Find the maximum value of \(f(x)=\dfrac{1}{4 x^{2}+2 x+1}\).

If \(\int \frac{x e^{x}}{(1+x)^{2}} d x=e^{x} f(x)+c\), then \(f(x)\) is equal to

The solution set of the system of inequalities \(5-4 x \leq-7\) or \(5-4 x \geq 7, x \in R\) is

An open hemispherical storage tank has radius 13 m. Oil flows into the tank such that the depth '\(h\)' of oil in the tank changes at the rate of 3 m / hr. When the depth \(h = 1\)m, the rate of change of the area of the top surface of the oil is

The medians \(A D\) and \(B E\) of a triangle with vertices \(A(0, b), B(0,0)\) and \(C(a, 0)\) are perpendicular to each other, if

\(\text { If } \operatorname{cosec}(90+A)+x \cos A \cot (90+A)=\sin (90+A) \text { then the value of } x \text { is }\)

The scalar components of a unit vector which is perpendicular to each of the vectors \(\hat{\imath}+2 \hat{\jmath}-\hat{k}\) and \(3 \hat{\imath}-\hat{\jmath}+2 \hat{k}\) are

If \(f\left(x^{5}\right)=5 x^{3}\), then \(f^{\prime}(x)=\)

Which of the following statements is/are true about the diagonal relationship of Be and Al ?
(i) Both react with NaOH to liberate hydrogen.
(ii) Their oxides are basic.
(iii) They are made passive by nitric acid.
(iv) Their carbides give acetylene on treatment with water.

The angle between the lines \(\frac{x-2}{3}=\frac{y-1}{1}=\frac{z+3}{-1}\) and \(\frac{x+1}{2}=\frac{y-1}{-1}=\frac{z}{2}\) is

Plane polarised light is passed through an analyser and the intensity of emerging light is reduced by \(75 \%\). Optical vibrations make an angle \(\theta\) with the axis of analyser, then \(\theta\) is

Two simple harmonic motions are represented by equations \(y_1=0.5 \sin \left[200 \pi t+\dfrac{\pi}{3}\right]\) and \(y_1=0.5 \cos \pi t\). The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is:

Given below are 4 equations showing Molar conductivities at infinite dilution of various electrolytes. Which one of them represents the correct equation?
\(\begin{aligned}
& \left(\Lambda_m^0\right)_{N a B r}-\left(\Lambda_m^0\right)_{N a C l}=\left(\Lambda_m^0\right)_{\mathrm{KCl}}-\left(\Lambda_m^0\right)_{\mathrm{KBr}} \\
& \left(\Lambda_m^0\right)_{\mathrm{HCl}}+\left(\Lambda_m^0\right)_{\mathrm{KOH}}-\left(\Lambda_m^0\right)_{\mathrm{KCl}}=\left(\Lambda_m^0\right)_{\mathrm{H}_2 \mathrm{O}} \\
& \left(\Lambda_m^0\right)_{\mathrm{KBr}}-\left(\Lambda_m^0\right)_{\mathrm{NaBr}}=\left(\Lambda_m^0\right)_{\mathrm{NaBr}}-\left(\Lambda_m^0\right)_{\mathrm{Nal}} \\
& \left(\Lambda_m^0\right)_{N H_4 \mathrm{Cl}}-\left(\Lambda_m^0\right)_{\mathrm{NH}_4 \mathrm{NO}_3}=\left(\Lambda_m^0\right)_{N H_4 \mathrm{Cl}}-\left(\Lambda_m^0\right)_{N H 4 B r}
\end{aligned}\)