If \(f(x)=|x-3|\), then \(f^{\prime}(3)\) is

The statement pattern \([p \rightarrow(q \rightarrow p)] \rightarrow[p \rightarrow(p \vee q)]\) is

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are three vectors such that \(|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=1\), \(\overline{\mathrm{c}}=\lambda(\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) and \(|\overline{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overline{\mathrm{b}}|=\frac{1}{\sqrt{2}},|\overline{\mathrm{c}}|=\frac{1}{\sqrt{6}}\), then the angle between \(\bar{a}\) and \(\bar{b}\) is

Let \(\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection at \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is equal to

Let \(\mathrm{G}\) be the centroid of a triangle \(\mathrm{ABC}\) and \(0 \mathrm{be}\) any other point in that plane, then \(\overline{\mathrm{OA}}+\overline{\mathrm{OB}}+\overline{\mathrm{OC}}+\overline{\mathrm{OG}}=\)

With usual notations, perimeter of a triangle \(\mathrm{ABC}\) is 6 times the arithmetic mean of since of its angles. If \(a=1\), then measure of angle \(\mathrm{A}=\)

A particular solution of \(3 \mathrm{e}^x \tan \mathrm{y} \mathrm{d} x+\left(1-\mathrm{e}^x\right) \sec ^2 \mathrm{y} \mathrm{dy}=0\) with \(y(1)=\frac{\pi}{4} \quad\) is

If \(\left|\frac{\mathrm{z}}{1+\mathrm{i}}\right|=2\), where \(\mathrm{z}=x+\mathrm{i} y, \mathrm{i}=\sqrt{-1}\) represents a circle, then centre ' \(C\) ' and radius ' \(r\) ' of the circle are

Calculate osmotic pressure of solution of 0.025 mole glucose in \(100 \mathrm{~mL}\) water at \(300 \mathrm{~K}\). \(\left[\mathrm{R}=0.082 \mathrm{~atm} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\right]\)

Which of the following is a solid trimer of formaldehyde?

The assets of a person are increasing at a rate proportional to the square root of the assets at a given time. His assets increase from Rupees 9 crores to Rupees 16 crores in 2 years, then his assets at the end of 5 more years will be

A diatomic molecule has moment of inertia 'I'. By applying Bohr's quantization condition, its rotational energy in the \(\mathrm{n}^{\text {th }}\) level is \([\mathrm{n} \geq 1]\) [h= Planck's constant \(]\)

The efficiency index of growth in plants represents ________.