Consider the following for real numbers α, β, γ and δ ? 1) sec α = 1/4 2) tan β = 20 3) cosec γ = 1/2 4) cos δ = 2 How many of the above statements are not possible?

Alpha, Beta and Gama went for a picnic at Jurassic Park, when they feel hungry they took out the stuff they brought. Alpha bought 6 sandwiches while Beta bought 4 and Gamma bought none. Both Alpha and Beta shared their sandwiches with Gamma such that each got the same amount. If Gamma paid a total amount of Rs. 10 to Alpha and Beta, how much of Rs. 10 should Alpha get ? (Assume that sandwiches and rupees can be split.)

A

8

B

6

C

3

D

None of these

The Taj Mahal Palace opened in Mumbai, in 1903. Alpha, Beta and Gamma three friends had a dinner at one of its famous restaurant. When the bill was received, Alpha paid 4/3 as much as Beta paid and Beta paid 1/4 as much as Gamma paid. What fraction of the bill did Beta pay?

A

3/11

B

5/11

C

3/19

D

5/19

$$\frac{{{\text{cos }}\alpha }}{{{\text{sin }}\beta }} = n$$ and $$\frac{{{\text{cos }}\alpha }}{{{\text{cos }}\beta }} = m,$$ then the value of $${\text{co}}{{\text{s}}^2}\beta $$ is?

A

$$\frac{{{m^2} - 1}}{{{n^2} - 1}}$$

B

$$\frac{{{m^2} - 3}}{{{n^2} - 4}}$$

C

$$\frac{{{m^2} + 3}}{{{n^2} + 3}}$$

D

$$\frac{{{n^2}}}{{{m^2} + {n^2}}}$$

What is the value of $\frac{{cosec\;\left( {78^\circ - \theta } \right) - \sec \left( {12^\circ \; + \;\theta } \right) - \tan \left( {67^\circ \; + \;\theta } \right)\; + \;\cot \left( {23^\circ - \theta } \right)}}{{\tan 13^\circ \tan 37^\circ \tan 45^\circ \tan 53^\circ \tan 77^\circ }}?$

A

-1

B

2

C

1

D

0

Consider the following remarks pertaining to the irrotational flow: 1. The Laplace equation of stream function $\frac{{{\delta ^2}{\rm{\Psi }}}}{{\delta {x^2}}} + \frac{{{\delta ^2}{\rm{\Psi }}}}{{\delta {y^2}}} = 0$ must be satisfied for the flow to be potential. 2. The Laplace equation for the velocity potential $\frac{{{\delta ^2}{\rm{\varphi }}}}{{\delta {x^2}}} + \frac{{{\delta ^2}{\rm{\varphi }}}}{{\delta {y^2}}} = 0$ must be satisfied to fulfil the orate . of mass conservation i.e continuity equation.. Which of the above statements is/are correct?

A

1 only

B

Both 1 and 2

C

2 only

D

Neither 1 or 2

In a two-electron atomic system having orbital and spin angular momenta $${l_1}{l_2}$$ and $${s_1}{s_2}$$ respectively, the coupling strengths are defined as $${\Gamma _{{l_1}{l_2}}},\,{\Gamma _{{s_1}{s_2}}},\,{\Gamma _{{l_1}{s_1}}},\,{\Gamma _{{l_2}{s_2}}},\,{\Gamma _{{l_1}{l_2}}}$$ and $${\Gamma _{{l_2}{s_1}}}.$$ For the jj coupling. scheme to be applicable, the coupling strengths must satisfy the condition

A

$${\Gamma _{{l_1}{l_2}}},\,{\Gamma _{{s_1}{s_2}}} > {\Gamma _{{l_1}{s_1}}},\,{\Gamma _{{l_2}{s_2}}}$$

B

$${\Gamma _{{l_1}{s_1}}},\,{\Gamma _{{l_2}{s_2}}} > {\Gamma _{{l_1}{l_2}}},\,{\Gamma _{{s_1}{s_2}}}$$

C

$${\Gamma _{{l_1}{s_2}}},\,{\Gamma _{{l_2}{s_1}}} > {\Gamma _{{l_1}{l_2}}},\,{\Gamma _{{s_1}{s_2}}}$$

D

$${\Gamma _{{l_1}{s_2}}},\,{\Gamma _{{l_2}{s_1}}} > {\Gamma _{{l_1}{s_1}}},\,{\Gamma _{{l_2}{s_2}}}$$

Consider the following for real numbers α, β, γ and δ: 1. sec α = 1/5 2. tan β = 15 3. cosec γ = 1/3 4. cos δ = 4 How many of the above statements are not possible?

A

One

B

Two