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Q1 Indices and Surds Evaluating Expressions Using Index Laws View

If $x = 1 + \sqrt[5]{2} + \sqrt[5]{4} + \sqrt[5]{8} + \sqrt[5]{16}$, then the value of $\left(1 + \frac{1}{x}\right)^{30}$ is
(A) 2
(B) 5
(C) 32
(D) 64

Q2 Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View

Let $j$ be a number selected at random from $\{1, 2, \ldots, 2024\}$. What is the probability that $j$ is divisible by 9 and 15?
(A) $\frac{1}{23}$
(B) $\frac{1}{46}$
(C) $\frac{1}{44}$
(D) $\frac{1}{253}$

Q3 Permutations & Arrangements Linear Arrangement with Constraints View

Let $S_n$ be the set of all $n$-digit numbers whose digits are all 1 or 2 and there are no consecutive 2's. (Example: 112 is in $S_3$ but 221 is not in $S_3$). Then the number of elements in $S_{10}$ is
(A) 512
(B) 256
(C) 144
(D) 89

Q4 Binomial Distribution Compute Exact Binomial Probability View

There are 30 True or False questions in an examination. A student knows the answer to 20 questions and guesses the answers to the remaining 10 questions at random. What is the probability that the student gets exactly 24 answers correct?
(A) $\frac{105}{2^9}$
(B) $\frac{105}{2^8}$
(C) $\frac{105}{2^{10}}$
(D) $\frac{4}{2^{10}}$

Q5 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View

Let $T$ be a right-angled triangle in the plane whose side lengths are in a geometric progression. Let $n(T)$ denote the number of sides of $T$ that have integer lengths. Then the maximum value of $n(T)$ over all such $T$ is
(A) 0
(B) 1
(C) 2
(D) 3

Q6 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View

Let $x_1, x_2, \ldots, x_n$ be non-negative real numbers such that $\sum_{i=1}^{n} x_i = 1$. What is the maximum possible value of $\sum_{i=1}^{n} \sqrt{x_i}$?
(A) 1
(B) $\sqrt{n}$
(C) $n^{3/4}$
(D) $n$

Q7 Laws of Logarithms Analyze a Logarithmic Function (Limits, Monotonicity, Zeros, Extrema) View

The precise interval on which the function $f(x) = \log_{1/2}\left(x^2 - 2x - 3\right)$ is monotonically decreasing, is
(A) $(-\infty, -1)$
(B) $(-\infty, 1)$
(C) $(1, \infty)$
(D) $(3, \infty)$

Q8 Circles Chord Length and Chord Properties View

The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

Q9 Sine and Cosine Rules Multi-step composite figure problem View

In $\triangle ABC$, $CD$ is the median and $BE$ is the altitude. Given that $\overline{CD} = \overline{BE}$, what is the value of $\angle ACD$?
(A) $\pi/3$
(B) $\pi/4$
(C) $\pi/5$
(D) $\pi/6$

Q10 Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View

If the points $z_1$ and $z_2$ are on the circles $|z| = 2$ and $|z| = 3$, respectively, and the angle included between these vectors is $60^\circ$, then the value of $\frac{|z_1 + z_2|}{|z_1 - z_2|}$ is
(A) $\sqrt{\frac{19}{7}}$
(B) $\sqrt{19}$
(C) $\sqrt{7}$
(D) $\sqrt{\frac{7}{19}}$

Q11 Number Theory Prime Number Properties and Identification View

Let $n \geqslant 1$. The maximum possible number of primes in the set $\{n+6, n+7, \ldots, n+34, n+35\}$ is
(A) 7
(B) 8
(C) 12
(D) 13

Q12 Combinations & Selection Distribution of Objects to Positions or Containers View

Suppose 40 distinguishable balls are to be distributed into 4 different boxes such that each box gets exactly 10 balls. Out of these 40 balls, 10 are defective and 30 are non-defective. In how many ways can the balls be distributed such that all the defective balls go to the first two boxes?
(A) $\frac{40!}{(10!)^4}$
(B) $\frac{30! \cdot 20!}{(10!)^5}$
(C) $\frac{20! \cdot 20!}{(10!)^5}$
(D) $\frac{30! \cdot 10!}{(10!)^4}$

Q13 Inequalities Integer Solutions of an Inequality View

The number of elements in the set $$\left\{x : 0 \leqslant x \leqslant 2,\, \left|x - x^5\right| = \left|x^5 - x^6\right|\right\}$$ is
(A) 2
(B) 3
(C) 4
(D) 5

Q14 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View

In a room with $n \geqslant 2$ people, each pair shakes hands between themselves with probability $\frac{2}{n^2}$ and independently of other pairs. If $p_n$ is the probability that the total number of handshakes is at most 1, then $\lim_{n \rightarrow \infty} p_n$ is equal to
(A) 0
(B) 1
(C) $e^{-1}$
(D) $2e^{-1}$

Q15 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of positive solutions to the equation $$e^x \sin x = \log x + e^{\sqrt{x}} + 2$$ is
(A) 0
(B) 1
(C) 2
(D) $\infty$

Q16 Number Theory GCD, LCM, and Coprimality View

Let $n > 1$ be the smallest composite integer that is coprime to $\frac{10000!}{9900!}$. Then
(A) $n \leqslant 100$
(B) $100 < n \leqslant 9900$
(C) $9900 < n \leqslant 10000$
(D) $n > 10000$

Q17 Inequalities Linear Programming (Optimize Objective over Linear Constraints) View

Let $P = \{(x, y) : x + 1 \geqslant y,\, x \geqslant -1,\, y \geqslant 2x\}$. Then the minimum value of $(x + y)$ where $(x, y)$ varies over the set $P$ is
(A) $-1$
(B) $-3$
(C) $3$
(D) $0$

Q18 Composite & Inverse Functions Counting Functions with Composition or Mapping Constraints View

Let $A = \{1, \ldots, 5\}$ and $B = \{1, \ldots, 10\}$. Then the number of ordered pairs $(f, g)$ of functions $f : A \rightarrow B$ and $g : B \rightarrow A$ satisfying $(g \circ f)(a) = a$ for all $a \in A$ is
(A) $\frac{10!}{5!} \times 5^5$
(B) $5^{10} \times 5!$
(C) $10! \times 5!$
(D) $\binom{10}{5} \times 10^5$

Q19 Sequences and Series Estimation or Bounding of a Sum View

Let $$S = \frac{1}{\sqrt{10000}} + \frac{1}{\sqrt{10001}} + \cdots + \frac{1}{\sqrt{160000}}$$ Then the largest positive integer not exceeding $S$ is
(A) 200
(B) 400
(C) 600
(D) 800

Q20 Inequalities Absolute Value Inequality View

The real number $x$ satisfies $$\frac{|x|^2 - |x| - 2}{2|x| - |x|^2 - 2} > 2$$ if and only if $x$ belongs to
(A) $(-2, -1) \cup (1, 2)$
(B) $(-2/3, 0) \cup (0, 2/3)$
(C) $(-1, -2/3) \cup (2/3, 1)$
(D) $(-1, 0) \cup (0, 1)$

Q21 Circles Circle-Line Intersection and Point Conditions View

Consider points of the form $\left(n, n^k\right)$, where $n$ and $k$ are integers with $n \geq 0$, $k \geq 1$. How many such points are strictly inside the circle of radius 10 with centre at the origin?
(A) 11
(B) 12
(C) 15
(D) 17

Q22 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View

Let $n > 1$, and let us arrange the expansion of $\left(x^{1/2} + \frac{1}{2x^{1/4}}\right)^n$ in decreasing powers of $x$. Suppose the first three coefficients are in arithmetic progression. Then, the number of terms where $x$ appears with an integer power, is
(A) 3
(B) 2
(C) 1
(D) 0

Q23 Sequences and Series Limit Evaluation Involving Sequences View

The limit $$\lim_{n \rightarrow \infty} \frac{2\log 2 + 3\log 3 + \cdots + n\log n}{n^2 \log n}$$ is equal to
(A) 0
(B) $1/4$
(C) $1/2$
(D) 1

Q24 Number Theory Quadratic Diophantine Equations and Perfect Squares View

Let $p < q$ be prime numbers such that $p^2 + q^2 + 7pq$ is a perfect square. Then, the largest possible value of $q$ is:
(A) 7
(B) 11
(C) 23
(D) 29

Q25 Exponential Functions Exponential Equation Solving View

The set of all real numbers $x$ for which $3^{2^{1-x^2}}$ is an integer has
(A) 3 elements
(B) 15 elements
(C) 24 elements
(D) infinitely many elements

Q26 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

Let $a, b, c$ be three complex numbers. The equation $$az + b\bar{z} + c = 0$$ represents a straight line on the complex plane if and only if
(A) $a = b$
(B) $\bar{a}c = b\bar{c}$
(C) $|a| = |b| \neq 0$
(D) $|a| = |b| \neq 0$ and $\bar{a}c = b\bar{c}$

Q27 Circles Area and Geometric Measurement Involving Circles View

In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals
(A) $\sqrt{\frac{3}{2}}$
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{3}$

Q28 Sequences and Series Convergence/Divergence Determination of Numerical Series View

For every increasing function $b : [1, \infty) \rightarrow [1, \infty)$ such that $$\int_1^\infty \frac{\mathrm{d}x}{b(x)} < \infty$$ we must have
(A) $\sum_{k=1}^{\infty} \frac{\sqrt{\log k}}{b(k)} < \infty$
(B) $\sum_{k=3}^{\infty} \frac{\log k}{b(\log k)} < \infty$
(C) $\sum_{k=1}^{\infty} \frac{e^k}{b\left(e^k\right)} < \infty$
(D) $\sum_{k=3}^{\infty} \frac{1}{\sqrt{b(\log k)}} < \infty$

Q29 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View

Consider the following two statements: (I) There exists a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(x^3 + x^5\right) = e^x - 100$. (II) There exists a continuous function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g\left(e^x\right) = x^3 + x^5$. Then
(A) Only (I) is correct.
(B) Only (II) is correct.
(C) Both (I) and (II) are correct.
(D) Neither (I) nor (II) is correct.

Q30 Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View

Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\int_{-1}^{1} \psi(x)\,\mathrm{d}x = 1$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Then $$\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{1-\varepsilon}^{1+\varepsilon} f(y)\,\psi\!\left(\frac{1-y}{\varepsilon}\right) \mathrm{d}y$$ equals
(A) $f(1)$
(B) $f(1)\psi(0)$
(C) $f'(1)\psi(0)$
(D) $f(1)\psi(1)$

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