Position of a \(2 \mathrm{~kg}\) mass moving along the \(X\)-axis is given by \(x=2 \cos [(2) t] \mathrm{m}\). Then maximum kinetic energy of the mass in joule is

Two towers \(A\) and \(B\), each of height \(20 \mathrm{~m}\) are situated a distance \(200 \mathrm{~m}\) apart. A body thrown horizontally from the top of the tower \(A\) with a velocity \(20 \mathrm{~ms}^{-1}\) towards the tower \(B\) hits the ground at point \(P\) and another body thrown horizontally from the top of tower \(B\) with a velocity \(30 \mathrm{~ms}^{-1}\) towards the tower \(A\) hits the ground at point \(Q\). If a car starting from rest from \(P\) reaches \(Q\) in 10 seconds, then the acceleration of the car is \(\left(\right.\) acceleration due to gravity \(\left.=10 \mathrm{~ms}^{-2}\right)\)

A particle of mass \(2.2 \times 10^{-30} \mathrm{~kg}\) and charge \(1.6 \times 10^{-19} \mathrm{C}\) is moving at a speed of \(10 \mathrm{~km} \mathrm{~s}^{-1}\) in a circular path of radius \(2.8 \mathrm{~cm}\) inside a solenoid. The solenoid has \(25 \frac{\text { turns }}{\mathrm{cm}}\) and its magnetic field is perpendicular to the plane of the particle's path. The current in the solenoid is
\(\left(\right.\) Take, \(\mu_0=4 \pi \times 10^{-7} \mathrm{Hm}^{-1}\) )

Find the dimension formula of \(\frac{a}{b}\) in the equation \(F=a \sqrt{x}+b t^2\), where \(F\) is a force, \(x\) is distance and \(t\) is time.

A wire of \(20 \Omega\) is immersed in ice. If \(10 \mathrm{~A}\) current is passed through this wire for 1 minute, ice completely melts. The mass of the ice is nearly \(\left(L_{\text {ice }}=79.7 \mathrm{calg}^{-1}\right)\)

A mass \(M\) attached to a horizontal spring executes simple harmonic motion with amplitude \(A_1\). When mass \(M\) passes mean position then a smaller mass \(m\) is attached to it and both of them together executing simple harmonic motion with amplitude \(A_2\). Then the value of \(\frac{A_1}{A_2}\) is

Potential difference between the points \(P\) and \(Q\) in the circuit shown is

Let \(\quad \overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \quad \overrightarrow{\mathbf{b}}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}} \quad\) and \(\overrightarrow{\mathbf{c}}=\lambda \hat{\mathbf{i}}+\hat{\mathbf{j}}+(2 \lambda-1) \hat{\mathbf{k}}\). If \(\overrightarrow{\mathbf{c}}\) is parallel to the plane containing \(\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}\), then \(\lambda\) is equal to

Which one of the following compounds will not show geometrical isomerism?

If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+x+1\) \(=0\), then the quadratic equation whose roots are \(\alpha^{2023}\) and \(\beta^{1012}\) is

If the coefficients of \(r\) th and \((r+1)\) th terms in the expansion of \((3+7 x)^{29}\) are equal, then \(r\) is equal to

If \(f:[0,2) \rightarrow[R\) is defined by
\(f(x)=\left\{\begin{array}{cl}1+\frac{2 x}{k} & \text { for } \quad 0 \leq x < 1 \\ k x & \text { for } 1 \leq x < 2\end{array}\right.\) where \(k>0\), and \(f\) is such that \(\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)\), then

The circle \(x^2+y^2-6 x-10 y+p=0\) neither intersects nor touch the coordinate axes and the point \((1,4)\) lies inside the circle. Then the range of possible values of ' \(p\) ' is