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

The point (4, 5) lies on the graph of $y = f(x)$. The graph is transformed to $y = 2f(x - 3) + 1$.

What are the coordinates of the image of (4, 5)?

Let $f(x) = x^2 e^{3x}$.

What is the value of $f'(1)$?

What is the total area enclosed between the curve $y = x^2 - 4$, the x-axis and the lines $x = 0$ and $x = 3$?

A function has derivative $f'(x) = (x - 1)^2(x + 2)$.

Which statement correctly describes the stationary points of $y = f(x)$?

How many solutions does the equation $2\sin(2x) = 1$ have in the domain $0 \le x \le 2\pi$?

Which expression gives the exact solution of $3^{x+1} = 5^x$?

A geometric series has first term 8 and a limiting sum of 24.

What is the common ratio?

Sam deposits $200 at the START of each month into an account that earns interest of 0.5% per month, compounded monthly. He makes 24 deposits.

Which expression gives the balance at the end of the 24th month?

For 40 students, x is hours of gaming per week (from 0 to 12) and y is a test score out of 60. The correlation coefficient is $r = -0.92$ and the least-squares regression line is $y = 58.4 - 3.1x$.

Which statement is correct?

The heights of students are normally distributed with a mean of 170 cm and a standard deviation of 8 cm.

Using the empirical rule (68%, 95%, 99.7%), approximately what percentage of students are between 162 cm and 186 cm tall?

The function $f(x) = kx(4 - x)$ for $0 \le x \le 4$, and $f(x) = 0$ otherwise, is a probability density function.

What is the value of k?

Which option correctly describes the graph of $y = 3\cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 1$?

A particle starts at the origin. Its velocity is $v = 6t - 3t^2$ m/s, where t is in seconds.

What is the total distance travelled in the first 3 seconds?

A discrete random variable X has the probability distribution below.

What is the standard deviation of X?