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QA1 Principle of Inclusion/Exclusion View

Each student in a small school has to be a member of at least one of THREE school clubs. It is known that each club has 35 members. It is not known how many students are members of two of the three clubs, but it is known that exactly 10 students are members of all three clubs. What is the largest possible total number of students in the school? What is the smallest possible total number of students in the school?

QA2 Vectors 3D & Lines Normal Vector Determination View

Let P be the plane containing the vectors $(6,6,9)$ and $(7,8,10)$. Find a unit vector that is perpendicular to $(2,-3,4)$ and that lies in the plane P. (Note: all vectors are considered as line segments starting at the origin $(0,0,0)$. In particular the origin lies in the plane P.)

QA3 Indefinite & Definite Integrals Definite Integral Evaluation by Parts View

Calculate the following two definite integrals. It may be useful to first sketch the graph. $$\int_{1}^{e^{2}} \ln|x|\, dx \qquad \int_{-1}^{1} \frac{\ln|x|}{|x|}\, dx$$

QA4 Probability Definitions Compute Cumulative or Complement Binomial Probability View

A fair die is thrown 100 times in succession. Find probabilities of the following events.
(i) 4 is the outcome of one or more of the first three throws.
(ii) Exactly 2 of the last 4 throws give an outcome divisible by 3 (i.e., outcome 3 or 6).

QA5 Curve Sketching Asymptote Determination View

Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions $$f(x) = \frac{x^{3}}{x^{2}-x} \qquad g(x) = \frac{x^{2}-x}{x^{3}} \qquad h(x) = \frac{x^{3}-x}{x^{3}+x}$$ (i) Does $f$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does $g$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(iii) Does $h$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

QA6 Function Transformations Monotonicity or convexity of transcendental functions View

Recall the function $\arctan(x)$, also denoted as $\tan^{-1}(x)$. Complete the sentence: $$\arctan(20202019) + \arctan(20202021) \quad\underline{\hspace{2cm}}\quad 2\arctan(20202020),$$ because in the relevant region, the graph of $y = \arctan(x)$ $\_\_\_\_$.
Fill in the first blank with one of the following: is less than / is equal to / is greater than. Fill in the second blank with a single correct reason consisting of one of the following phrases: is bounded / is continuous / has positive first derivative / has negative first derivative / has positive second derivative / has negative second derivative / has an inflection point.

QA7 Factor & Remainder Theorem Determine coefficients or parameters from root conditions View

The polynomial $p(x) = 10x^{400} + ax^{399} + bx^{398} + 3x + 15$, where $a, b$ are real constants, is given to be divisible by $x^{2}-1$.
(i) If you can, find the values of $a$ and $b$. Write your answers as $a =$ $\_\_\_\_$, $b =$ $\_\_\_\_$. If it is not possible to decide, state so.
(ii) If you can, find the sum of reciprocals of all 400 (complex) roots of $p(x)$. Write your answer as sum $=$ $\_\_\_\_$. If it is not possible to decide, state so.

QA8 Number Theory Divisibility and Divisor Analysis View

For a positive integer $n$, let $D(n) =$ number of positive integer divisors of $n$. For example, $D(6) = 4$ because 6 has four divisors, namely $1, 2, 3$ and $6$. Find the number of $n \leq 60$ such that $D(n) = 6$.

QA9 Binomial Theorem (positive integer n) Evaluation of a Finite or Infinite Sum View

Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.

QA10 Number Theory Modular Arithmetic Computation View

Note that $25 \times 16 - 19 \times 21 = 1$. Using this or otherwise, find positive integers $a, b$ and $c$, all $\leq 475 = 25 \times 19$, such that

$a$ is $1 \bmod 19$ and $0 \bmod 25$,
$b$ is $0 \bmod 19$ and $1 \bmod 25$, and
$c$ is $4 \bmod 19$ and $10 \bmod 25$.

(Recall the mod notation: since 13 divided by 5 gives remainder 3, we say 13 is $3 \bmod 5$.)

QB1 7 marks Circles Chord Length and Chord Properties View

[7 points] Suppose $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are points on a circle such that AC and BD are diameters of that circle. Suppose $\mathrm{AB} = 12$ and $\mathrm{BC} = 5$. Let P be a point on the arc of the circle from A to B (the arc that does not contain points C and D). Let the distances of P from $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D be $a, b, c$ and $d$ respectively. Find the values of $\frac{a+b}{c+d}$ and $\frac{a-b}{d-c}$. You may assume $d \neq c$ so the second ratio makes sense.

QB2 7 marks Complex Numbers Arithmetic Roots of Unity and Cyclotomic Properties View

[7 points] Let $z = e^{\left(\frac{2\pi i}{n}\right)}$. Here $n \geq 2$ is a positive integer, $i^{2} = -1$ and the real number $\frac{2\pi}{n}$ can also be considered as an angle in radians.
(i) Show that $\displaystyle\sum_{k=0}^{n-1} z^{k} = 0$.
(ii) Show that $\displaystyle\sum_{k=0}^{8} \cos(40k+1)^{\circ} = 0$, i.e., $\cos(1^{\circ}) + \cos(41^{\circ}) + \cos(81^{\circ}) + \cos(121^{\circ}) + \cdots + \cos(241^{\circ}) + \cos(281^{\circ}) + \cos(321^{\circ}) = 0$.

QB3 10 marks Variable acceleration (1D) Geometric Related Rates with Distance or Angle View

[10 points] A spider starts at the origin and runs in the first quadrant along the graph of $y = x^{3}$ at the constant speed of 10 unit/second. The speed is measured along the length of the curve $y = x^{3}$. The formula for the curve length along the graph of $y = f(x)$ from $x = a$ to $x = b$ is $\ell = \int_{a}^{b} \sqrt{1 + f'(x)^{2}}\, dx$. As the spider runs, it spins out a thread that is always maintained in a straight line connecting the spider with the origin. What is the rate in unit/second at which the thread is elongating when the spider is at $\left(\frac{1}{2}, \frac{1}{8}\right)$?
You should use the following names for variables. At any given time $t$, the spider is at the point $\left(u, u^{3}\right)$, the length of the thread joining it to the origin in a straight line is $s$ and the curve length along $y = x^{3}$ from the origin till $\left(u, u^{3}\right)$ is $\ell$. You are asked to find $\frac{ds}{dt}$ when $u = \frac{1}{2}$. (Do not try to evaluate the integral for $\ell$: it is unnecessary and any attempt to do so will not get any credit because a closed formula in terms of basic functions does not exist.)

QB4 12 marks Proof Existence Proof View

[12 points] Throughout this problem we are interested in real valued functions $f$ satisfying two conditions: at each $x$ in its domain, $f$ is continuous and $f(x^{2}) = f(x)^{2}$. Prove the following independent statements about such functions. The hints below may be useful.
(i) There is a unique such function $f$ with domain $[0,1]$ and $f(0) \neq 0$.
(ii) If the domain of such $f$ is $(0, \infty)$, then ($f(x) = 0$ for every $x$) OR ($f(x) \neq 0$ for every $x$).
(iii) There are infinitely many such $f$ with domain $(0, \infty)$ such that $\int_{0}^{\infty} f(x)\, dx < 1$.
Hints: (1) Suppose a number $a$ and a sequence $x_{n}$ are in the domain of a continuous function $f$ and $x_{n}$ converges to $a$. Then $f(x_{n})$ must converge to $f(a)$. For example $f(0.5^{n}) \rightarrow f(0)$ and $f(2^{\frac{1}{n}}) \rightarrow f(1)$ if all the mentioned points are in the domain of $f$. In parts (i) and (ii) suitable sequences may be useful. (2) Notice that $f(x) = x^{r}$ satisfies $f(x^{2}) = f(x)^{2}$.

QB5 12 marks Roots of polynomials Existence or counting of roots with specified properties View

[12 points] Consider polynomials $p(x)$ with the following property, called $(\dagger)$. $(\dagger)$ If $r$ is a root of $p(x)$, then $r^{2} - 4$ is also a root of $p(x)$.
(i) We want to find every quadratic polynomial of the form $p(x) = x^{2} + bx + c$ such that $p(x)$ has two distinct roots, has integer coefficients and has property $(\dagger)$. Prove that there are exactly two such polynomials and list them.
(ii) It is also true that there are exactly two cubic polynomials of the form $p(x) = x^{3} + ax^{2} + bx + c$ with the property $(\dagger)$ such that $p(x)$ shares no root with the polynomials you found in part (i). Explain fully how you will prove this along with the method to find the polynomials, but do not try to explicitly find the polynomials.

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