TMUA 2021 Paper 2

Find the value of $${\int }_1^4\left(3\sqrt{x}+\frac{4}{x^2}\right)\mathrm{d}x$$

$A(0,2)$ and $C(4,0)$ are opposite vertices of the square $ABCD$. What is the equation of the straight line through $B$ and $D$?

A student is chosen at random from a class. Each student is equally likely to be chosen. Which of the following conditions is/are necessary for the probability that the student wears glasses to equal $\frac{4}{15}$?

I. Exactly 11 students in the class do not wear glasses.

II. The number of students in the class is divisible by 3.

III. The class contains 30 students, and 8 of them wear glasses.

Consider the following claim about positive integers $a$, $b$ and $c$: if $a$ is a factor of $bc$, then $a$ is a factor of $b$ or $a$ is a factor of $c$. Which of the following provide(s) a counterexample to this claim?

I. $a=5,b=10,c=20$

II. $a=8,b=4,c=4$

III. $a=6,b=7,c=12$

On which line is the first error in the following argument?

A. ${\sin }^{2}x+{\cos }^{2}x=1$ for all values of $x$.

B. Therefore $\cos x=\sqrt{1-{\sin }^{2}x}$ for all values of $x$.

C. Hence $1+\cos x=1+\sqrt{1-{\sin }^{2}x}$ for all values of $x$.

D. Thus $(1+\cos x{)}^2={\left(1+\sqrt{1-{\sin }^{2}x}\right)}^2$ for all values of $x$.

E. Substituting $x=\pi$ gives $0=4$.

Consider the following two statements about the polynomial $\mathrm{f}(x)$:

$P$: $\mathrm{f}(x)=0$ for exactly three real values of $x$

$Q$: ${\mathrm{f}}^{\prime }(x)=0$ for exactly two real values of $x$

Which one of the following is correct?

A circle has equation $(x-9{)}^2+(y+2{)}^2=4$. A square has vertices at $(1,0)$, $(1,2)$, $(-1,2)$ and $(-1,0)$. A straight line bisects both the area of the circle and the area of the square. What is the $x$-coordinate of the point where this straight line meets the $x$-axis?

Consider the following statement about the polynomial $\mathrm{p}(x)$, where $a$ and $b$ are real numbers with $a<b$:

$(\ast )$ There exists a number $c$ with $a<c<b$ such that ${\mathrm{p}}^{\prime }(c)=0$.

Which one of the following is true?

Consider the following statements about a polynomial $\mathrm{f}(x)$:

I. $\mathrm{f}(x)=p{x}^3+q{x}^2+rx+s$, where $p\ne 0$.

II. There is a real number $t$ for which ${\mathrm{f}}^{\prime }(t)=0$.

III. There are real numbers $u$ and $v$ for which $\mathrm{f}(u)\mathrm{f}(v)<0$.

Which of these statements is/are sufficient for the equation $\mathrm{f}(x)=0$ to have a real solution?

| | Statement I is sufficient | Statement II is sufficient | Statement III is sufficient | |---|---|---|---| | A | Yes | Yes | Yes | | B | Yes | Yes | No | | C | Yes | No | Yes | | D | Yes | No | No | | E | No | Yes | Yes | | F | No | Yes | No | | G | No | No | Yes | | H | No | No | No |

The first seven terms of a sequence of positive integers are: $$u_1=15,u_2=21,u_3=30,u_4=37,u_5=44,u_6=51,u_7=59.$$ Consider the following statement about this sequence: $(\ast )$ If $n$ is a prime number, then $u_n$ is a multiple of 3 or $u_n$ is a multiple of 5. What is the smallest value of $n$ that provides a counterexample to $(\ast )$?

A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers: Show that if $a^2-4b^3\ge 0$ then there exist real numbers $x$ and $y$ such that $a=xy(x+y)$ and $b=xy$. Consider the following attempt: $$(x-y{)}^2\ge 0\text{\hspace{1em}(I)}$$ $$\text{so}{x}^2+y^2-2xy\ge 0\text{\hspace{1em}(II)}$$ $$\text{so}(x+y{)}^2-4xy\ge 0\text{\hspace{1em}(III)}$$ $$\text{so}{x}^2{y}^2(x+y{)}^2-4x^3{y}^3\ge 0\text{\hspace{1em}(IV)}$$ $$\text{so}{a}^2-4b^3\ge 0\text{\hspace{1em}(V)}$$ Which of the following best describes this attempt?

Which of the following statements about polynomials $\mathrm{f}$ and $\mathrm{g}$ is/are true?

I. If $\mathrm{f}(x)\ge \mathrm{g}(x)$ for all $x\ge 0$, then ${\int }_0^x\mathrm{f}(t)\mathrm{d}t\ge {\int }_0^x\mathrm{g}(t)\mathrm{d}t$ for all $x\ge 0$.

II. If $\mathrm{f}(x)\ge \mathrm{g}(x)$ for all $x\ge 0$, then ${\mathrm{f}}^{\prime }(x)\ge {\mathrm{g}}^{\prime }(x)$ for all $x\ge 0$.

III. If ${\mathrm{f}}^{\prime }(x)\ge {\mathrm{g}}^{\prime }(x)$ for all $x\ge 0$, then $\mathrm{f}(x)\ge \mathrm{g}(x)$ for all $x\ge 0$.

A region $R$ in the $(x,y)$-plane is defined by the simultaneous inequalities $$y-x<3$$ $$y-x^2<1$$ Which of the following statements is/are true for every point in $R$?

I. $-1<x<2$

II. $(y-x)(y-x^2)<3$

III. $y<5$

Consider the following simultaneous equations, where $p$ is a real number: $$p\cdot 2^x+{\log }_{2}y=2$$ $$2^x+{\log }_{2}y=1$$ What is the complete range of $p$ for which these simultaneous equations have a real solution $(x,y)$?

A circle has equation $$x^2+ax+y^2+by+c=0$$ where $a$, $b$ and $c$ are non-zero real constants. Which one of the following is a necessary and sufficient condition for the circle to be tangent to the $y$-axis?

$p$ and $q$ are real numbers, and the equation $$x|x|=px+q$$ has exactly $k$ distinct real solutions for $x$. Which one of the following is the complete list of possible values for $k$?

Consider the following functions defined for $x>1$: $$\mathrm{f}(x)={\log }_2\left({\log }_2\sqrt{x}\right)$$ $$\mathrm{g}(x)={\log }_2\left(\sqrt{{\log }_{2}x}\right)$$ Which one of the following is true for all values of $x>1$?

A student chooses two distinct real numbers $x$ and $y$ with $0<x<y<1$. The student then attempts to draw a triangle $ABC$ with: $$AB=1,\sin A=x,\sin B=y.$$ Which of the following statements is/are correct?

I. For some choice of $x$ and $y$, there is exactly one triangle the student could draw.

II. For some choice of $x$ and $y$, there are exactly two different triangles the student could draw.

III. For some choice of $x$ and $y$, there are exactly three different triangles the student could draw.

(Note that congruent triangles are considered to be the same.)

The angle $\theta$ can take any of the values $1^{\circ },2^{\circ },3^{\circ },\dots ,359^{\circ },360^{\circ }$. For how many of these values of $\theta$ is it true that $$\sin \theta \sqrt{1+\sin \theta }\sqrt{1-\sin \theta }+\cos \theta \sqrt{1+\cos \theta }\sqrt{1-\cos \theta }=0$$

A sequence of functions ${\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3,\dots$ is defined by $${\mathrm{f}}_1(x)=|x|$$ $${\mathrm{f}}_{n+1}(x)=|{\mathrm{f}}_n(x)+x|\text{for }n\ge 1.$$ Find the value of $${\int }_{-1}^1{\mathrm{f}}_{99}(x)\mathrm{d}x$$