(a) Draw a qualitatively accurate sketch of the unique bounded region R in the first quadrant that has maximum possible finite area with boundary described as follows. R is bounded below by the graph of $y = x^2 - x^3$, bounded above by the graph of an equation of the form $y = kx$ (where $k$ is some constant), and R is entirely enclosed by the two given graphs, i.e., the boundary of the region R must be a subset of the union of the given two graphs (so R does not have any points on its boundary that are not on these two graphs). Clearly mark the relevant point(s) on the boundary where the two given graphs meet and write the coordinates of every such point. (b) Consider the solid obtained by rotating the above region R around $Y$-axis. Show how to find the volume of this solid by doing the following: Carefully set up the calculation with justification. Do enough work with the resulting expression to reach a stage where the final numerical answer can be found mechanically by using standard symbolic formulas of algebra and/or calculus and substituting known values in them. Do not carry out the mechanical work to get the final numerical answer.
(a) Find the domain of the function $g(x)$ defined by the following formula. $$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$ Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations. (b) $g(1000)$. (c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope. (d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope. (e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.
(a) For non-negative numbers $a, b, c$ and any positive real number $r$ prove the following inequality and state precisely when equality is achieved. $$a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r(c-a)(c-b) \geq 0$$ Hint: Assuming $a \geq b \geq c$ do algebra with just the first two terms. What about the third term? What if the assumption is not true? (b) As a special case obtain an inequality with $a^4 + b^4 + c^4 + abc(a+b+c)$ on one side. (c) Show that if $abc = 1$ for positive numbers $a, b, c$, then $$a^4 + b^4 + c^4 + a^3 + b^3 + c^3 + a + b + c \geq \frac{a^2+b^2}{c} + \frac{b^2+c^2}{a} + \frac{c^2+a^2}{b} + 3.$$
Find all solutions of the following equation where it is required that $x, k, y, n$ are positive integers with the exponents $k$ and $n$ both $> 1$. $$20x^k + 24y^n = 2024$$
A test developed to detect Covid gives the correct diagnosis for $99\%$ of people with Covid. It also gives the correct diagnosis for $99\%$ of people without Covid. In a city $\frac{1}{1000}$ of the population has Covid. If the probability is $x\%$, then your answer should be the integer closest to $x$. E.g., for probability $\frac{1}{3} = 33.33\ldots\%$, you should type 33 as your answer. For probability $\frac{2}{3}$ you should type 67 as your answer. What is the probability that a randomly selected person tests positive? (We assume that in our random selection every person is equally likely to be chosen.) [2 points]
A test developed to detect Covid gives the correct diagnosis for $99\%$ of people with Covid. It also gives the correct diagnosis for $99\%$ of people without Covid. In a city $\frac{1}{1000}$ of the population has Covid. If the probability is $x\%$, then your answer should be the integer closest to $x$. E.g., for probability $\frac{1}{3} = 33.33\ldots\%$, you should type 33 as your answer. For probability $\frac{2}{3}$ you should type 67 as your answer. Suppose that a randomly selected person tested positive. What is the probability that this person has Covid? [2 points]
Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find an integer $n$ with the least possible absolute value such that $p(x)$ has a real root between $n$ and $n+1$. Write this number along with your reason as per the given instruction. [2 points] Instruction for (6): Write two numbers separated by a comma: value of $n$, number of the theorem below that justifies this answer. E.g., if you think that $n=5$ because of the factor theorem, then type $\mathbf{5,1}$ as your answer with no space, full stop or any other punctuation.
Two mighty frogs jump once per unit time on the number line as described in the question. The first frog is at $x = 2^i$ at time $t = i$. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]
Two mighty frogs jump once per unit time on the number line as described in the question. The second frog starts at $x=0$ and jumps $i+1$ steps to the right just after $t=i$, so that at times $t=0,1,2,3,\ldots$ this frog is at positions $x=0,1,3,6,\ldots$ respectively. How many numbers of the form $7n+1$ (with $n$ an integer) does the frog visit from $t=0$ to $t=99$ (both endpoints included)? [3 points]
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$. Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation. $\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024\}$. [1 point] Options:
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$. Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation. $\{Q \mid \mathbf{u} \cdot \mathbf{v} = -2024\sqrt{\mathbf{v} \cdot \mathbf{v}}\}$. [2 points] Options:
Let $O=(0,0,0)$, $P=(19,5,2024)$ and $Q=(x,y,z)$ be points in 3-dimensional space where $Q$ is an unknown point. Consider vector $\mathbf{u} = \overrightarrow{OP} = 19\hat{i} + 5\hat{j} + 2024\hat{k}$ and unknown vector $\mathbf{v} = \overrightarrow{OQ} = x\hat{i} + y\hat{j} + z\hat{k}$. Instruction: for the specified set choose the correct option describing it and type in the number of that option. E.g., if you think the given set is a line, enter $\mathbf{3}$ as your answer with no full stop or any other punctuation. $\{Q \mid \mathbf{u} \cdot \mathbf{v} = 2024(\mathbf{v} \cdot \mathbf{v})\}$. [2 points] Options:
An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$. Write the number of even positive integers that are factors of $2024^2$. [2 points]
An integer $d$ is called a factor of an integer $n$ if there is an integer $q$ such that $n = qd$. In particular the set of factors of $n$ contains $n$ and contains 1. You are given that $2024 = 8 \times 11 \times 23$. Write the number of ordered pairs $(a,b)$ of positive integers such that $a^2 - b^2 = 2024^2$. If there are infinitely many such pairs, write the word infinite as your answer. [3 points]
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$. Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,4)$ and $(2,3)$. [1 point]
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$. Find the number of good paths from $(0,0)$ to $(6,6)$ that pass through both the points $(1,2)$ and $(3,4)$. [2 points]
A good path is a sequence of points in the $XY$ plane such that in each step exactly one of the coordinates increases by 1 and the other stays the same. E.g., $$(0,0),(1,0),(2,0),(2,1),(3,1),(3,2),(3,3)$$ is a good path from the origin to $(3,3)$. It is a fact that there are exactly 924 good paths from the origin to $(6,6)$. Find the number of good paths from $(0,0)$ to $(6,6)$ such that neither of the two points $(1,2)$ and $(3,4)$ occurs on the path, i.e., the path must miss both of the points $(1,2)$ and $(3,4)$. [3 points]