DP Mathematics: Analysis and Approaches Questionbank

Find $f\left(1\right)$.

Solve $f\left(x\right)=0$.

The graph of $f$ has a local minimum at point $\text{A}$.

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

Write down the zero of $f\left(x\right)$.

Write down the coordinates of the local minimum point.

Consider the function $g\left(x\right)=3-x$.

Solve $f\left(x\right)=g\left(x\right)$.

Sketch the curve $y=f\left(x\right)$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

Find the time taken for the water to have a temperature of $50 °\text{C}$. Give your answer correct to the nearest second.

Sketch the graph of $T$ versus $t$, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.

The model for the temperature of the water can also be expressed in the form $T=T_0{a}^{\frac{t}{10}}$ for $t\ge 0$ and $a$ is a positive constant.

Find the exact value of $a$.

Find the $x$-coordinates of the points of intersection of the two curves.

Find the $x$-coordinate of $\text{P}$.

Find the value of $k$.

By considering the graph of $y=g\left(x\right)-f\left(x\right)$, or otherwise, solve $f\left(x\right)<g\left(x\right)$ for $x\in \mathrm{ℝ}$.

Find the value of $k$, giving your answer correct to four significant figures.

Sketch the graph of $y=f\left(x\right)$ for $-3\le x\le 3$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.

Consider the function $g\left(x\right)=x^3-3cx+d$ for $x\in \mathrm{ℝ}$ and where $c , d\in \mathrm{ℝ}$.

Find all conditions on $c$ and $d$ such that the graph of $y=g\left(x\right)$ has exactly one $x$-axis intercept, explaining your reasoning.

Sketch the graph of $y=g^{-1}\left(x\right)$, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

State the cross-sectional radius of the bowl at this point.

Sketch the graph of $f$ on the grid below.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

Find the value of $x$ for which $f\prime \left(x\right)=0$.

Given that the areas of the two shaded regions are equal, show that $\theta =2 \sin \theta$.

Find an expression, in terms of $x$ for the travel time $T$, from $\text{A}$ to $\text{B}$, passing through $\text{D}$.

Hence determine the value of $\theta$.

the boat travels directly to $\text{B}$.

the boat is taken from $\text{A}$ to $\text{P}$, and the bus from $\text{P}$ to $\text{B}$.

Find the value of $x$ so that $T$ is a minimum.

Write down the minimum value of $T$.

Find the smallest possible value of $c$.

Write down the minimum total cost for this journey.

Find the value of $a$.

Find the value of $d$.

Find the new value of $x$ so that the total cost $C$ to travel from $\text{A}$ to $\text{B}$ via $\text{D}$ is a minimum.

Show that $b=\frac{\mathrm{\pi }}{6}$.

Find the height of the water at $12:00$.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

Determine the value of $m$.

Show that $b=\frac{\mathrm{\pi }}{6}$.

Find the value of $a$.

Find the value of $d$.

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

Given that $P\left(\left|X-m\right|\le a\right)=0.3$, determine the value of $a$.

Find the smallest possible value of $c$.

Find the height of the water at $12:00$.

By varying the value of $b$, suggest two key features common to these curves.

Hence, find the coordinates of the rational point $\text{Q}$ where this tangent intersects $C$, expressing each coordinate as a fraction.

The point $\text{S}(-1 , 1)$ also lies on $C$. The line $\text{[QS]}$ intersects $C$ at a further point. Determine the coordinates of this point.

By considering each curve from part (a), identify two key features that would distinguish one curve from the other.

Write down the equation of the vertical asymptote.

On the set of axes below, sketch the graph of $y=f\left(x\right)$.

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.

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

Write down the equation of the horizontal asymptote.

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

Write down the equation of the vertical asymptote.

Write down the equation of the horizontal asymptote.

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

Plant $B$.

Plant $A$ correct to three significant figures.

Plant $B$.

Plant $A$ correct to three significant figures.

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

Sketch the graph of $f″$, the second derivative of $f$. Indicate clearly the $x$-intercept and the $y$-intercept.

Sketch the curve $y=f\left(x\right)$, clearly indicating the coordinates of the endpoints.

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

Find the largest possible domain $D$ for $f$ to be a function.

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Explain why $f$ is an even function.

Explain why the inverse function $f^{-1}$ does not exist.

Find the inverse function $g^{-1}$ and state its domain.

Find $g^{\prime }(x)$.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

State the range of $f$.

Solve the inequality $\left|3x\arccos (x)\right|>1$.

Write down the $y$-intercept of the graph.

Find $f^{\prime }(x)$.

Show that $a=8$.

Find $f(2)$.

Write down the $x$-coordinates of these two points;

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

Write down the number of possible solutions to the equation $f(x)=5$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Express this volume in $\text{c}{\text{m}}^3$.

Find $\frac{\text{d}A}{\text{d}r}$.

Find the value of $f\left(x\right)$ when $x=\frac{1}{2}$.

Find the function, $P\left(x\right)$ , that would define this footpath on the map.

State the domain of $P$.

Find the coordinates of the bridges relative to the centre of Orangeton.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

Determine the amount that he will have in his account after 3 years. Give your answer correct to two decimal places.

Find the difference between the cost of the bicycle and the amount of money in Tommaso’s account after 3 years. Give your answer correct to two decimal places.

After $m$ complete months Tommaso will, for the first time, have enough money in his account to buy the bicycle.

Find the value of $m$.

Show that $b=0.3-a$.

Find the difference between the greatest possible expected value and the least possible expected value.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

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