# Questions by Arjun

1
Consider the following system of equivalences of integers, $x \equiv 2 \text{ mod } 15$ $x \equiv 4 \text{ mod } 21$ The number of solutions in $x$, where $1 \leq x \leq 315$, to the above system of equivalences is $0$ $1$ $2$ $3$
2
The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is $2$ $0$ $1$ none of the above
1 vote
3
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy $f(x^2+1)=f(x)g(x),$ for every $x \in \mathbb{R}$, then $f$ has exactly one real root $x_0$ such that $f’(x_0) \neq 0$ $f$ has exactly one real root $x_0$ such that $f’(x_0) = 0$ $f$ has $m$ distinct real roots $f$ has no real root
1 vote
4
Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then, $X \lt1$ $X\gt3/2$ $1\lt X\lt 3/2$ none of the above holds
1 vote
5
The set of complex numbers $z$ satisfying the equation $(3+7i)z+(10-2i)\overline{z}+100=0$ represents, in the complex plane, a straight line a pair of intersecting straight lines a point a pair of distinct parallel straight lines
6
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
1 vote
7
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$
1 vote
8
Let $\omega$ denote a complex fifth root of unity. Define $b_k =\sum_{j=0}^{4} j \omega^{-kj},$ for $0 \leq k \leq 4$. Then $\sum_{k=0}^{4} b_k \omega ^k$ is equal to $5$ $5 \omega$ $5(1+\omega)$ $0$
9
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
10
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ equals $f(x,A)+f(x,B)$ $f(x,A)+f(x,B)\: – 1$ $f(x,A)+f(x,B)\: – f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\: – f(x,B) \mid$
11
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to $-1$ $1$ $0$ does not converge
1 vote
12
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
13
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
1 vote
14
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is $0$ $1/32$ $15/32$ $10/32$
15
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$
16
The set $\{x \: : \begin{vmatrix} x+\frac{1}{x} \end{vmatrix} \gt6 \}$ equals the set $(0,3-2\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $(- \infty, -3-2\sqrt{2}) \cup (-3+2 \sqrt{2}, \infty)$ $(- \infty, 3-2\sqrt{2}) \cup (3+2\sqrt{2}, \infty)$ $(- \infty, -3-2\sqrt{2}) \cup (-3+2 \sqrt{2},3-2\sqrt{2}) \cup (3+2 \sqrt{2}, \infty )$
17
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $K-xy$ $K+xy$ none of the above
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is non-empty none of the above
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals $\sqrt{2/3}$ $\sqrt{3/2}$ $3/ \sqrt{2}$ $\sqrt{2}/3$
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x,$ for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$