Three infinitely long wires with linear charge density \(\lambda\) are placed along the \(x-a x i s, y\)-axis and \(z-\) axis respectively. Which of the following denotes an equipotential surface?

Statement \(I :\)Two forces \((\overrightarrow{{P}}+\overrightarrow{{Q}})\) and \((\overrightarrow{{P}}-\overrightarrow{{Q}})\) where \(\overrightarrow{{P}} \perp \overrightarrow{{Q}}\), when act at an angle \(\theta_{1}\) to each other, the magnitude of their resultant is \(\sqrt{3\left({P}^{2}+{Q}^{2}\right)}\), when they act at an angle \(\theta_{2}\), the magnitude of their resultant becomes \(\sqrt{2\left({P}^{2}+{Q}^{2}\right)}\). This is possible only when \(\theta_{1}<\theta_{2}\). Statement \(II :\) In the situation given above. \(\theta_{1}=60^{\circ} \text { and } \theta_{2}=90^{\circ}\) In the light of the above statements, choose the most appropriate answer from the options given below

In an experiment for estimating the value of focal length of converging mirror, image of an object placed at \(40\,cm\) from the pole of the mirror is formed at distance \(120\,cm\) from the pole of the mirror. These distances are measured with a modified scale in which there are \(20\) small divisions in \(1\,cm\). The value of error in measurement of focal length of the mirror is \(1 / K\) \(cm\). The value of \(K\) is  \(..........\)

In order to determine the Young's Modulus of a wire of radius \(0.2\, cm\) (measured using a scale of least count \(=0.001\, cm )\) and length \(1 \,m\) (measured using a scale of least count \(=1\, mm\) ), a weight of mass \(1\, kg\) (measured using a scale of least count \(=1 \,g\) ) was hanged to get the elongation of \(0.5\, cm\) (measured using a scale of least count \(0.001\, cm\) ). What will be the fractional error in the value of Young's Modulus determined by this experiment? (in \(\%\))

As shown in the figure, in an experiment to determine Young's modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of \(45^{\circ}\) with the load axis. The length of wire is \(62.8\,cm\) and its diameter is \(4\,mm\). The Young's modulus is found to be \(x \times\) \(10^4\,Nm ^{-2}\). The value of \(x\) is

As per given figure, a weightless pulley \(P\) is attached on a double inclined frictionless surface. The tension in the string (massless) will be (if \(g=\) \(10\,m / s ^2\) )

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

Let \(\mathrm{f}: N \rightarrow N\) be a function such that \(\mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n})\) for every \(\mathrm{m}, \mathrm{n} \in N\). If \(\mathrm{f}(6)=18\) then \(\mathrm{f}(2) \cdot \mathrm{f}(3)\) is equal to :

Let a line \(L_1\) pass through the origin and be perpendicular to the lines
\(L_2: \vec{r} = (3+t)\hat{i} + (2t-1)\hat{j} + (2t+4)\hat{k}\) and
\(L_3: \vec{r} = (3+2s)\hat{i} + (3+2s)\hat{j} + (2+s)\hat{k}\), \(t, s \in \mathbb{R}\).
If \((a, b, c)\), \(a \in \mathbb{Z}\), is the point on \(L_3\) at a distance of \(\sqrt{17}\) from the point of intersection of \(L_1\) and \(L_2\), then \((a+b+c)^2\) is equal to ________.

Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be differentiable function such that \(f(\mathrm{x})=1-2 \mathrm{x}+\int_0^x e^{x-t} f(t) \mathrm{dt}\) for all \(\mathrm{x} \in[0, \infty)\).
Then the area of the region bounded by \(\mathrm{y}=f(\mathrm{x})\) and the coordinate axes is

There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is

Given \(A\) and \(B\) are input terminals. Logic \(1\, = \,> 5\, V\) Logic \(0\, =\, < 1\, V\) Which logic gate operation, the above circuit does?

Lets \(S=\{z \in C:|z-1|=1\) and \((\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2}\}\). Let \(\mathrm{z}_1, \mathrm{z}_2\) \(\in S\) be such that \(\left|z_1\right|=\max _{z \in S}|z|\) and \(\left|z_2\right|=\min _{z \in S}|z|\). Then \(\left|\sqrt{2} z_1-z_2\right|^2\) equals :