ENGAA 2023 Section 1, Part B (Mathematics questions)

$x^2-x-6$ is a factor of $x^3+a{x}^2+2x+b$, where $a$ and $b$ are real constants.

What is the value of $a+b$?

An arithmetic progression has first term $a$ and common difference $d$.

The sum of the first 9 terms plus the sum of the first 10 terms is equal to the sum of the first 11 terms.

Which of the following is a correct expression for $a$ in terms of $d$?

The first three terms of a convergent geometric progression are:$$2p,p-3,p-7$$What is the sum to infinity of this progression?

The diagram shows a circle with centre $O$.

What is the value of $x$?

The function$$f(x)=\sqrt{2}{x}^2-6x+4$$can be written in the form$$f(x)=p(x+q{)}^2+r$$where $p$, $q$ and $r$ are constants.

What is the value of $p(r-q)$?

Which of the following is equal to$${\int }_{-\left(3-\sqrt{5}\right)}^{3-\sqrt{5}}\frac{x^2}{3-\sqrt{5}}dx$$

A function $f$ is defined by$$f(x)=\frac{a}{x}+\frac{b}{x^2}$$where $a$ and $b$ are constants.

It is given that $f^{\prime }(1)=2$ and $f^{\prime \prime }(-1)=-2$

What is the value of $a+b$?

Find the real value of $x$ that satisfies$${\log }_3\left(x^2+3x+2\right)=2+{\log }_3\left(x^2+2x\right)$$

What is the constant term in the simplified binomial expansion of ${\left(\frac{2}{x}+\frac{x}{4}\right)}^8$?

Given that$$x^2+y^2=1$$what is the greatest possible value of $2x+3y$?