TMUA 2022 Paper 2

Determine the number of stationary points on the curve with equation $$y=3x^4+4x^3+6x^2-5$$

Find the coefficient of the $x^5$ term in the expansion of $$(1+x{)}^5\times \sum _{i=0}^5{x}^i$$

Consider the following statement about the positive integer $n$: $$\text{if }n\text{ is prime, }\text{then }{n}^2+2\text{ is }\text{not}\text{ prime}$$ Which of the following is a counterexample to this statement?

• I: $n=2$
• II: $n=3$
• III: $n=4$

The point $P$ has coordinates $(p,q)$, and the equation of a circle is $$x^2+2fx+y^2+2gy+h=0$$ where $f$, $g$, $h$, $p$ and $q$ are all real constants. Let $L$ be the distance between the centre of the circle and the point $P$. Which one of the following is sufficient on its own to be able to calculate $L$?

A straight line $L$ passes through $(1,2)$. Let P be the statement: if the $y$-intercept of $L$ is negative, then the $x$-intercept of $L$ is positive. Which of the following statements must be true?

• I: P
• II: the converse of P
• III: the contrapositive of P

A list consists of $n$ integers. Consider the following statements: P: $n$ is odd. Q: The median of the list is one of the numbers in the list. Which one of the following is true?

Consider the following claim: The difference between two consecutive positive cube numbers is always prime. Here is an attempted proof of this claim:

• I: $(x+1{)}^3=x^3+3x^2+3x+1$
• II: Taking $x$ to be a positive integer, the difference between two consecutive cube numbers can be expressed as $(x+1{)}^3-x^3=3x^2+3x+1$
• III: It is impossible to factorise $3x^2+3x+1$ into two linear factors with integer coefficients because its discriminant is negative.
• IV: Therefore for every positive integer value of $x$, the integer $3x^2+3x+1$ cannot be factorised.
• V: Hence, the difference between two consecutive cube numbers will always be prime.

Which of the following best describes this proof?

A selection, $S$, of $n$ terms is taken from the arithmetic sequence $1,4,7,10,\dots ,70$. Consider the following statement: $(\ast )$ There are two distinct terms in $S$ whose sum is 74. What is the smallest value of $n$ for which $(\ast )$ is necessarily true?

Consider the following statement: $(\ast )$ For all real numbers $x$, if $x<k$ then $x^2<k$. What is the complete set of values of $k$ for which $(\ast )$ is true?

Which of the following statements is/are true?

• I: For all real numbers $x$ and for all positive integers $n$, $x<n$
• II: For all real numbers $x$, there exists a positive integer $n$ such that $x<n$
• III: There exists a real number $x$ such that for all positive integers $n$, $x<n$

The diagram shows a kite $PQRS$ whose diagonals meet at $O$, with $OP=x$, $OQ=y$, $OR=x$, $OS=z$. Which of the following is necessary and sufficient for angle $SPQ$ to be a right angle?

Place the following integrals in order of size, starting with the smallest. $$P={\int }_0^1{2}^{\sqrt{x}}dxQ={\int }_0^1{2}^{x}dxR={\int }_0^1{\left(\sqrt{2}\right)}^{x}dx$$

Consider the statement $(\ast )$ about a real number $x$: $(\ast )$ There exists a real number $y$ such that $x-xy+y$ is negative. For how many real values of $x$ is $(\ast )$ true?

Consider the two inequalities: $$|x+5|<|x+11||x+11|<|x+1|$$ Which one of the following is correct?

The real numbers $x$, $y$ and $z$ are all greater than 1, and satisfy the equations $${\log }_{x}y=z\text{and}{\log }_{y}z=x$$ Which one of the following equations for ${\log }_{z}x$ must be true?

In this question, $a_1,\dots ,a_{100}$ and $b_1,\dots ,b_{100}$ and $c_1,\dots ,c_{100}$ are three sequences of integers such that $$a_n\le b_n+c_n$$ for each $n$. Which of the following statements must be true?

• I: $\min (a_1,\dots ,a_{100})\le \min (b_1,\dots ,b_{100})+\min (c_1,\dots ,c_{100})$
• II: $\min (a_1,\dots ,a_{100})\ge \min (b_1,\dots ,b_{100})+\min (c_1,\dots ,c_{100})$
• III: $\max (a_1,\dots ,a_{100})\le \max (b_1,\dots ,b_{100})+\max (c_1,\dots ,c_{100})$

A student answered the following question: $a$ and $b$ are non-zero real numbers. Prove that the equation $x^3+a{x}^2+b=0$ has three distinct real roots if $27b\left(b+\frac{4a^3}{27}\right)<0$. Here is the student's solution:

• I: We differentiate to get $\frac{dy}{dx}=3x^2+2ax$, so the stationary points are at $\left(0,b\right)$ and $\left(-\frac{2a}{3},b+\frac{4a^3}{27}\right)$.
• II: If $27b\left(b+\frac{4a^3}{27}\right)<0$, then $b$ and $b+\frac{4a^3}{27}$ must have opposite signs, and so one of the stationary points is above the $x$-axis and one is below.
• III: If the cubic has three distinct real roots then one of the stationary points is above the $x$-axis and one is below.
• IV: Hence if $27b\left(b+\frac{4a^3}{27}\right)<0$, then the equation has three distinct real roots.
• V: Hence, if $27b\left(b+\frac{4a^3}{27}\right)<0$, then the equation $x^3+a{x}^2+b=0$ has three distinct real roots.

Which one of the following options best describes the student's solution?

P, Q, R and S show the graphs of $$y=(\cos x{)}^{\cos x},y=(\sin x{)}^{\sin x},y=(\cos x{)}^{\sin x}\text{and}y=(\sin x{)}^{\cos x}$$ for $0<x<\frac{\pi }{2}$ in some order. Which row in the following table correctly identifies the graphs?

A polygon has $n$ vertices, where $n\ge 3$. It has the following properties: every vertex of the polygon lies on the circumference of a circle $C$; the centre of the circle $C$ is inside the polygon; the radii from the centre of the circle $C$ to the vertices of the polygon cut the polygon into $n$ triangles of equal area. For which values of $n$ are these properties sufficient to deduce that the polygon is regular?

The functions $f_1$ to $f_5$ are defined on the real numbers by $$f_1(x)=\cos x,f_2(x)=\sin (\cos x),f_3(x)=\cos (\sin (\cos x)),$$ $$f_4(x)=\sin (\cos (\sin (\cos x))),f_5(x)=\cos (\sin (\cos (\sin (\cos x))))$$ where all numbers are taken to be in radians. These functions have maximum values $m_1,m_2,m_3,m_4$ and $m_5$ respectively. Which one of the following statements is true?