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Questions by Arjun

2 votes
2 answers
641
ISI2015-MMA-7
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when divided by $3\}$. Then the number of elements in $\mathcal{R}$ is $40$ $36$ $34$ $33$
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ and $y$ have the same remainder when d...
1 votes
3 answers
642
ISI2015-MMA-8
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals $n^2$ $n^3$ $2^n$ $3^n$
Let $A$ be a set of $n$ elements. The number of ways, we can choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$, equals$n^2$$n^3$$2^n$$3^n$
3 votes
2 answers
644
ISI2015-MMA-10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
The value of the infinite product$$P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$$$1$$2/3$$7/...
2 votes
2 answers
645
ISI2015-MMA-11
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals $854$ $153$ $160$ none of the above
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals$854$$153$$160$none of the above
1 votes
1 answer
646
ISI2015-MMA-12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ is $-524/65$ $524/65$ $-1/65$ $1/65$
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ i...
1 votes
2 answers
647
ISI2015-MMA-13
The number of real roots of the equation $2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{-x} \text{ is }$ $0$ $1$ $2$ infinitely many
The number of real roots of the equation$$2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{-x} \text{ is }$$$0$$1$$2$infinitely many
1 votes
1 answer
648
ISI2015-MMA-14
Consider the following system of equivalences of integers, $x \equiv 2 \text{ mod } 15$ $x \equiv 4 \text{ mod } 21$ The number of solutions in $x$, where $1 \leq x \leq 315$, to the above system of equivalences is $0$ $1$ $2$ $3$
Consider the following system of equivalences of integers,$$x \equiv 2 \text{ mod } 15$$$$x \equiv 4 \text{ mod } 21$$The number of solutions in $x$, where $1 \leq x \leq...
1 votes
1 answer
649
ISI2015-MMA-15
The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is $2$ $0$ $1$ none of the above
The number of real solutions of the equations $(9/10)^x = -3+x-x^2$ is$2$$0$$1$none of the above
2 votes
1 answer
651
ISI2015-MMA-17
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
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
2 votes
1 answer
652
ISI2015-MMA-18
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
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 linea pair of intersecting straig...
0 votes
2 answers
653
ISI2015-MMA-19
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 $2$ $2e$ $2 \pi$ $2i$
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 ...
1 votes
3 answers
654
ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals$e^{-1}$$e^{-1/2}$$e^{-2}$$1$
1 votes
1 answer
655
ISI2015-MMA-21
Let $\omega$ denote a complex fifth root of unity. Define $b_k =\sum_{j=0}^{4} j \omega^{-kj},$ for $0 \leq k \leq 4$. Then $ \sum_{k=0}^{4} b_k \omega ^k$ is equal to $5$ $5 \omega$ $5(1+\omega)$ $0$
Let $\omega$ denote a complex fifth root of unity. Define $$b_k =\sum_{j=0}^{4} j \omega^{-kj},$$ for $0 \leq k \leq 4$. Then $ \sum_{k=0}^{4} b_k \omega ^k$ is equal to$...
0 votes
1 answer
656
ISI2015-MMA-22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$equals $1$does not ...
0 votes
2 answers
658
ISI2015-MMA-24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to $-1$ $1$ $0$ does not converge
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to$-1$$1$$0$does not converge
1 votes
1 answer
659
ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals$1$$0$$e^{-8/3}$$e^{4/9}$
0 votes
1 answer
660
ISI2015-MMA-26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to$\infty$$0$$\log_e 2$$1$
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