Mathematics Tes...

A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:

If \(\overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{b}}\) are two vectors such that \(|\overrightarrow{\mathrm{a}}|=\frac{1}{\sqrt{3}},|\overrightarrow{\mathrm{b}}|=2\) and \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\) is a unit vector, then find the angle between \(\vec{a}\) and \(\vec{b}\)

If point \(A\) is in equilibrium under the action of the applied forces, the values of tension. \(\mathrm{T}_{\mathrm{AC}}=\) ?

A bullet fired at an angle of \(30^{\circ}\) with the horizontal hits the ground \(3.0 \mathrm{~km}\) away. By adjusting its angle of projection, find maximum range \(\left(\mathrm{R}_{\max }\right)\) is achieved by the bullet? Assume the muzzle speed to be fixed, and neglect air resistance.

A medicine is known to be 50% effective to cure a patient. If the medicine is given to 4 patients, what is the probability that at least one patient is cured by this medicine?

The equation \(x^{2}+y^{2}+2 g x+2 f y+c=0\) always represents a circle whose centre is \((-g,-f)\) and radius is \(\sqrt{g^{2}+f^{2}-c}\). If \(g^{2}+f^{2}=c\), then in this case, the circle is called as:

Find the equation of the plane passing through the points A (1, 0, 2), B (2, 1, 1) and C (-1, 2, 1)?

Consider a truss \(PQR\) loaded at \(P\) with a force \(F\) as shown in the figure. The tension in the member \(QR\) is:

Find the value of \({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+2 \times\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}\right)+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2}\)?