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Practice

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Engineering

Let $f : (0,\infty) \to R$ and $F(x) = \int_0^x t f(t) dt$ . If $F(x^2) = x^2 + x^3$ , then $\sum_{r=1}^{12} f(r^2)$ is equal to :

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Given below are two statements: Statement I: Chlorine can easily combine with oxygen to form oxides; and the product has a tendency to explode. Statement II: Chemical reactivity of an element can be determined by its reaction with oxygen and halogens. In the light of the above statements, choose the correct answer from the options given below

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A metal wire of uniform mass density having length L and mass M is bent to form a semicircular arc and a particle of mass m is placed at the centre of the arc. The gravitational force on the particle by the wire is: [Diagram: Semicircular arc with mass m at center]

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Questions

Questions

Let f:(0,∞)→Rf : (0,\infty) \to Rf:(0,∞)→R and F(x)=∫0xtf(t)dtF(x) = \int_0^x t f(t) dtF(x)=∫0x​tf(t)dt. If F(x2)=x2+x3F(x^2) = x^2 + x^3F(x2)=x2+x3, then ∑r=112f(r2)\sum_{r=1}^{12} f(r^2)∑r=112​f(r2) is equal to :

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Let $f : (0,\infty) \to R$ and $F(x) = \int_0^x t f(t) dt$ . If $F(x^2) = x^2 + x^3$ , then $\sum_{r=1}^{12} f(r^2)$ is equal to :