In a radioactive material, the activity at time t₁ is R₁ and at a later time t₂, it is R₂. If the decay constant of the material is λ, then:

1. $R_1=R_2{e}^{\lambda (t_1\mathit{+}{t}_2)}$

2. $R_1=R_2{e}^{-\lambda (t_1-t_2)}$

3. $R_1=R_2(t_1-t_2)$

4. $R_1=R_2$

The ionization potential of the hydrogen atom is 13.6 V. Hydrogen atoms in the ground state are excited by monochromatic radiation of photon energy 12.1 eV. According to Bohr’s theory, the spectral lines emitted by hydrogen will be:

1. two

2. three

3. four

4. one

The potential energy of a long spring when stretched by \(2\) cm is \(U\). If the spring is stretched by \(8\) cm, the potential energy stored in it is:
1. \(4U\)
2. \(8U\)
3. \(16U\)
4. \(U/4\)

For a projectile projected at angles \((45^{\circ}-\theta)\) and \((45^{\circ}+\theta)\), the horizontal ranges described by the projectile are in the ratio of:
1. \(1:1\)
2. \(2:3\)
3. \(1:2\)
4. \(2:1\)

A body of mass 3 kg is under a constant force which causes a displacement s in metres in it, given by the relation s = $\frac{1}{3}$ t², where t is in sec. Work done by the force in 2 sec is:

1. $\frac{5}{19}J$

2. $\frac{3}{8}J$

3. $\frac{8}{3}J$

4. $\frac{19}{5}J$

The velocity \(v\) of a particle at time \(t\) is given by \(v=at+\dfrac{b}{t+c}\), where \(a,\) \(b\) and \(c\) are constants. The dimensions of \(a,\) \(b\) and \(c\) are respectively:
1. \(\left[{LT}^{-2}\right],[{L}] \text { and }[{T}]\)
2. \( {\left[{L}^2\right],[{T}] \text { and }\left[{LT}^2\right]}  \)
3. \( {\left[{LT}^2\right],[{LT}] \text { and }[{L}]}  \)
4. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)

A microscope is focused on a mark on a piece of paper and then a slab of glass of thickness 3 cm and a refractive index 1.5 is placed over the mark. How should the microscope be moved to get the mark in focus again?

1. 1 cm upward

2. 4.5 cm downward

3. 1 cm downward

4. 2 cm upward

\(300 ~\text{J}\) of work is done in sliding a \(2~\text{kg}\) block up an inclined plane of height \(10~\text{m}\). Taking \(g=\) \(10\) m/s², work done against friction is:
1. \(200 ~\text{J}\)
2. \(100 ~\text{J}\)
3. \(\text{zero}\)
4. \(1000 ~\text{J}\)

A transistor is operated in a common emitter configuration at constant collector voltage V_c = 1.5 V such that a change in the base current from 100 μA to 150 μA produces a change in the collector current from 5 mA to 10 mA. The current gain (β) is:

1. 67

2. 75

3. 100

4. 50