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Let A = \sum_{n=1}^{10000} \dfrac{1}{\sqrt{n}} Determine [A]. Here [.] denotes the greatest integer function. ...
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Find all continuous functions f:\mathbb{R}\to [1,\infty) for which there exist an a\in\mathbb{R} and k , a positive integer, such that f(x)f(2x)\cdots f(nx)\le an^k for every real number x and positive integer n. ...
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Let f:[0,2]→R be defined by f(x)=\sqrt{x^3+2-2\sqrt{x^3+1}}+\sqrt{x^3+10-6\sqrt{x^3+1}} Find the indefinite integral of f w.r.t x. ...
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Given that \int_{-\infty}^{\infty} e^{-x^2}\ \mathrm dx=\sqrt{\pi} evaluate the integral I=\int_0^\infty \dfrac{1}{\sqrt{x}}\ e^{-x}\ \mathrm dx ...
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1) Prove: \left|\begin{array}{cccc} 1+a_1 & 1 & \cdots & 1\\1 & 1+a_2 & \cdots & 1\\\vdots & \vdots & \ddots & \vdots \\1 & 1 & \cdots & 1+a_n\end{array}\right| = a_1a_2\cdots a_n \left(1+\dfrac{1}{a_1}+ \dfrac{1}{a_2}+\ldots ...
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1) Evaluate the limit: \lim_{n\to\infty} |\sin (\pi\sqrt{n^2+n+1})| 2) Prove that \lim_{n\to \infty}n^2 \int_0^{1/n} x^{x+1}\ \mathrm dx=\dfrac{1}{2} 3) Find the real parameters m and n such that the graph of the function f(x ...
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A ball is bouncing vertically up and down. It has a velocity v0 when it strikes the ground. The acceleration due to gravity is slowly reduced by 10% during a very long interval of time. Assuming no air resistance and perfectl ...
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Consider a rectangular basin with a very wide mouth. When a tap above the basin is opened, it fills in time T1. With the basin filled completely and the tap closed, opening the plug at the bottom drains the sink in time T2. W ...
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Let 0 < a ≤ b. Define a1 = a+b/2 , b1 = a1 b and for each integer k≥1, ak+1 = ak + bk /2 , bk+1 = ak+1 bk . Find the limit of the sequence {bn}. That is find \lim_{n\to\infty} b_n ...
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Suppose you drill a narrow tube (with cross sectional area A) from the surface of the earth down to the center. Then line the cylindrical wall of the tube with a frictionless coating. Then fill the tube back up with the dirt ...
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Coming back after a long time..... The question: Prove that a point can be found which is equidistant from each of the following four points: A (a m1, a/m1), B (a m2, a/m2), C (a m3, a/m3), and D (a/m1m2m3, a m1m2m3) ...
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Find ax5 + by5 where a,b,x,y are real numbers satisfying ax+by = 3 ax2+by2 = 7 ax3+by3 = 16 ax4+by4 = 42 ...
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Find the following limit for \alpha >1 : \lim_{n\to \infty} n\left(\dfrac{1^\alpha + 2^\alpha + \ldots + n^\alpha}{n^{\alpha+1}}-\dfrac{1}{\alpha +1}\right) Here n is a natural number. ...
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Let R be the set of real numbers. Suppose f : R → R be a continuous and periodic function with period T > 0. Prove that for every a < b, \lim_{n\to \infty} \int_a^b f(nx)\ \mathrm dx = \dfrac{b-a}{T}\int_0^T f(x)\ \ma ...
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Evaluate \int_{-1}^1\dfrac{\ln(13-6x)}{\sqrt{1-x^2}}\ \mathrm dx ...
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A given amount of (approximately ideal) gas is held in a spherical container. For what gas pressure the weight of the container will be minimal? ...
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The cube shown below has 5 faces grounded. The sixth side, insulated from the others, is held at a potential V0. What is the potential at the center of the cube and why? *Image* ...
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2n points are chosen on a circle. In how many ways can one join pairs of points by non-intersecting chords? ...
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A ball of mass M moving with velocity V0 on a frictionless surface strikes the first of two identical balls, each of mass m = 2 kg, connected by a massless spring with spring constant k = 1 kg/s2 (see Figure). Consider the co ...
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A body starts sliding (from rest) without friction down a hill of height h. In the beginning the body has potential energy and no kinetic energy; at the end the body has only kinetic energy. It follows from energy conservatio ...
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Consider a drop of conducting liquid in gravity free space. Its radius is R while the surface tension of the liquid is T. Now the drop is supplied some charge Q. For what values of Q is the drop stable. ...
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Let |S| denote the number of elements in the set S while n(S) denote the number of subsets of S including itself and the null set. If A, B, C are sets for which n(A) + n(B) + n(C) = n(A U B U C) and |A|=|B|=100, find the mini ...
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Prove the following: For every real valued function f differentiable on an interval [a,b] not containing 0 and for all pairs x1 ≠x2 in [a,b], there exists a point ξ in (x1, x2) such that \dfrac{x_1f(x_2)-x_2f(x_1)}{x_1-x_2 ...
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Suppose f be a continuously differentiable function on [a,b] and twice differentiable at x=a with f''(a) being non-zero; that is, the limit \lim_{x\to a^+}\dfrac{f'(x)-f'(a)}{x-a} exists and is non-zero. Applying LMVT to f in ...
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Find the limit: \lim_{x\to\infty}\left(\sin\sqrt{x+1}-\sin\sqrt{x}\right) (you must supply a proof for your answer.) ...
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Which is greater in the following pairs? (i) e^\pi or \pi^e (that's old) (ii) 2^{\sqrt{2}} or e (iii) \ln 8 or 2. In each case your answer must be accompanied by a proof. ...
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Q1) Prove that if a function f is continuous on [a,b], differentiable on (a,b) and f(a)=f(b)=0, then for any \alpha\in\mathbb{R} , there exists some c\in (a,b) such that \alpha f(c)+f'(c)=0 Q2) Let f and g be functions contin ...
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Let A={1, 2, 3, ..., n} Find the number of functions f : A → A having the property that f(f(x)) = f(x). P.S. Edited a bit. ...
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The energy released by the fission of uranium nuclei would be higher if the uranium nucleus split into three parts rather than into two parts. Despite this, the fission of uranium only produces two nuclei. Why is this? ...
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Find: \lim_{x\to\infty} \left(\dfrac{ex}{2}+x^2\left\{\left(1+\dfrac{1}{x}\right)^x-e\right\}\right) Note that {.} is NOT the fractional part. ...
