Recent questions tagged isi2015-mma

3 votes
2 answers
2
If $a,b$ are positive real variables whose sum is a constant $\lambda$, then the minimum value of $\sqrt{(1+1/a)(1+1/b)}$ is$\lambda \: – 1/\lambda$$\lambda + 2/\lambda...
3 votes
3 answers
3
Let $x$ be a positive real number. Then$x^2+\pi ^2 + x^{2 \pi} x \pi+ (\pi + x) x^{\pi}$$x^{\pi}+\pi^x x^{2 \pi} + \pi ^{2x}$$\pi x +(\pi+x)x^{\pi} x^2+\pi ^2 + x^{2 \...
1 votes
2 answers
5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is$2^n$$2^{n+1}$$2^{n-1}$$2^{2n}$
2 votes
2 answers
7
1 votes
3 answers
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$$3^n$
3 votes
2 answers
10
The value of the infinite product$$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$$7/...
2 votes
2 answers
11
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals$854$$153$$160$none of the above
1 votes
1 answer
12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(3-2i)$ are two two roots of this polynomial then the value of $a$ i...
1 votes
2 answers
13
The number of real roots of the equation$$2 \cos \left( \frac{x^2+x}{6} \right) = 2^x +2^{-x} \text{ is }$$$0$$1$$2$infinitely many
1 votes
1 answer
14
1 votes
1 answer
15
2 votes
1 answer
17
Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then,$X \lt1$$X\gt3/2$$1\lt X\lt 3/2$none of the above holds
2 votes
1 answer
18
0 votes
2 answers
19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is$2$$2e$$2 ...
1 votes
3 answers
20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals$e^{-1}$$e^{-1/2}$$e^{-2}$$1$
1 votes
1 answer
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$...
0 votes
1 answer
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 ...
0 votes
2 answers
24
1 votes
1 answer
25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals$1$$0$$e^{-8/3}$$e^{4/9}$
0 votes
1 answer
26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to$\infty$$0$$\log_e 2$$1$
1 votes
2 answers
27
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$$1/32$$...
1 votes
1 answer
28
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{11...