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If \(\int e^{\sin x}\left[\frac{x \cos ^{3} x-\sin x}{\cos ^{2} x}\right] d x=e^{\sin x} \cdot f(x)+c\), where \(c\) is constant of integration, then \(f(x)=\)
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Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a twice continuously differentiable function such that \(f(0)=f(1)=f^{\prime}(0)=0 .\) Then
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Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a twice continuously differentiable function such that \(f(0)=f(1)=f^{\prime}(0)=0\). Then
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Let \(\mathrm{f}:[\mathrm{a}, \mathrm{b}] \rightarrow \mathbb{R}\) be such \(\mathrm{f}\) is differentiable in \((\mathrm{a}, \mathrm{b}), \mathrm{f}\) is continuous at \(\mathrm{x}=\mathrm{a} \& \mathrm{x}=\mathrm{b}\) and moreover \(\mathrm{f}(\mathrm{a})=0=\mathrm{f}(\mathrm{b}) .\) Then
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Let \(f(x)=3 x^{10}-7 x^{8}+5 x^{6}-21 x^{3}+3 x^{2}-7\). Then \(\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{h^{3}+3 h}\)
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Let \(f:[a, b] \rightarrow \mathbb{R}\) be differentiable on \([a, b] \& k \in \mathbb{R} .\) Let \(f(a)=0=f(b)\) Also let \(J(x)=f^{\prime}(x)+k f(x)\). Then
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The domain of definition of \(f(x)=\sqrt{\frac{1-|x|}{2-|x|}}\) is
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Let \(f_{1}(s)=e^{x}, f_{2}(x)=e^{f(x)}, \ldots \ldots . f_{n+1}(x)=e^{f(x)}\) for all \(n \geq 1\). The for any fixed \(n, \frac{d}{d x} f_{n}(x)\) is
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The approximate value of \(\sin 31^{\circ}\) is
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White phosphorus \(\mathrm{P}_{4}\) has the following characteristics:
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Some basic concepts in chemistry
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Among the following the extensive variables are
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Some basic concepts in chemistry
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Which statements are correct for the peroxide ion?
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Some basic concepts in chemistry
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\([\mathrm{X}]+\) dil. \(\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow[\mathrm{Y}]:\) Colourless, suffocating gas \([\mathrm{Y}]+\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}+\mathrm{H}_{2} \mathrm{SO}_{4} \rightarrow\) Green colouration of solution
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During a reversible adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio \(\frac{C_{p}}{C_{v}}\) for the gas is
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Some basic concepts in chemistry
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A metal \(M\) (specific heat \(0.16\) ) forms a metal chloride with \(=65 \%\) chlorine present in it. The formula of the metal chloride will be
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The number of possible organobromine compounds, which can be obtained in the allylic bromination of 1 - butene with \(\mathrm{N}\) - bromosuccinimide is
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$$ \begin{aligned} &{[\mathrm{P}] \stackrel{\mathrm{Br}_{2}}{\longrightarrow} \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Br}_{2} \underset{\mathrm{NH}_{3}}{\stackrel{\mathrm{NaNH}_{2}}{\longrightarrow}}[\mathrm{Q}]} \\ &{[\mathrm{Q}] \stackrel{20 \% \mathrm{H}_{2} \mathrm{SO}_{4}}{\mathrm{Hg}^{2}+\Delta}[\mathrm{R}] \stackrel{\mathrm{Zn}-\mathrm{Hg} / \mathrm{HCl}}{\longrightarrow}[\mathrm{S}]} \end{aligned} $$ The species \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) respectively are
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Which of the following is least thermally stable?
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At room temperature, the reaction between water and fluorine produces
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Out of the following outer electronic configurations of atoms, the highest oxidation state is achieved by which one ?
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