Equation of the plane passing through the point \((1,2,3)\) and parallel to the plane \(2 x+3 y-4 z=0\)

The integral \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x \mathrm{~d} x\) is equal to

Let \(\mathrm{A}=\lim _{\mathrm{x} \rightarrow 0^{+}}\left(1+\tan ^2 \sqrt{x}\right)^{\frac{1}{2 x}}\), then \(\log _{\mathrm{e}} \mathrm{A}=\)

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

The slopes of the lines represented by \(6 x^2+2 \mathrm{~h} x \mathrm{y}+\mathrm{y}^2=0\) are in the ratio \(2: 3\), then \(\mathrm{h}=\)

Let A and B be two events such that the probability that exactly one of them occurs is \(\frac{2}{5}\) and the probability that A or B occurs is \(\frac{1}{2}\), then the probability of both of them occur together is

If \(\tan \theta=2\) and \(\theta\) lies in the third quadrant, then the value of \(\sec \theta\) is

With usual notations, in \(\triangle \mathrm{ABC}\), if \(\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5\) and \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{k+7}{30}\),
then \(\mathrm{k}=\)

The maximum value of the objective function \(\mathrm{z}=2 \mathrm{x}+3 \mathrm{y}\) subject to the constraints \(\mathrm{x}+\mathrm{y} \leq 5,2 \mathrm{x}+\mathrm{y} \geq 4\) and \(\mathrm{x} \geq 0, \mathrm{y} \geq 0\) is

The dimensions of Planck's constant is the same as the product of

A gas is compressed at constant temperature. Its molecules gain

The mass of earth is 81 times the mass of the moon and the distance between their centres is \(\mathrm{R}\). The distance from the centre of the earth where gravitational force will be zero is

When exposed to sunlight, thin films of oil on water often exhibit brilliant colours due to the phenomenon of