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

The graph of $y = f(x)$ is transformed by a dilation by a factor of 2 from the y-axis, followed by a reflection in the x-axis, followed by a translation of 3 units in the positive direction of the y-axis.

What is the rule of the image?

Let $f(x) = \sqrt{x - 2}$ and $g(x) = x^2 + 1$, each defined on its maximal domain.

What is the maximal domain of $f(g(x))$?

Let $f: [1, \infty) \to R$, $f(x) = (x - 1)^2 + 2$.

Which option gives the inverse function?

What is the equation of the tangent to the graph of $y = x \log_e(x)$ at the point where $x = e$?

What is the average value of $f(x) = \sin(x)$ over the interval $[0, \pi]$?

For which values of k does the graph of $y = x^3 - 3x + k$ have exactly two x-intercepts?

What is the value of $\displaystyle\int_0^{\log_e 2} e^{2x}\,dx$?

The random variable X has a binomial distribution with n = 10 and p = 0.3.

What is $\Pr(X \ge 2)$, correct to four decimal places?

The random variable X has a binomial distribution with mean 6 and variance 4.2.

What are the values of n and p?

A continuous random variable X has probability density function $f(x) = \frac{3x^2}{8}$ for $0 \le x \le 2$, and $f(x) = 0$ elsewhere.

What is the median of X?

The random variable X is normally distributed with mean 50. It is known that $\Pr(X < 42) = 0.1$.

What is $\Pr(42 < X < 58)$?

In a random sample of 400 voters, 240 support a proposal.

Using $z \approx 1.96$, which is an approximate 95% confidence interval for the proportion of all voters who support the proposal?

A survey gives an approximate 95% confidence interval for a population proportion. The survey is repeated with a random sample four times as large, and the sample proportion is the same.

How does the width of the new approximate 95% confidence interval compare with the original?

Which function satisfies $f(x + y) = f(x)f(y)$ for all real x and y?