DP Mathematics: Analysis and Approaches Questionbank

The first term in an arithmetic sequence is $4$ and the fifth term is ${\log }_2 625$.

Find the common difference of the sequence, expressing your answer in the form ${\log }_2 p$, where $p\in \mathrm{\mathbb{Q}}$.

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 the common ratio of the sequence.

Find the volume of the smallest slice of pie.

The apple pie has a volume of $61 425 {\text{cm}}^3$.

Find the total number of slices Mia can cut from this pie.

For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.

Find Jane’s total earnings at the start of her $10$th year of employment. Give your answer correct to the nearest dollar.

Show that $H_n=2400n+67 600$.

Find the value of $N$.

Given that $J_n$ follows a geometric sequence, state the value of the common ratio, $r$.

Given that the grand prize is not won and the grand prize continues to double, write an expression in terms of $n$ for the value of the grand prize in the $n\text{th}$ week of the lottery.

Show that $a=8$.

Write down an expression for $f^{-1}\left(x\right)$.

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

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

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Find the first term of the sequence, $u_1$.

Find $S_{\infty }$.

Find the least value of $n$ such that $S_{\infty }-S_n<0.001$.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Hence or otherwise, show that the series is convergent.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Find Gemma’s annual salary for the year 2021, to the nearest dollar.

Write down the distance Rosa runs in the third training session;

Write down the distance Rosa runs in the $n$th training session.

Find the value of $k$.

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

Find the distance Carlos runs in the fifth month of training.

Calculate the total distance Carlos runs in the first year.

Calculate the café’s total profit for the first 12 weeks.

Calculate the tea-shop’s total profit for the first 12 weeks.

Write down the common ratio of the sequence.

Find the value of $u_5$.

Find the smallest value of $n$ for which $u_n$ is less than ${10}^{-3}$.

Find the value of $r$, the common ratio of the sequence.

Find the value of $n$ for which $u_n=2$.

Find the sum of the first 30 terms of the sequence.

Calculate, in CAD, the total amount John pays for the bicycle.

Find the value of the bicycle during the 5th year. Give your answer to two decimal places.

Calculate, in years, when the bicycle value will be less than 50 USD.

Find the total amount John has paid to insure his bicycle for the first 5 years.

John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.

Find the value of the common ratio for this sequence.

Find the distance that the post is driven into the ground by the eighth strike of the hammer.

Find the total depth that the post has been driven into the ground after 10 strikes of the hammer.

A geometric sequence has $u_4=-70$ and $u_7=8.75$. Find the second term of the sequence.

Find the value of $r$.

Find the value of $u_{10}$.

Find the least value of $n$ such that $S_n>5543$.

Find the least value of n for which S_n > 163.

Given that $q=8d^2$, find the other two real roots.

Find the values of θ which give the greatest value of the sum.

Find the sum of the first 8 terms.

Show that $abc=8$.

Find the common ratio.

Find an expression for r in terms of θ.

The following diagram shows [CD], with length $b\text{ cm}$, where $b>1$. Squares with side lengths $k\text{ cm},k^2\text{ cm},k^3\text{ cm},\dots$, where , are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.

The total sum of the areas of all the squares is $\frac{9}{16}$. Find the value of $b$.

Find the sum to infinity of this sequence.

Find the common ratio.

Show that one of the real roots is equal to 1.

Consider a geometric sequence where the first term is 768 and the second term is 576.

Find the least value of $n$ such that the $n$th term of the sequence is less than 7.

Show that $\text{P}(A\cup B)=\text{P}(A)+\text{P}(A^{\prime }\cap B)$.