The number of solutions of $$x + [x] = 2013$$, where $$[x]$$ is greatest integer less than or equal to $$x$$, is
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The number of solutions of $$x + [x] = 2013$$, where $$[x]$$ is greatest integer less than or equal to $$x$$, is
Write $$x = n + f$$ with $$n = [x]$$ and $$0 \le f < 1$$. Then $$2n + f = 2013$$, so $$f = 2013 - 2n$$ is an integer, which forces $$f = 0$$. But then $$2n = 2013$$, which is impossible since 2013 is odd. So there are no solutions.
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The complete solution set of $$\sqrt{x - 2} < 2$$ is
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State True or False. The number of elements in the set $$\{x \mid x(x + 8)(x + 58) = 2013, x \text{ is an integer}\}$$ is 1.
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If $$a$$, $$b$$, $$c$$ are integers such that $$c > 0$$. Determine condition(s) on $$a$$, $$b$$, $$c$$ to put $$a + b\sqrt{c}$$ in the form $$(\sqrt{p} \pm \sqrt{q})^2$$ where $$p$$ and $$q$$ are integers.
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Find all integers $$n$$ such that $$\left|2n^3 - 6n^2 + 4n + 3\right|$$ is prime.
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A particle starting from rest accelerates uniformly and travels a distance $$d$$ for some time. It then travels uniformly at the speed acquired after the initial acceleration for an equal time, after which it decelerates and comes to rest. The magnitude of the deceleration is twice the initial acceleration. If total distance travelled is $$nd$$, find $$n$$.
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Two mirrors AB and CD are placed as shown with their reflecting surfaces towards each other. A ray of light incident on AB at $$45^\circ$$ on reflection from AB and CD emerges parallel to the original direction. Through what angle (in degrees) does the ray reflected by CD turn if<br>i) AB is turned in a clock wise sense through a small angle $$5^\circ$$.<br>ii) CD is turned in anticlockwise sense through a small angle $$5^\circ$$.
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Find the equivalent resistance between A and B. Also find the equivalent resistance between A and B if the two resistances of $$6R$$ are removed.
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If a nanogram ($$10^{-9}\text{ g}$$) of mass were to be completely converted into electrical energy and used to power a $$40\text{ W}$$ lamp, find the time (in seconds) for which the lamp can be run. Assume the mass is converted at a uniform rate to produce the power required by the lamp.
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