DP Mathematics: Analysis and Approaches Questionbank

Find the times when $P$ comes to instantaneous rest.

Find an expression for $s$ in terms of $t$.

Find the maximum displacement of $P$, in metres, from its initial position.

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

Show that, at these times, $\tan 6t=2$.

Hence show that $\frac{v_2}{v_1}=\frac{v_3}{v_2}=-{\text{e}}^{-\frac{\mathrm{\pi }}{2}}$.

Find an expression for the velocity of $P_1$ at time $t$.

Particle $P_2$ also moves in a straight line. The position of $P_2$ is given by $\mathit{r}=\left(\begin{array}{c}-1\\ 6\end{array}\right)+t\left(\begin{array}{c}4\\ -3\end{array}\right)$.

The speed of $P_1$ is greater than the speed of $P_2$ when $t>q$.

Find the value of $q$.

Let $s_1$ be the distance travelled by the particle from $t=0$ to $t=T$ and let $s_2$ be the distance travelled by the particle from $t=T$ to $t=1$.

Show that $s_2>s_1$.

Find the value of $T$.

Find the smallest value of $t$ for which the particle is at rest.

Find the total distance travelled by the particle.

Find the acceleration of the particle when $t=7$.

Find the value of $q$.

Find the displacement of particle A from the origin when $t=q$.

Find the distance of particle A from the origin when $t=10$.

Find the value of $d$.

By solving an appropriate differential equation, show that the particle’s velocity at time $t$ is given by $v\left(t\right)=(1+v_0){\text{e}}^{-t}-1$.

Find the value of $p$.

A second particle, particle B, travels along the same straight line such that its velocity is given by $v\left(t\right)=14-2t$, for $t\ge 0$.

When $t=k$, the distance travelled by particle B is equal to $d$.

Find the value of $k$.

Show that the time $T$ taken for the particle to reach $s_{\text{max}}$ satisfies the equation ${\text{e}}^T=1+v_0$.

By solving an appropriate differential equation and using the result from part (b) (i), find an expression for $s_{\text{max}}$ in terms of $v_0$.

By using the result to part (b) (i), show that $v\left(T-k\right)={\text{e}}^k-1$.

Deduce a similar expression for $v(T+k)$ in terms of $k$.

Hence, show that $v\left(T-k\right)+v\left(T+k\right)\ge 0$.

Show that the distance of $P$ from $\mathrm{O}$ at this time is $\frac{88}{27}$ metres.

Find the value of $t$ when $P$ reaches its maximum velocity.

Sketch a graph of $v$ against $t$, clearly showing any points of intersection with the axes.

Find the total distance travelled by $P$.

Find the value of $t$ when the particle is at rest.

Find the acceleration of the particle when it changes direction.

Find the total distance travelled by the particle.

Determine when the particle changes its direction of motion.

Find the acceleration of the particle at the instant it first changes direction.

Find the maximum distance of the particle from O.

Show that $f^{\prime }\left(0\right)=0$.

By differentiating the above equation twice, show that

$$\left(1-x^2\right)f^{\left(4\right)}\left(x\right)-5x{f}^{\left(3\right)}\left(x\right)-4f^{″}\left(x\right)=0$$

where $f^{\left(3\right)}\left(x\right)$ and $f^{\left(4\right)}\left(x\right)$ denote the 3rd and 4th derivative of $f\left(x\right)$ respectively.

Hence show that the Maclaurin series for $f\left(x\right)$ up to and including the term in $x^4$ is $x^2+\frac{1}{3}{x}^4$.

Use this series approximation for $f\left(x\right)$ with $x=\frac{1}{2}$ to find an approximate value for ${\pi }^2$.

A particle moves in a straight line such that at time $t$ seconds $(t⩾0)$, its velocity $v$, in $\text{m}{\text{s}}^{-1}$, is given by $v=10t{\text{e}}^{-2t}$. Find the exact distance travelled by the particle in the first half-second.

Determine the first time t₁ at which P has zero velocity.

Find an expression for the acceleration of P at time t.

Find the value of the acceleration of P at time t₁.

Find $t_1$ and $t_2$.

Find the displacement of the particle when $t=t_1$

Find his velocity when $t=15$.

Calculate the vertical distance Xavier travelled in the first 10 seconds.

Determine the value of $h$.

A particle moves along a horizontal line such that at time $t$ seconds, $t$ ≥ 0, its acceleration $a$ is given by $a$ = 2$t$ − 1. When $t$ = 6 , its displacement $s$ from a fixed origin O is 18.25 m. When $t$ = 15, its displacement from O is 922.75 m. Find an expression for $s$ in terms of $t$.

Write down the values of $t$ when $a=0$.

Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.

Find an expression for the velocity of P at time $t$.

Find the total distance travelled by P when its velocity is increasing.

Find the coordinates of A.

For the graph of $f$, write down the amplitude.

For the graph of $f$, write down the period.

Hence, write $f\left(x\right)$ in the form $p\text{cos}\left(x+r\right)$.

Find the maximum speed of the ball.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

Find the initial velocity of P.

Find the maximum speed of P.

Write down the number of times that the acceleration of P is 0 m s⁻² .

Find the acceleration of P when it changes direction.

Find the total distance travelled by P.

Note:     In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$, is given by $v(t)=\frac{2t^2-4t}{t^2-2t+2}$, for $0⩽t⩽5$. The following diagram shows the graph of $v$

There are $t$-intercepts at $(0,0)$ and $(2,0)$.

Find the maximum distance of the particle from A during the time $0⩽t⩽5$ and justify your answer.

Write down the first value of $t$ at which P changes direction.

Find the total distance travelled by P, for $0⩽t⩽8$.

A second particle Q also moves along a straight line. Its velocity, $v_{\text{Q}}\text{ m}{\text{s}}^{-1}$ after $t$ seconds is given by $v_{\text{Q}}=\sqrt{t}$ for $0⩽t⩽8$. After $k$ seconds Q has travelled the same total distance as P.

Find $k$.

Find when the particle is at rest.

Find the acceleration of the particle when $t=2$.

Find the total distance travelled by the particle.

Find the initial velocity of $P$.

Find the value of $p$.

(i)     Find the value of $q$.

(ii)     Hence, find the speed of P when $t=q$.

(i)     Find the total distance travelled by P between $t=1$ and $t=p$.

(ii)     Hence or otherwise, find the displacement of P from A when $t=p$.

Find the value of $s\left(4\right)-s\left(2\right)$.

Find the total distance travelled in the first 5 seconds.

Find the initial displacement of particle A from point P.

Find the value of $t$ when particle A first reaches point P.

Find the value of $t$ when particle A first changes direction.

Find the total distance travelled by particle A in the first 3 seconds.

Given that particles A and B start at the same point, find the displacement function $s_{\text{B}}$ for particle B.

Find the other value of $t$ when particles A and B meet.

Determine the coordinates of point $\text{A}$ and the coordinates of point $\text{B}$.

Hence, write down the maximum speed of the body.

the value of $t_1$.

the distance travelled between $t=0$ and $t=t_1$.

the acceleration when $t=t_1$.

Find the distance travelled in the first 30 seconds.

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.