For all real \(x\), the minimum value of \(\frac{1-x+x^2}{1+x+x^2}\) 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 service station manager sells gas at an average of ₹ 100 per hour on a rainy day, ₹ 150 per hour on a dubious day, ₹ 250 per hour on a fair day and ₹ 300 on a clear sky. If weather bureau statistics show the probabilities of weather as follows, then his mathematical expectation is

Out of 100 people selected at random, 10 have common cold. If five persons selected at random from the group, then the probability that at most one person will have common cold is

If \(f^{\prime}(x)=k(\cos x+\sin x)\) and \(f(0)=9, f\left(\frac{\pi}{2}\right)=15\), then \(f(x)=\)

If \(\frac{1}{6} \sin \theta, \cos \theta, \tan \theta\) are in G.P., then the general solution of \(\theta\) is

The number of different 9 digit number that can be formed, from the digits of the number 445577888 by rearranging its digits, so that the odd digits occupy even positions, are

A long solenoid has 1500 turns. When a current of \(3.5 \mathrm{~A}\) flows through it, the magnetic flux linked with each turn of solenoid is \(2.8 \times 10^{-3}\) weber. The self-inductance of solenoid is

Which of the following is Rosenmund reduction?

During the electrolysis of fused \(\mathrm{NaCl}\), the product obtained at anode is

Moment of inertia of a solid sphere about its diameter is 'I'. It is then casted into 27 small spheres of same diameter. The moment of inertia of each small sphere about its diameter is

If \(\mathrm{T}\) is the half-life of a radioactive substance then its instantaneous rate of change of activity is proportional to

The pressure and density of a diatomic gas \(\left(\gamma=\frac{7}{5}\right)\) changes adiabatically from \((\mathrm{P}, \rho)\) to \(\left(\mathrm{P}^{\prime}, \rho^{\prime}\right)\). If \(\frac{\rho^{\prime}}{\rho}=32\) then \(\frac{\mathrm{P}^{\prime}}{\mathrm{P}}\) should be