = e^{2x}$ at the point where $x = 0$ crosses the x-axis at which point?

Let $f(x) = x^3 - 6x^2 + 9x + 1$.

For which values of x is the graph of $y = f(x)$ concave up?

A rectangular garden bed is built against a straight wall, so fencing is needed on only three sides. There is 20 m of fencing.

What is the maximum possible area of the garden bed?

What is the area of the region bounded by the curve $y = \frac{1}{x}$, the x-axis and the lines $x = 1$ and $x = e^2$?

What is the area of the region enclosed between the graphs of $y = x$ and $y = x^2$?

Let $F(x) = \displaystyle\int_0^x (t^2 + 1)\,dt$.

What is the value of $F'(2)$?

A discrete random variable X has $E(X) = 3$ and $\text{Var}(X) = 2$. Let $Y = 4X - 1$.

What are $E(Y)$ and $\text{Var}(Y)$?

The random variable X has a binomial distribution with $n = 5$ and $p = 0.4$.

What is $P(X = 2)$?

A continuous random variable X has probability density function $f(x) = 2x$ for $0 \le x \le 1$, and $f(x) = 0$ elsewhere.

What is $P(X > 0.5)$?

Ali scored 78 in a maths test (class mean 70, standard deviation 5) and 85 in an English test (class mean 75, standard deviation 8).

Relative to each class, in which test did Ali perform better?

In a large population, the proportion of people with a certain trait is $p = 0.2$. Random samples of size $n = 100$ are taken.

What is the standard deviation of the sample proportion $\hat{P}$?

A survey gives an approximate 95% confidence interval of (0.42, 0.50) for the proportion p of students who walk to school.

Which interpretation is correct?