A convex mirror of radius of curvature \(30 \mathrm{~cm}\) forms an image that is half the size of the object. The object distance is :

The ratio of escape velocity of a planet to the escape velocity of earth will be:Given : Mass of the planet is \(16\) times mass of earth and radius of the planet is \(4\) times the radius of earth.

The angular speed of truck wheel is increased from \(900\, rpm\) to \(2460\, rpm\) in \(26\) seconds. The number of revolutions by the truck engine during this time is (Assuming the acceleration to be uniform).

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

A particle of mass \(20\,g\) is released with an initial velocity \(5\,m/s\) along the curve from the point \(A,\) as shown in the figure. The point \(A\) is at height \(h\) from point \(B.\) The particle slides along the frictionless surface. When the particle reaches point \(B,\) its angular momentum about \(O\) will be ......... \(kg - m^2/s\). [Take \(g = 10\,m/s^2\) ]

Two point dipoles of dipole moment \({\vec P_1}\) and \({\vec P_2}\) are at a distance \(x\) from each other and  \({\vec P_1}\) || \({\vec P_2}\). The force between the dipoles is

The zener diode has a \(V _{ z }=30\, V\). The current passing through the diode for the following circuit is \(\ldots \ldots \ldots \,mA.\)

A conducting wire of length \( l\) area of cross-section \(A\) and electric resistivity \(\rho\) is connected between the terminals of a battery. \(A\) potential difference \(V\) is developed between its ends, causing an electric current.If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be

The plot given below is of the average power delivered to an \(LRC\) circuit versus frequency. The quality factor of the circuit is

Let \(a , b , c\) be three distinct positive real numbers such that \((2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}\) and \(b^{\log _e 2}=a^{\log _e c}\). Then \(6 a+5 b c\) is equal to \(........\).

The area of the region bounded by \(y^{2}=8 x\) and \(y^{2}=16(3-x)\) is equal to

Let \(f(x)=x^5+2 x^3+3 x+1, x \in R\), and \(g(x)\) be a function such that \(g(f(x))=x\) for all \(x \in R\). Then \(\frac{g(7)}{g^{\prime}(7)}\) is equal to :

In a screw gauge, there are \(100\) divisions on the circular scale and the main scale moves by \(0.5\,mm\) on a complete rotation of the circular scale. The zero of circular scale lies \(6\) divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, \(4\) linear scale divisions are clearly visible while \(46^{\text {th }}\) division the circular scale coincide with the reference line. The diameter of the wire is \(...........\times 10^{-2}\,mm\)