DP Mathematics: Analysis and Approaches Questionbank

Find the times when the particle’s acceleration is $-1.9 {\text{m\hspace{0.05em}s}}^{-2}$.

Find the particle’s acceleration when its speed is at its greatest.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

Find the values of $t$ when $h_A\left(t\right)=h_B\left(t\right)$.

For $0\le t\le 9$, find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$.

After $10$ years, the population of marsupials is $3P_0$. It is known that $N=4P_0$.

Find the value of $k$ for this population model.

Find the coordinates of $\text{B}$.

Hence, solve the inequality $0<\frac{2x-1}{x+1}<2$.

Solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Determine the length of time between the first airplane arriving at $\text{P}$ and the second airplane arriving at $\text{P}$.

Find the coordinates of $\text{P}$.

Find the period of $f$.

Find the value of $x$ for which the functions have the greatest difference.

Find the value of $b$.

Hence, find the value of $f$(6).

Find the value of $p$ and the value of $q$.

The quadratic equation $x^2-2kx+(k-1)=0$ has roots $\alpha$ and $\beta$ such that ${\alpha }^2+{\beta }^2=4$. Without solving the equation, find the possible values of the real number $k$.

Find the area of the region enclosed by the graph of $f$ and the line $L$.

Solve f '(x) = f "(x).

Given that x_{k + 1} = x_k + a, find a.

Hence find the value of n such that $\sum _{k=1}^n{x}_k=861$.

Show that $\cos \theta =\frac{3}{4}$.

Given that $\tan \theta >0$, find $\tan \theta$.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Find the value of $p$ and of $q$.

Hence, find the area of the region enclosed by the graphs of $f$ and $g$.

Find the value of $p$.

The following diagram shows part of the graph of $f$.

The region enclosed by the graph of $f$, the $x$-axis and the lines $x=-p$ and $x=p$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Consider the vectors a = $\left(\begin{array}{c}3\\ 2p\end{array}\right)$ and b = $\left(\begin{array}{c}p+1\\ 8\end{array}\right)$.

Find the possible values of p for which a and b are parallel.

Show that $\text{OC}=r\text{cos}\theta$.

Find the area of triangle OBC in terms of $r$ and θ.

Given that the area of triangle OBC is $\frac{3}{5}$ of the area of sector OAB, find θ.

Find $q$.

Find $p$.

Write down the probability of drawing three blue marbles.

Explain why the probability of drawing three white marbles is $\frac{1}{6}$.

The bag contains a total of ten marbles of which $w$ are white. Find $w$.

Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.

Jill plays the game nine times. Find the probability that she wins exactly two prizes.

Write down the coordinates of $\text{B}$.

Given that $f^{\prime }\left(a\right)=\frac{1}{\text{ln}p}$, find the equation of $L_1$ in terms of $x$, $p$ and $q$.

The line $L_2$ is tangent to the graph of $g$ at $\text{A}$ and has equation $y=\left(\text{ln}p\right)x+q+1$.

The line $L_2$ passes through the point $\left(-2\text{, }-2\right)$.

The gradient of the normal to $g$ at $\text{A}$ is $\frac{1}{\text{ln}\left(\frac{1}{3}\right)}$.

Find the equation of $L_1$ in terms of $x$.

$\overrightarrow{\text{OA}}\bullet \overrightarrow{\text{OC}}$.

$\overrightarrow{\text{OB}}\bullet \overrightarrow{\text{OC}}$.

Given that $\text{A}\hat{\text{O}}\text{C}=\text{B}\hat{\text{O}}\text{C}$, show that $k=7$.

Calculate the area of triangle $\text{AOC}$.

Find an expression for the velocity, $v$ m s⁻¹, of the rocket during the first stage.

Find the distance that the rocket travels during the first stage.

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

Find the point of intersection of $L_1$ and $L_2$.

Write down a direction vector for $L_3$.

$L_3$ passes through the intersection of $L_1$ and $L_2$.

Write down a vector equation for $L_3$.

Calculate the probability a flight is not late.

Find the value of $m$.

Calculate the probability that at least $7$ of these flights are on time.

Given that at least $7$ of these flights are on time, find the probability that exactly $10$ flights are on time.

SpeedWay increases the number of flights from city $\text{A}$ to city $\text{B}$ to $20$ flights each week, and improves their efficiency so that more flights are on time. The probability that at least $19$ flights are on time is $0.788$.

A flight is chosen at random. Calculate the probability that it is on time.