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A \(1 \, \text{kg}\) mass is suspended from the ceiling by a rope of length \(4 \, \text{m}\). A horizontal force \(F\) is applied at the mid point of the rope so that the rope makes an angle of \(45^\circ\) with respect to the vertical axis as shown in figure. The magnitude of \(F\) is:
(Assume that the system is in equilibrium and \(g = 10 \, \text{m/s}^{2}\))

If \( \vec{a} \) and \( \vec{b} \) make angle \( \cos^{-1}\!\left(\dfrac{5}{9}\right) \) with each other, then \( \lvert \vec{a} + \vec{b} \rvert = \sqrt{2} \, \lvert \vec{a} - \vec{b} \rvert \) for \( \lvert \vec{a} \rvert = n \, \lvert \vec{b} \rvert \). The integer value of \( n \) is ________.

The resultant of two vectors \( \vec{A} \) and \( \vec{B} \) is perpendicular to \( \vec{A} \) and its magnitude is half that of \( \vec{B} \). The angle between vectors \( \vec{A} \) and \( \vec{B} \) is…..\( ^\circ \).

Two forces \( \vec{F}_{1} \) and \( \vec{F}_{2} \) are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between \( \vec{F}_{1} \) and \( \vec{F}_{2} \) is \( \cos^{-1}\!\left( \frac{1}{n} \right) \). The value of \( |n| \) is ________.

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JEE Mains JEE Advance BITSAT
JEE Mains | PYQ