Three masses \(m,\,2\,m\) and \(3\,m\) are moving in \(x-y\) plane with speed \(3\,u,\,2\,u\,\) and \(u\) respectively as shown in figure. The three masses collide at the same point at \(P\) and stick together. The velocity of resulting mass will be

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is \(6.4\) m. The speed of the bullets from the gun is ______ m/s.
(take \(g=10\) m/s\(^2\))

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

A particle is doing simple harmonic motion of amplitude \(0.06 \mathrm{~m}\) and time period \(3.14 \mathrm{~s}\). The maximum velocity of the particle is _______ \(\mathrm{cm} / \mathrm{s}\).

A particle is moving along a circular path with a constant speed of \(10\,ms^{-1}.\) What is the magnitude of the change in velocity of the particle, when it moves through an angle of \(60^{o}\) around the centre of the circle .......... \(m/s\)

A uniform rod of length \('l'\) is pivoted at one of its ends on a vertical shaft of negligible radius When the shaft rotates at angular speed \(\omega\) the rod makes an angle \(\theta\) with it (see figure). To find \(\theta\) equate the rate of change of angular momentum (direction going into the paper ) \(\frac{ m \ell^{2}}{12} \omega^{2} \sin \theta \cos \theta\) about the centre of mass \((CM)\) to the torque provided by the horizontal and vertical forces \(F_{H}\) and \(F_{V}\) about the CM. The value of \(\theta\) is then such that:

A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm. By applying an external torque of \(25 \pi \mathrm{Nm}\) for 40 s, the speed increases to 2100 rpm. The diameter of the disk is ________ m.

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 ________.

If \((1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\), then \(\dfrac{9a_9}{a_{10}}\) is equal to __________.

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is \(\left(\frac{x}{100}\right) \% .\) If the relative errors in measuring the mass and the diameter are \(6.0 \%\) and \(1.5 \%\) respectively, the value of \(x\) is

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is \(10\) and one of the foci is at \((0, 5\sqrt 3 )\), then the length of its latus rectum is

If \(7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\frac{1}{7^3}(5+3 \alpha)+\ldots \infty\), then the value of \(\alpha\) is :

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