MATHEMATICS - OBJECTIVE -3

11. If three distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all three of them are divisible by both 2 and 3 

(a) 4/55                 (b) 4/35

(c) 4/33                 (d) 4/1155

 Answer (d)

Solution method: A distinct number that is randomly chosen from a population of big size with a small fixed sample lot is called as Bayer's approximation of probability ratios. This is represented by

          nCr=  n! / r! ( n - r)!

There are 16 digits divisible by 6 ( common factor of 2 and 3 = 2x3 =6) So r=16, N=100

Number of ways in which 3 numbers are selected from these 16 numbers is  16C3

Number of ways in which 3 numbers are selected randomly from 100 numbers is 100C3 

 nCr = ( 16 x 15 x 14 / 3 x 2 x 1 )

         _______________________   = 4 / 1155

         ( 100 x 99 x 98 / 3 x 2 x 1 )

Explanation: This type of analysis is done on a huge set of numbers ,within which ,a smaller subset is selected for probability analysis. This can be considered as the Bayer theorem corollary,since probability theory can be used to negotiate plans for immediate future .This is used in market speculation in current world. As shown above, a definite answer to probability of numbers indicates futuristic ways on market trading in stocks and business prediction in today's competitive world. So here ,it shows that probability is indeed a science of fairly good amount of accuracy .

Chevalier De Mere's problem ( a classic case)shows that Bayer's theory is not fully accurate. This needs approximations as per the above case. 

CHEMISTRY - OBJECTIVE -4

 10. The methods chiefly used for the extraction of lead and tin from their ores, respectively are 

(a) self reduction and carbon reduction

(b) self reduction and electrolytic reduction

(c) carbon reduction and self reduction 

(d) cyanide process and carbon reduction

answer (a)

Answer explanation : metals are extracted from their ores using different chemical processes. Among them the most important is carbon reduction. Coal has different types of reduction methods like carbon reduction, metal oxide reduction, hydrogen reduction and so forth. This has to be analysed as per the valence layer of electrons in the atom of the metal which is oxide coated in nature. Hence the removal of oxide using self reduction has also been involved .In this method, Lead and tin are immersed in water for more than 30 hours ; then it will automatically get removed of the oxide which bubbles out through water. And this type of oxide removal is used in steel manufacturing since it is cheapest and best. 

CHEMISTRY - OBJECTIVE - 3

9. Benzamide on treatment with POCL3 gives 

(a) aniline                     (b) benzonitrile

(c) chlorobenzene          (d) benzyl amine

answer (b)

Answer explanation :Benzamide (C6H5CONH2) on treatment with POCL3 (POCL3 is Phosphorous oxy chloride ) gives C6H5-C≡N (triple bond).

This is called Benzonitrile. nitrile has triple bond for amino group

Nitrogen has electronic configuration 2,2,3 (atomic number of nitrogen is 7). outer 'p' orbital has 3 electrons (1s2,2s2,2p3)

Nitrogen has multiple energy level electrons since it is an inert gas .Inert gas means slow moving electrons inside atoms.So, 'p' orbital has three electrons, since nucleus attracts towards inside more, the inner layer electrons (in s layer) in the valence covalent bond in the case of Nitrogen and Hydrogen .

Hydrogen has 1 atomic number and 1 electron mates with oxygen to form water H2O. Oxygen has atomic number 8,it has 8 electrons .1s,2s,2p levels has 2,2,4 electrons respectively. So, 4 electrons shall part from oxygen atom. Thus, four valence electrons are required for oxide formation for any inert gas .

Carbon has atomic number 6(1S2,2S2,2P2). CN has one single carbon mating with three Nitrogen atoms    ( triple bond). 

 Nitrogen has 3 electrons in the outer layer, out of which 1 mates with carbon .Hence C≡N molecule has Nitrogen atom in which 1 electron is shared with carbon, and 2 free electrons remain free for each Nitrogen atom.So CH3-CH2-C≡N has 6 electrons free for three Nitrogen atoms.

PHYSICS - PROBLEM SOLVING -4

 8. The piston cylinder arrangement shown contains a diatomic gas at temperature 300K.The cross-sectional area of the cylinder is 1m². Initially the height of the piston above the base of the cylinder is 1 m. The

temperature is now raised to 400K at constant pressure. Find the new height of the piston above the base of the cylinder.

 If the piston is now brought back to its original height without any heat loss, find the new eqilibrium temperature of the gas. You can leave the answer in fraction.

Solution :  

At equilibrium,  constant pressure, 

PV =nRT  

P1V1 = P2V2

 But here Pressure is constant ,P

So, Px A xh1 =nRxT1

and Px Ax h2 =nRxT2 ,that is h1T2=h2T1

So, 1 x 400 = h2 x 300

h2 = 1 x 400/300 = 4/3 m

Now let us solve the second part of the above question.

 For adiabatic process.

$T_f{V}_f^{\gamma -1}=T_i{V}_i^{\gamma -1}$
$T_f=T_i(\frac{V_i}{V_f}{)}^{\gamma -1}=(400)(\frac{h_i}{h_f}{)}^{1.4-1}$
$=400(\frac{4}{3}{)}^{0.4}$=448.8 K

(temperature as final and initial has been given)

EXPLANATION:

The Automobile piston has been shown in the figure.This has all the characteristics of the above problem.Hence inorder to achieve perfect combustion ,one needs to achieve gamma ( γ )value of 1.4 ( the value for diatomic air )

Inorder to achieve this value ,a piston ring is attached to the grooves of the cylinder making it air-tight design ,so that atmospheric pressure is maintained inside cylinder at the point of exhaust valve opening and subsequent closure .This design has been used by engineers all around the world in automotive engines with minimum requirement of servicing of piston. 

Aircraft engines uses a special kind of rubber sealant to cover the piston grooves inside cylinder so that when plane takes a right turn, the exhaust valve in the right hand portion of cylinder exhausts dry air which exerts pressure towards right side .Thus the cylinder of aircraft engine has around twelve valve which has inlet and exhaust pairs of valves. The degree turn around is pressed by the co-pilot and corresponding outlet valve opens for pushing down exhaust gas jet which makes it surge ahead with immense power from downward push from the exhaust gas push-out in the direction concerned as per valve emission in the exhaust port as inclined,top,down,forward,rearward,inclined right,inclined left and so forth.                                                                                                 

   
        

PHYSICS - OBJECTIVE -3

 7. A ray of light is incident on an equilateral glass prism placed on a horizontal table. For minimum deviation which of the following is true ?

(a)  PQ is horizontal

(b)  QR is horizontal  

(c)  RS is horizontal

(d)  Either PQ or RS is                    horizontal

answer (b)

Answer explanation : θ1 ,θ2 are incident Angles. This along with the refractive index,n, decides the equation of refraction and total internal reflection

                n1Sinθ1  = n2Sinθ2

Q is measured with respect to horizontal.

If QR is green light ,then RS is yellow,orange,red  and PQ is violet,indigo,blue  among white light.

MATHEMATICS - OBJECTIVE -2

6. Area of triangle formed by the lines x + y = 3 and angle bisectors of the pair of straight lines  x² - y²+2y=1 is

(a) 2 sq units           (b) 4 sq units

(c) 6 sq units            (d) 8 sq units

answer (a)

Answer explanation: x+y =3 so for plotting x-y graph,at x=0,y=3  &  at y=0,x=3 (using this co-ordinates graph has been plotted)

Now to plot line for next equation ,x² - y²+2y=1

at x=0,y=1(first solution) and x=3,y=4 (second solution). let us plot graph using first solution,x=0,y=1 Only this solution bisects the first equation line x+y=3 ;so this is only the valid solution  as per the question above.

area of triangle formed by bisection as shown in figure =1/2 x base x height 

=1/2 x 2 x (3-1)

= 1/2 x 2 x2

 =2 sq units

This type of graph is used in probability analysis ,where one solution gives the right idea about the situation .Hence the graph shown above often indicates the algebrical solution rather than the angular aspect .Probability analysis is maximum at the above plotted solution ,while the second solution indicates lack of confidence as per probability distribution of the selected sample lot .Hence this graph is a probability analysis study of the above said algebric equation along with the given parameters.

MATHEMATICS - OBJECTIVE -1

5. The sides of a triangle are in the ratio 1 :√3: 2,then the angles of the triangle are in the ratio 

(a) 1:3:5                               (b) 2:3:4 

(c) 3:2:1                               (d) 1:2:3

answer (d)

Answer:explanation

hypotenuse = √(a² + b²) 

                  = √(1² + √3²) = 2 (since 2 is the biggest,among 3)

Sin θ = base/hypotenuse = 1/2 =0.5

So, Angle θ = 30°

We know that second angle is 90° ( right angled traingle)

Hence, the third angle is 180 - (30+90) =60°

So answer is 30:60:90 = 1:2:3

 This indicates that a shadow and its source can be compared using ratios and the length of the structure can be estimated from the shadow of its full structure.

Ratio analysis is often used in trigonometry and algebric equations as Simultaneous equations are solved in a digital Scientific calculator using Ratio analysis method .For this ,if it is a third degree equation, solution can be obtained by three equations containing three variables. This method has been mentioned in ' Abacus ',solution techniques ,where easy solutions are found using counting beads in an abacus board .

CHEMISTRY - OBJECTIVE -2

  4. 

C2 is rotated anti - clockwise 120 degrees about C2 - C3 bond. The resulting conformer is 

(a) partially eclipsed  (b) eclipsed          (c) gauche                (d) staggered

answer - (c)

Answer explanation : answer is gauche. gauche means less energy configuration . Other configurations involved are staggered,eclipse and Anti conformers.

This means gauche position of atoms in molecular structure has less stress-strain factor than other types. Anti conformer has least energy among all .

Now, this can be explained in below fashion.The human DNA conform has been called mutation in molecular genetics study.So the twisting moment force inside molecular structure causes ' shape drift ' in the molecular structure and this happens in high energy molecular chemistry which is involved in cryogenics and radioactive experiments in molecular chemistry

CHEMISTRY -OBJECTIVE-1

 3.   2-phenylpropene on acidic hydration gives

(a) 2-phenyl-2-propanol (b) 2-phenyl-1-propanol  

(c) 3-phenyl-1-propanol (d) 1-phenyl-2-propanol

Answer : (a)

Answer explanation: Organic chemistry is based on living compounds, molecules,atoms and ions .Chemicals can be classified as Inorganic or Organic .Inorganic is non-living structures of chemicals while organic are living compounds or singular molecular structures. 

Now Propene is CH3=CH2-CH3 ( Here '=' indicates double bond ; this means propene is jointly connected through methyl group (CH3) and Benzyl group (C6H5-CH2)

Now as shown above, CH3=CH2 can be written as C2H5 which is four carbon short of benzyl group (C6H6 is Benzene).Here we can see that CH3=CH2 is a covalent bond and non-ionic in nature .Hence ionisation can happen only on right,CH3 side of the above compound.

Hence hydration through acidic solution (which is acidic hydration) happens only on right side of the chemical structure ,that is CH3. Hence CH3 becomes CH2-OH ,which is hydroxyl group added to the CH3 replacing one hydrogen ion in the outer 's' orbit  (s,p,d,f are the orbit shapes)

The following chart explains the organic chemistry configuration types in the right perspective 

Thus the above chart shows the difference between different types of ions , atoms,molecules ,compounds and atomic particles of lower order in the Periodic predecession of Chemicals 

Now let us consider a totally different scenario,which has all of the above types of chemical structures .This is known as Markonikov's rule ,which states that bonds,molecules,ions and compounds can be added or substracted from a chemical under reaction with another chemical( under right conditions),which results in complete new arrangement of the atomic structure inside the molecule ,whose valence electrons flyoff from one chemical and adds to another due to electro magnetic attraction between these two. 

 ((  In Nuclear science, this activity is done using an instrument called 'Cyclotron' which when rotated at great speed makes the atom lose one electron and change into another chemical ( which is called radio isotopy) This has been used in Alchemy for metal transformation . Thus deeper study of Organic chemistry leads to Alchemical insights as described above.  ))