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
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) $\frac{1}{253}$
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
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) $\frac{4}{2^{10}}$
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) $n$
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) $(3, \infty)$
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) $2\pi/3$
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) $\pi/6$
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) $\sqrt{\frac{7}{19}}$
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) $\frac{30! \cdot 10!}{(10!)^4}$
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
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) $2e^{-1}$
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) $n > 10000$
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) $0$
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) $\binom{10}{5} \times 10^5$
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
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
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
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
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 $\bar{a}c = b\bar{c}$
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) $\sqrt{3}$