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Q1 Straight Lines & Coordinate Geometry Locus Determination View

A rod slides with its ends on two coordinate axes. A point $P$ divides the rod in the ratio $1:2$. Find the locus of $P$.

Q2 Number Theory Quadratic Diophantine Equations and Perfect Squares View

Find the number of integer solutions to $x^2 + y^2 = 2007$.

Q3 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Find the number of intersection points of $y = \log x$ and $y = x^2$.

Q4 Sequences and Series Evaluation of a Finite or Infinite Sum View

Find $\lim_{n \to \infty} u_n$ where $u_n = \dfrac{1}{2} + \dfrac{2}{2^2} + \dfrac{3}{2^3} + \dfrac{4}{2^4} + \cdots + \dfrac{n}{2^n}$.

Q5 Complex numbers 2 Roots of Unity and Cyclotomic Properties View

Let $w$ be a primitive cube root of unity. Simplify $\dfrac{1}{z-3} + \dfrac{1}{z-3w} + \dfrac{1}{z-3w^2}$.

Q6 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View

Let $f:\{1,2,3,4\} \to \{1,2,3,4\}$ be a function such that $f(i) \neq i$ for all $i$ (i.e., a derangement). Find the number of such functions.

Q7 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View

Let $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \leq 0$. Which of the following is true?
(A) $f$ is not differentiable at $x = 0$
(B) $f$ is differentiable at $x = 0$ but $f'$ is not differentiable at $x = 0$
(C) $f$ is differentiable at $x = 0$ and $f'$ is differentiable at $x = 0$
(D) $f$ is differentiable everywhere and $f'$ is also differentiable everywhere

Q8 Number Theory Properties of Integer Sequences and Digit Analysis View

Find the last digit of $9! + 3^{9966}$.

Q9 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

Let $f(x) = \dfrac{2x^2 + 3x + 1}{2x - 1}$. Find the maximum and minimum values of $f$ on $[2, 3]$.

Q10 Conic sections Locus and Trajectory Derivation View

From a point $P(h,k)$, two tangents are drawn to the parabola $y^2 = 4ax$ which are perpendicular to each other. Find the locus of $P$.

Q12 Number Theory Quadratic Diophantine Equations and Perfect Squares View

Find the number of positive integer solutions to $2^a - 5^b \cdot 7^c = 1$.

Q13 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

Find the locus of $z$ satisfying $|z - ia| = \text{Im}(z) + 1$, where $a$ is a real constant.

Q14 Trig Graphs & Exact Values View

Which of the following is true about $\tan(\sin x)$?
(A) $\tan(\sin x) = 1$ has solutions
(B) $\tan(\sin x) \geq 1$ for some $x$
(C) $\tan(\sin x) < 1$ for all $x$
(D) $\tan(\sin x)$ never attains the value $1$

Q15 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Let $f(x) = \dfrac{x^2}{x-1}$. Which of the following is true?
(A) $f$ is neither one-one nor onto
(B) $f$ is one-one and onto
(C) $f$ is one-one but not onto
(D) $f$ is onto but not one-one

Q16 Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

Let $f$ be a periodic function with period $1$, and let $g(t) = \int_0^t f(x)\,dx$. Define $h(t) = \lim_{n\to\infty} \dfrac{g(t+n)}{n}$. Which of the following is true about $h(t)$?
(A) $h(t)$ depends on $t$
(B) $h(t)$ is not defined for all $t$
(C) $h(t)$ is defined for all $t \in \mathbb{R}$ and is independent of $t$
(D) None of the above

Q17 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Let $a_1 = 24^{1/3}$ and $a_{n+1} = (a_n + 24)^{1/3}$. Find the integer part of $a_{100}$.

Q18 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

If $xy = 1$, find the minimum value of $\dfrac{4}{4-x^2} + \dfrac{9}{9-y^2}$.

Q19 Sequences and series, recurrence and convergence Convergence proof and limit determination View

Find $\lim_{n\to\infty}\left(1 + \dfrac{1}{n}\right)^n$.

Q20 Curve Sketching Number of Solutions / Roots via Curve Analysis View

Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true?
(A) $f$ has exactly two real roots
(B) $f$ has no real roots
(C) $f$ has four real roots
(D) $f$ has exactly two real roots, one of which is $-1$

Q21 Permutations & Arrangements Linear Arrangement with Constraints View

In how many ways can $10$ A's and $6$ B's be arranged in a row such that no two B's are adjacent and no two A's are adjacent? (More precisely: find the number of arrangements of $10$ A's and $6$ B's such that between any two consecutive B's there is at least one A, and between any two consecutive A's there is at least one B.)

Q22 Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View

Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true?
(A) $f$ is differentiable everywhere except at $x = 0$
(B) $f$ is continuous but not differentiable at $x = 0$
(C) $f$ is differentiable everywhere
(D) None of the above

Q23 Circles Circle Equation Derivation View

Find the equation of the circle with $AB$ as diameter, where $A$ and $B$ are the intercepts of the line $2x + 3y = k$ on the coordinate axes.

Q24 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

Let $x_n = \dfrac{1}{(2a)^n}$ for $a > 0$. Which of the following is true?
(A) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$
(B) $x_n \to 0$ if $0 < a < 1/2$ and $x_n \to \infty$ if $a > 1/2$
(C) $x_n \to 1$ for all $a > 0$
(D) $x_n \to \infty$ if $0 < a < 1/2$ and $x_n \to 0$ if $a > 1/2$

Q25 Trig Graphs & Exact Values View

For $\theta \in (0, \pi/2)$, which of the following is true?
(A) $\cos(\sin\theta) < \cos\theta$
(B) $\cos(\sin\theta) < \sin(\cos\theta)$
(C) $\cos(\sin\theta) > \cos\theta$
(D) $\cos(\sin\theta) > \sin(\cos\theta)$

Q26 Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View

A room is in the shape of a rectangular box. The shortest path along the surface from one corner $A$ to the opposite corner $B$ has length $\sqrt{29}$ (given the relevant dimensions are $5$ and $2$). Find this shortest distance.

Q27 Sequences and Series Estimation or Bounding of a Sum View

Find the integer part of $S = \displaystyle\sum_{k=2}^{9999} \dfrac{1}{\sqrt{k}}$.

Q28 Inequalities Set Operations Using Inequality-Defined Sets View

Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true?
(A) $A = B$
(B) $A \subset B$ (A is a proper subset of B)
(C) $B \subset A$ (B is a proper subset of A)
(D) Neither $A \subset B$ nor $B \subset A$

Q29 Combinations & Selection Combinatorial Identity or Bijection Proof View

Using AM-GM inequality, find a lower bound for $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right)$, and determine which of the following holds:
(A) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) < \left(\dfrac{2^{10}}{11}\right)^{11}$
(B) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) = \left(\dfrac{2^{10}}{11}\right)^{11}$
(C) $\left({}^{10}C_0\right)\left({}^{10}C_1\right)\cdots\left({}^{10}C_{10}\right) > \left(\dfrac{2^{10}}{11}\right)^{11}$
(D) None of the above

Q30 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View

Let $s, sr, sr^2, sr^3$ be the roots of $x^4 + ax^3 + bx^2 + cx + d = 0$ (roots in geometric progression). Show that $c^2 = a^2 d$.

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