An engine operates by taking \(n\,moles\) of an ideal gas through the cycle \(ABCDA\) shown in figure. The thermal efficiency of the engine is : (Take \(C_v =1 .5\, R\), where \(R\) is gas constant)

A long cylindrical volume contains a uniformly distributed charge of density \(\rho\). The radius of cylindrical volume is \(R\). A charge particle \((q)\) revolves around the cylinder in a circular path. The kinetic of the particle is

The correct truth table for the given input data of the following logic gate is:

The electrostatic potential due to an electric dipole at a distance '\(r\)' varies as _______.

Statement \(I:\) If three forces \(\vec{F}_{1}, \vec{F}_{2}\) and \(\vec{F}_{3}\) are represented by three sides of a triangle and \(\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement \(II:\) A triangle made up of three forces \(\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}\) and \(\overrightarrow{{F}}_{3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

A body of weight \(200 \mathrm{~N}\) is suspended form a tree branch thought a chain of mass \(10 \mathrm{~kg}\). The branch pulls the chain by a force equal to _______. (if \(g=10 \mathrm{~m} / \mathrm{s}^2\) )

A body of mass \(5\, kg\) under the action of constant force \(\overrightarrow F  = {F_x}\hat i + {F_y}\hat j\) has velocity at \(t\,= 0\, s\) as \(\overrightarrow v  = \left( {6\hat i - 2\hat j\,m/s} \right)\)  and at \(t\, = 10\,s\) as \(\overrightarrow v  = +6\hat j\,m/s\). The force \(\overrightarrow F \) is 

Let \(\alpha \in(0,1)\) and \(\beta=\log _{ c }(1-\alpha)\). Let \(P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots . .+\frac{x^n}{n}, x \in(0,1)\) Then the integral \(\int \limits_0^\alpha \frac{ t ^{50}}{1- t } dt\) is equal to

The area of the region enclosed by the curve \(y=x^3\) and its tangent at the point \((-1,-1)\) is

Let \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}\) and \(\overrightarrow{ c }=2 \hat{ i }-\hat{ j }+4 \hat{ k }\). If a vector \(\overrightarrow{ d }\) satisfies \(\overrightarrow{ d } \times \overrightarrow{ b }=\overrightarrow{ c } \times \overrightarrow{ b }\) and \(\overrightarrow{ d } \cdot \overrightarrow{ a }=24\), then \(|\overrightarrow{ d }|^2\) is equal to \(.........\).

A particle moves in a straight line so that its displacement \(x\) at any time \(t\) is given by \(x^2=1+t^2\). Its acceleration at any time \(\mathrm{t}\) is \(\mathrm{x}^{-\mathrm{n}}\) where \(\mathrm{n}=\) _______.

For real numbers \(a, b (a> b >0)\), let Area \(\left\{(x, y): x^{2}+y^{2} \leq a^{2}\right.\) and \(\left.\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \geq 1\right\}=30 \pi\) and  Area \(\left\{(x, y): x^{2}+y^{2} \geq b^{2}\right.\) and \(\left.\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1\right\}=18 \pi\)  Then the value of \((a-b)^{2}\) is equal to

Let \(f(x)=\begin{cases} e^{x-1}, & x<0 \\ x^2-5x+6, & x \geq 0 \end{cases}\) and \(g(x)=f(|x|)+|f(x)|\). If the number of points where \(g\) is not continuous and is not differentiable are \(\alpha\) and \(\beta\) respectively, then \(\alpha+\beta\) is equal to ______