The area (in sq. units) of the region described by \(\left\{(x, y) / y^2 \leq 2 x\right.\) and \(\left.y \geq(4 x-1)\right\}\) is

Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be three non-coplanar vectors and \(\overline{\mathrm{p}}, \overline{\mathrm{q}}, \overline{\mathrm{r}}\) defined by the relations
\(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}\)
then the value of the expression \((\bar{a}+\bar{b}) \cdot \bar{p}+(\bar{b}+\bar{c}) \cdot \bar{q}+(\bar{c}+\bar{a}) \cdot \bar{r}\) is equal to

A line makes an angle of \(45^{\circ}\) with \(x\) -axis and congruent angles with \(y\) and \(z\) -axes, then the direction cosines of the line are

The approximate value of \(\cot ^{-1}(1 \cdot 001)\) is

If \(y=(\sin x)^{\tan x}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

The value of \(\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}\) is

If \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi\), then the value of \(x^{2025}+x^{2026}+x^{2027}\) is

If the line, \(\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3}\) lies in the plane, \(\ell x+m y-z=9\), then \(\ell^2+m^2\) is equal to

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at \(\mathrm{t}=0\). The number of bacteria is increased by \(20 \%\) in 2 hours. If the population of bacteria is 2000 after \(\frac{\mathrm{k}}{\log \left(\frac{6}{5}\right)}\) hours, then \(\left(\frac{\mathrm{k}}{\log 2}\right)^2\) is

The magnetic flux near the axis and inside the air core solenoid of length 80 cm carrying current ' I ' is \(1.57 \times 10^{-6} \mathrm{~Wb}\). Its magnetic moment will be [cross-sectional area of a solenoid is very small as compared to its length,\(\mu_0=4 \pi \times 10^{-7} \text { SI unit] }(\pi=3.14)\)

A pipe \(\mathrm{P}_{\mathrm{C}}\) closed at one end and pipe \(\mathrm{P}_{\mathrm{O}}\) open at both ends are vibrating in second overtone. They are in resonance with a given tuning fork. The ratio of the length of pipe \(\mathrm{P}_{\mathrm{C}}\) vs \(\mathrm{P}_{\mathrm{O}}\) is (Neglect end correction)

If the position vectors of the point \(\mathrm{A}\) and \(\mathrm{B}\) are \(3 \hat{i}+\hat{j}+2 \hat{k}\) and \(\hat{i}-2 \hat{j}-4 \hat{k}\) respectively, then the equation of the plane through \(\mathrm{B}\) and perpendicular to \(\mathrm{AB}\) is

A \(20 \mu \mathrm{F}\) capacitor is connected to \(45 \mathrm{~V}\) battery through a circuit whose resistance is \(2000 \Omega\). What is the final charge on the capacitor?