JEE (Advanced) 2021 Paper-2
Let $$g_{i} : \left[\frac{\pi}{8},\frac{3\pi}{8}\right] \rightarrow R, i = 1,2$$, and $$f:\left[\frac{\pi}{8},\frac{3\pi}{8}\right] \rightarrow R$$ be function such that
$$g_{1}(x) = 1, g_{2}(x) = |4x-\pi|$$ and $$f(x) = \sin^{2} x$$, for all $$x \epsilon \left[\frac{\pi}{8},\frac{3\pi}{8}\right]$$
Define
$$S_{i} = \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} f(x)\cdot g_{i}(x) dx, i- 1, 2$$
he value of $$\frac{48S_{2}}{\pi^{2}}$$ is _______.
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let $$M = \left\{(x, y) \epsilon R \times R โถ x^{2} + y^{2} \leq r^{2} \right\}$$
where r > 0. Consder the geometric progression $$a_{n} = \frac{1}{2^{n-1}}, n = 1, 2, 3, ...$$ Let $$S_{0}=0$$ and, for $$n \geq 1$$, let $$S_{n}$$ denote the sume of the first n terms of this progression . For $$n \geq 1$$, Let$$C_{n}$$ denote the circle with center $$\left(S_{n-1},S_{n-1}\right)$$ and radius $$a_{n}$$.
Consider M with $$r = \frac{1025}{513}$$. Let k be the number of all those circle $$C_{n}$$ that are inside M. Let l be the maximum posible number of circle among these k circle such that no two circle intersect. Then
Consider $$M$$ with $$r = \frac{(2^{199}-1)\sqrt{2}}{2^{198}}$$. The number of all those circles $$D_{n}$$ that are inside M is
Let $$\psi: [0, \infty) \rightarrow R, \psi: [0, \infty) \rightarrow R, f:[0, \infty) \rightarrow R$$ and $$g:[0, \infty) \rightarrow R$$ be functions such that $$f(0)=g(0)=0$$,
$$\psi:(x)=e^{-x} + x, x \geq 0$$,
$$\psi:(x)=e^{2} - 2x - 2e^{-x} + 2 x \geq 0$$,
$$f(x)= \int_{-x}^{x} (|t| - t^{2})e^{-t^{2}} dt, x > 0$$,
and
$$g(x) = \int_{0}^{x^{2}} \sqrt{t} e^{-t} dt, x > 0.$$
For the following questions answer them individually
A number is chosen at random from the set {1, 2, 3, โฆ , 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is ___ .
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Let E be the ellipse $$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$$. For any three distinct points ๐,Q and $$Q^{'}$$ on E, let M(P, Q) be the mid-point of the line segment joining P and Q, and $$M(P, Q^{'})$$ be the mid-point of the line segment joining P and $$Q^{'}$$. Then the maximum possible valu of the distance between M(P, Q) and $$M(P, Q^{'})$$ as P, Q and $$Q^{'}$$ vary on E, is
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For any real number $$x$$, let [$$x$$] denote the largest integer less than or equal to $$x$$. If $$I = \int_{0}^{10}\left[\sqrt{\frac{10x}{x + 1}}\right] dx$$, then the value of 9I is _______.
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One end of a horizontal uniform beam of weight ๐ and length ๐ฟ is hinged on a vertical wall at point O and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point Q, at a height ๐ฟ above the hinge at point O. A block of weight $$\alpha W$$ is attached at the point P of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of ($$2\sqrt{2}$$)W. Which of the following statement(s) is(are) correct?
