Recent questions tagged isi2015-mma

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31
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32
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}$$\sqrt{2}/3$
1 votes
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33
If $f(x)$ is a real valued function such that $$2f(x)+3f(-x)=15-4x,$$ for every $x \in \mathbb{R}$, then $f(2)$ is$-15$$22$$11$$0$
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34
If $f(x) = \dfrac{\sqrt{3}\sin x}{2+\cos x}$, then the range of $f(x)$ isthe interval $[-1, \sqrt{3}/2]$the interval $[- \sqrt{3}/2, 1]$the interval $[-1, 1]$none of the ...
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35
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then$f$ and $g$ agree at no points$f$ and $g$ agree at exactly one point$f$ and $g$ agree at exactly two points$f$ and $g$ agr...
1 votes
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36
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37
Let $a$ be a non-zero real number. Define$$f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$$ for $x \in \mathbb{R}$....
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38
A real $2 \times 2$ matrix $M$ such that $$M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1- \varepsilon \end{pmatrix}$$exists for all $\varepsilon 0$does not exist for any $\vare...
7 votes
3 answers
39
The eigenvalues of the matrix $X = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ are$1,1,4$$1,4,4$$0,1,4$$0,4,4$
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40
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ by$$\textbf{A} = \begin{...
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41
Let $k$ and $n$ be integers greater than $1$. Then $(kn)!$ is not necessarily divisible by$(n!)^k$$(k!)^n$$n! \cdot k! \cdot$$2^{kn}$
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45
Angles between any pair of $4$ main diagonals of a cube are$\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$$\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$$\pi/2$none of the ...
0 votes
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46
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then$(h,k) = (0,0)$$(h,k) = (1/8, -1/16)$$...
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48
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49
The polar equation $r=a \cos \theta$ representsa spirala parabolaa circlenone of the above
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50
Let$$\begin{array}{} V_1 & = & \frac{7^2+8^2+15^2+23^2}{4} – \left( \frac{7+8+15+23}{4} \right) ^2, \\ V_2 & = & \frac{6^2+8^2+15^2+24^2}{4} – \left( \frac{6+8+15+24...
3 votes
1 answer
51
A permutation of $1,2, \dots, n$ is chosen at random. Then the probability that the numbers $1$ and $2$ appear as neighbour equals$\frac{1}{n}$$\frac{2}{n}$$\frac{1}{n-1}...
1 votes
1 answer
54
If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is$\log \frac{1+x}{1-x}$$\frac{x}{1-x} + \log(1+x)$$\frac{1}{1-x} ...
1 votes
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57
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then$\underset{n \to \infty}{\lim} a_n$ does not exist$\underset{n \t...
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58
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59
In the Taylor expansion of the function $f(x)=e^{x/2}$ about $x=3$, the coefficient of $(x-3)^5$ is$e^{3/2} \frac{1}{5!}$$e^{3/2} \frac{1}{2^5 5!}$$e^{-3/2} \frac{1}{2^5 ...