Recent questions tagged cat2003-2

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81
The number of roots common between $x^3 + 3 x^2 + 4x + 5 = 0$ and $x^3 + 2 x^2 + 7x +3 =0$ is$0$$1$$2$$3$
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82
Let $\text{ABCDEF}$ be a regular hexagon. What is the ratio of the area of the triangle $\text{ACE}$ to that of the hexagon $\text{ABCDEF}?$$\frac{1}{3}$$\frac{1}{2}$$\fr...
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83
Consider the sets $T_n = \{n, n + 1, n + 2, n + 3, n + 4\}$, where $n = 1, 2, 3,\dots, 96.$ How many of these sets contain $6$ or any integral multiple thereof (i.e., any...
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84
The infinite sum $1 + \frac{4}{7} + \frac{9}{7^2} + \frac{16}{7^3} + \frac{25}{7^4} + \dots$ equals$\frac{27}{14}$$\frac{21}{13}$$\frac{49}{27}$$\frac{256}{147}$
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89
If $|b| \geq 1$ and $x =\; – |a| b$, then which one of the following is necessarily true?$a – xb < 0$$a – xb \geq 0$$a – xb 0$$a – xb \leq 0$
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91
If $13x + 1 < 2$ and $z,$ and $z + 3 = 5y^2$, then$x$ is necessarily less than $y$$x$ is necessarily greater than $y$$x$ is necessarily equal to $y$None of the above is n...
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92
If n is such that $36 \leq n \leq 72$ then $x = \frac{n^2 + 2 \sqrt{n}(n+4) +16}{n+4\sqrt{n}+4}$ satisfies$20 < x < 54$$23 < x < 58$$25 < x < 64$$28 < x < 60$
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94
Let $x$ and $y$ be positive integers such that $x$ is prime and $y$ is composite. Then,$y – x$ cannot be an even integer$xy$ cannot be an even integer.$\frac{x+y}{x}$ c...
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97
If $a, a + 2$ and $a + 4$ are prime numbers, then the number of possible solutions for $a$ is:onetwothreemore than three