A conical pendulum of length \(1\,m\) makes an angle \(\theta \, = 45^o\) w.r.t. \(Z-\) axis and moves in a circle in the \(XY\) plane.The radius of the circle is \(0.4\, m\) and its centre is vertically below \(O\). The speed of the pendulum, in its circular path, will be ..... \(m/s\) (Take \(g\, = 10\, ms^{-2}\))

The value of current I in the electrical circuit as given below, when potential at A is equal to the potential at B, will be ____ A.

Two bodies of the same mass are moving with the same speed, but in different directions in a plane. They have a completely inelastic collision and move together thereafter with a final speed which is half of their initial speed. The angle between the initial velocities of the two bodies (in degree) is.

A small ball of mass \(m\) is thrown upward with velocity \(u\) from the ground. The ball experiences a resistive force \(mkv ^{2}\) where \(v\) is its speed. The maximum height attained by the ball is 

A solid steel ball of diameter 3.6 mm acquired terminal velocity \(2.45 \times 10^{-2} \mathrm{~m} / \mathrm{s}\) while falling under gravity through an oil of density \(925 \mathrm{~kg} \mathrm{~m}^{-3}\). Take density of steel as \(7825 \mathrm{~kg} \mathrm{~m}^{-3}\) and g as 9.8 \(\mathrm{m} / \mathrm{s}^2\). The viscosity of the oil in SI unit is

The de-Broglie wavelength associated with an electron and a proton were calculated by accelerating them through same potential of \(100\, V\). What should nearly be the ratio of their wavelengths ? \(\left( m _{ P }=1.00727\, u , m _{ e }=0.00055 \,u \right)\)

A person lives in a high-rise building on the bank of a river \(50\, m\) wide. Across the river is a well lit tower of height \(40\, m\). When the person, who is at a height of \(10\, m\), looks through a polarizer at an appropriate angle at light of the tower reflecting from the river surface, he notes that intensity of light coming from distance \(X\) from his building is the least and this corresponds to the light coming from light bulbs at height \('Y'\) on the tower. The values of \(X\) and \(Y\) are respectively close to (refractive index of water \(  \simeq  \frac{4}{3})\)

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

The general displacement of a simple harmonic oscillator is \(x = A \sin \omega t\). Let \(T\) be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when \(t=\frac{T}{\beta}\). The value of \(\beta\) is \(.........\)

The point of intersection \(C\) of the plane \(8 x+y+2 z=0\) and the line joining the points \(A (-3,-6,1)\) and \(B (2,4,-3)\) divides the line segment \(AB\) internally in the ratio \(k : 1\). If \(a , b , c\) \((| a |,| b |,| c |\) are coprime) are the direction ratios of the perpendicular from the point \(C\) on the line \(\frac{1- x }{1}=\frac{ y +4}{2}=\frac{z+2}{3}\), then \(|a + b + c|\) is equal to \(.............\).

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(m+n\) is equal to :

Two slabs with square cross section of different materials \((1,2)\) with equal sides \((l)\) and thickness \(\mathrm{d}_1\) and \(\mathrm{d}_2\) such that \(\mathrm{d}_2=2 \mathrm{~d}_1\) and \(l \gt \mathrm{d}_2\). Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is \(\theta_2=2 \theta_1\). If the shear moduli of material 1 is \(4 \times 10^9 \mathrm{~N} / \mathrm{m}^2\), then shear moduli of material 2 is \(\mathrm{x} \times 10^9 \mathrm{~N} / \mathrm{m}^2\), where value of \(x\) is ________.

If \(A\) is a symmetric matrix and \(B\) is a skew-symmetrix matrix such that \(A + B = \left[ {\begin{array}{*{20}{c}}
2&3\\
5&{ - 1}
\end{array}} \right]\) , then \(AB\) is equal to