A capacitor is discharging through a resistor \(R\). Consider in time \(t _{1}\), the energy stored in the capacitor reduces to half of its initial value and in time \(t_{2}\), the charge stored reduces to one eighth of its initial value. The ratio \(t_{1} / t_{2}\) will be ................

Given below are two statements: one is labelled as Assertion\((A)\) and the other is labelled as Reason \((R).\) \(Assertion\) \((A)\) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading. \(Reason\) \((R)\) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling. In the light of the above statements, choose the correct answer from the options given below :

Match List \(- I\) (Electromagnetic wave type) with List \(- II\) (Its association/application) and select the correct option from the choices given below the lists

લિસ્ટ\(-I\)	લિસ્ટ\(-II\)
\(a\). Infrared waves	\(i\). To treat muscular strain
\(b\). Radio waves	\(ii\). For broadcasting
\(c\). \(X-\)rays	\(iii\). To detect fracture of bones
\(d\). Ultraviolet rays	\(iv\). Absorbed by the ozone layer of the atmosphere

\(a\)         \(b\)         \(c\)         \(d\)

The magnetic field vector of an electromagnetic wave is given by \({B}={B}_{o} \frac{\hat{{i}}+\hat{{j}}}{\sqrt{2}} \cos ({kz}-\omega {t})\); where \(\hat{i}, \hat{j}\) represents unit vector along \({x}\) and \({y}\)-axis respectively. At \(t=0\, {s}\), two electric charges \(q_{1}\) of \(4\, \pi\) coulomb and \({q}_{2}\) of \(2 \,\pi\) coulomb located at \(\left(0,0, \frac{\pi}{{k}}\right)\) and \(\left(0,0, \frac{3 \pi}{{k}}\right)\), respectively, have the same velocity of \(0.5 \,{c} \hat{{i}}\), (where \({c}\) is the velocity of light). The ratio of the force acting on charge \({q}_{1}\) to \({q}_{2}\) is :-

At some instant, a radioactive sample \(S_1\) having an activity \(5\,\mu Ci\)  has twice the number of nuclei as another sample \(S_2\)  which has an activity of  \(10\,\mu Ci.\)  The halflives of \(S_1\)  and  \(S_2\) are

A particle is moving in a straight line. The variation of position ' \(x\) ' as a function of time ' \(t\) ' is given as \(x=\left(t^3-6 t^2+20 t+15\right) m\). The velocity of the body when its acceleration becomes zero is _______.

Interference fringes are observed on a screen by illuminating two thin slits \(1 \,mm\) apart with a light source \((\lambda=632.8\, nm )\). The distance between the screen and the slits is \(100\, cm\). If a bright fringe is observed on a screen at a distance of \(1.27\, mm\) from the central bright fringe, then the path difference between the waves, which are reaching this point from the slits is close to\(....\mu m\)

A uniform electric field of \(10\,N / C\) is created between two parallel charged plates (as shown in figure). An electron enters the field symmetrically between the plates with a kinetic energy \(0.5\,eV\). The length of each plate is \(10\,cm\). The angle \((\theta)\) of deviation of the path of electron as it comes out of the field is  \(.........\)(in degree).

The total number of words (with or without meaning) that can be formed out of the letters of the word '\(DISTRIBUTION\)' taken four at a time, is equal to ...........

Let the set of all \(a \in R\) such that the equation \(\cos 2 x+a \sin x=2 a-7\) has a solution be \([p, q]\) and \(\mathrm{r}=\tan 9^{\circ}-\tan 27^{\circ}-\frac{1}{\cot 63^{\circ}}+\tan 81^{\circ}\), then \(pqr\) is equal to ...........

10 kg of ice at \(-10^{\circ}C\) is added to 100 kg of water to lower its temperature from \(25^{\circ}C.\) Consider no heat exchange to surroundings. The decrement to the temperature of water is ___________ \(^{\circ}C.\)
(specific heat of ice \(= 2100\ J/Kg.^{\circ}C\), specific heat of water \(= 4200\ J/Kg.^{\circ}C\), latent heat of fusion of ice \(= 3.36 \times 10^{5}\ J/Kg\))

Two balls \(A\) and \(B\) are placed at the top of \(180 \,m\) tall tower. Ball \(A\) is released from the top at \(t =0 \,s\). Ball \(B\) is thrown vertically down with an initial velocity \(u\)' at \(t=2\, s\). After a certain time, both balls meet \(100 \,m\) above the ground. Find the value of \(u\) in ............... \(ms ^{-1}\). [use \(g =10 \,ms ^{-2}\) ]

Let \(O\) be the origin. Let \(\overline{ OP }= x \hat{ i }+ y \hat{ j }-\hat{ k }\) and \(\overline{ OQ }=-\hat{ i }+2 \hat{ j }+3 x \hat{ k }, x , y \in R , x >0,\) be such that \(|\overline{ PQ }|=\sqrt{20}\) and the vector \(\overline{ OP }\) is perpendicular to \(\overline{ OQ }\). If \(\overline{ OR }=3 \hat{ i }+ z \hat{ j }-7 \hat{ k }, z \in R ,\) is coplanar with \(\overline{ OP }\) and \(\overline{ OQ },\) then the value of \(x ^{2}+ y ^{2}+ z ^{2}\) is equal to ...... .