### Mathematics
1. **Differentiation and Integration:**
– Explain the process of finding the derivative of a composite function using the chain rule.
– Solve: (int (2x^3 – 3x^2 + 4x – 5) dx).
2. **Vectors and Three-Dimensional Geometry:**
– Find the equation of the plane passing through the points ((1, 2, 3)) and perpendicular to the line joining the points ((1, 0, 0)) and ((0, 1, 0)).
### Physics
3. **Electrostatics:**
– Derive the expression for the electric field due to a dipole on its axial line.
– Explain Gauss’s law and use it to derive the electric field intensity due to a uniformly charged spherical shell.
4. **Optics:**
– Explain the principle of interference and derive the condition for constructive and destructive interference in the case of Young’s double-slit experiment.
### Chemistry
5. **Chemical Kinetics:**
– Derive the integrated rate equation for a first-order reaction. What is the significance of the half-life of a reaction?
6. **Coordination Compounds:**
– Discuss the crystal field theory of transition metal complexes. How does it explain the color of complex compounds?
### Biology
7. **Genetics and Evolution:**
– Explain the Mendelian principles of inheritance. How do dihybrid crosses help in understanding the law of independent assortment?
8. **Biotechnology:**
– Describe the process of recombinant DNA technology. What are its applications in medicine and agriculture?
### Economics
9. **Macroeconomics:**
– Explain the concept of the multiplier effect in the context of income and output. How does it impact the economy during a recession?
10. **Indian Economic Development:**
– Discuss the major reforms introduced in India after 1991. How have these reforms impacted the Indian economy?
These questions cover a range of fundamental topics across core subjects. Answering them will help students gain a deeper understanding of the subjects and prepare them well for their exams.
Solution
### Mathematics
1. **Differentiation and Integration:**
– **Chain Rule for Differentiation:**
The chain rule is used to differentiate composite functions. If you have \( y = f(g(x)) \), then the derivative of ( y ) with respect to ( x ) is given by:
[
frac{dy}{dx} = f'(g(x)) cdot g'(x)
]
For example, if ( y = sin(3x^2) ), let ( u = 3x^2 ). Then ( y = sin(u) ). The derivative ( dy/dx ) is:
[
frac{dy}{dx} = cos(u) cdot frac{du}{dx} = cos(3x^2) cdot 6x = 6x cos(3x^2)
]
– **Integration:**
Solve (int (2x^3 – 3x^2 + 4x – 5) dx):
[
int (2x^3 – 3x^2 + 4x – 5) dx = \frac{2x^4}{4} – \frac{3x^3}{3} + \frac{4x^2}{2} – 5x + C
]
Simplifying, we get:
[
frac{x^4}{2} – x^3 + 2x^2 – 5x + C
]
2. **Vectors and Three-Dimensional Geometry:**
– **Equation of a Plane:**
The direction ratios of the line joining points ((1, 0, 0)) and ((0, 1, 0)) are ( (1-0, 0-1, 0-0) = (1, -1, 0) ).
The normal to the plane will have direction ratios proportional to ( (1, -1, 0) ).
The equation of the plane passing through point ((1, 2, 3)) and having normal direction ratios ( (1, -1, 0) ) is:
[
1(x – 1) – 1(y – 2) + 0(z – 3) = 0
]
Simplifying, we get:
[
x – y + 1 = 0 text{ or } x – y = -1
]
### Physics
3. **Electrostatics:**
– **Electric Field of a Dipole:**
The electric field ( E ) at a point on the axial line of an electric dipole is given by:
[
E = frac{1}{4 pi epsilon_0} cdot frac{2p}{r^3}
]
where ( p ) is the dipole moment and ( r ) is the distance from the center of the dipole to the point.
– **Gauss’s Law:**
Gauss’s law states that the total electric flux through a closed surface is equal to (frac{q_{text{enc}}}{epsilon_0}), where (q_{text{enc}}) is the charge enclosed.
For a uniformly charged spherical shell:
[
E = frac{1}{4 pi epsilon_0} cdot frac{Q}{r^2} text{ for } r > R
]
[
E = 0 text{ for } r < R
]
4. **Optics:**
– **Young’s Double-Slit Experiment:**
The condition for constructive interference is:
[
Delta x = n \lambda \quad (n = 0, 1, 2, \ldots)
]
The condition for destructive interference is:
[
Delta x = (n + \frac{1}{2}) \lambda \quad (n = 0, 1, 2, \ldots)
]
where ( Delta x ) is the path difference and ( lambda ) is the wavelength of light.
### Chemistry
5. **Chemical Kinetics:**
– **First-Order Reaction:**
For a first-order reaction ( A to B ), the rate law is:
[
frac{d[A]}{dt} = -k[A]
]
Integrating, we get:
[
[A] = [A]_0 e^{-kt}
]
The half-life ( t_{1/2} \) is:
[
t_{1/2} = frac{ln 2}{k}
]
6. **Coordination Compounds:**
– **Crystal Field Theory:**
Crystal Field Theory explains the splitting of d-orbitals in transition metal complexes. In an octahedral field, the d-orbitals split into ( e_g ) and ( t_{2g} ) sets. The difference in energy between these sets is called crystal field splitting energy ( Delta_0 ).
The color of the complexes arises due to d-d transitions, where an electron jumps from a lower-energy ( t_{2g} ) orbital to a higher-energy ( e_g ) orbital, absorbing a photon of visible light.
### Biology
7. **Genetics and Evolution:**
– **Mendelian Principles:**
Mendel’s principles of inheritance include the Law of Segregation and the Law of Independent Assortment. The dihybrid cross, involving two different traits, shows that alleles of different genes assort independently.
For example, crossing ( RrYy times RrYy ) (round and yellow seeds) results in a phenotypic ratio of 9:3:3:1.
8. **Biotechnology:**
– **Recombinant DNA Technology:**
This process involves:
1. Isolation of DNA
2. Cutting of DNA at specific sites using restriction enzymes
3. Insertion of the DNA fragment into a vector
4. Introduction of the vector into a host cell
5. Selection and screening of recombinant cells
Applications include gene therapy, production of insulin, and genetically modified crops.
### Economics
9. **Macroeconomics:**
– **Multiplier Effect:**
The multiplier effect refers to the phenomenon where an initial change in spending leads to a larger change in aggregate income. The size of the multiplier is given by:
[
text{Multiplier} = \frac{1}{1 – MPC}
]
where ( MPC ) is the marginal propensity to consume. During a recession, government spending can have a multiplied impact on the economy, boosting aggregate demand and output.
10. **Indian Economic Development:**
– **Economic Reforms of 1991:**
Major reforms included liberalization, privatization, and globalization (LPG). These reforms opened up the Indian economy to foreign investment, reduced tariffs, and deregulated industries.
Impact:
– Increased GDP growth rates
– Expansion of the services sector
– Rise in foreign direct investment (FDI)
– Enhanced global integration of the Indian economy