Class 12 Physics Important Questions with Solutions Maharashtra Board 2025-26

XII HSC Physics Important Question Bank (2025-26)

ROTATIONAL DYNAMICS

  • 1) Distinguish between centripetal and centrifugal force. [2M]
  • 2) What is banking of road, obtain an expression for max and min safety speed of vehicles along curve horizontal road. [4M]
  • 3) Draw neat labelled diagram and derive Expression for conical pendulum. [3M]
  • 4) Derive expression for vertical circular motion. [3M]
  • 5) State and prove perpendicular axis theorem. [3M]
  • 6) State and prove parallel axis theorem. [4M]
  • 7) State and Prove law of conservation of angular momentum. [3M]
  • 8) Define Radius of Gyration and write its significance. [2M]
  • 9) Derive expression for kinetic energy of a Rolling body. [3M]

MECHANICAL PROPERTIES OF FLUIDS

  • 1) Define Intermolecular force, Adhesive and Cohesive force, range of molecules. [1M each]
  • 2) What is surface energy? Obtain relation between surface tension and surface energy. [3M]
  • 3) Define Surface tension, state its S.I. unit and dimension. [3M]
  • 4) Define angle of contact? State its four characteristics. [3M]
  • 5) Derive Laplace's law (Excess Pressure). [4M]
  • 6) Define Capillary action and derive expression for rise and fall of liquid in the capillary tube. [3M]
  • 7) Define critical velocity, Reynolds number, coefficient of velocity. [1M each]
  • 8) Stoke's law, terminal velocity. [1M each]

KINETIC THEORY OF GASES

  • 1) Derive the expression for pressure exerted by the gas. [4M]
  • 2) Define RMS velocity. [1M]
  • 3) Write short note on: Ferry's black body draw a neat labelled diagram. [3M]
  • 4) State and explain wien's displacement law? [3M]
  • 5) State Stefan's law. [1M]
  • 6) Define Emissive power and coefficient of Emission of body. [1M each]
  • 7) State and prove Kirchhoff's law of heat radiation. [3M]
  • 8) Derive Mayer's Relation. [3M]

THERMODYNAMICS

  • 1) State first law of thermodynamic. [1M]
  • 2) Thermodynamics Equilibrium. [2M]
  • 3) Heat Engine. [4M]
  • 4) Carnot Cycle. [4M]
  • 5) Distinguish between thermal processes. [2M]
  • 6) Derive expression for work done of Isothermal and adiabatic process. [3M]

OSCILLATIONS

  • 1) Define SHM? State its differential Equation? [2M]
  • 2) Obtain expression for acceleration, Velocity and displacement. [4M]
  • 3) Composition of two SHM's. [4M]
  • 4) State and derive expression for kinetic energy and potential energy. [3M]
  • 5) Define simple pendulum, derive expression for the period of motion of simple pendulum on which factor it depends upon? [3M]
  • 6) Distinguish free and forced vibration. [2M]
  • 7) Damp Oscillation. [2M]
  • 8) Define second's pendulum? [2M]

SUPERPOSITION OF WAVES

  • 1) Derive equation for stationary wave. (3M)
  • 2) Conditions for Nodes and Antinodes. (2M)
  • 3) Derive the Expression for beats. (3M)
  • 4) Laws of vibrating string. (3M)
  • 5) Explain phenomenon for production of beats. (2M)
  • 6) Show that only odd harmonics are present in pipe closed at one end. (3M)
  • 7) Show that odd and even harmonics are present for pipe open at both the ends. (3M)

WAVE OPTICS

  • 1) Postulates of Huygen's wave theory of light. (2M)
  • 2) Derive the laws of refraction of light using Huygen's principle. (3M)
  • 3) Explain what is meant by polarization. (2M)
  • 4) Derive Malus laws. (3M)
  • 5) What is Brewster's law? Derive the formula for Brewster angle. (3M)
  • 6) Describe YDSE experiment. (4M)
  • 7) Condition for constructive and destructive interference. (2M)
  • 8) Condition for obtaining good interference pattern. (2M)
  • 9) What are Fraunhofer and Fresnel diffractions. (2M)
  • 10) Resolving power. (3M)
  • 11) Explain Rayleigh's criterion. (2M)

ELECTROSTATICS

  • 1) Obtain expression for electric field intensity due to uniformly charged spherical shell or hollow sphere. (3M)
  • 2) Obtain an expression for electric field intensity due to an infinitely long straight charged wire or charged conducting cylinder. (3M)
  • 3) State Gauss law. (1M)
  • 4) Obtain an expression for electric field due to an infinite charged plane sheet. (3M)
  • 5) Derive an expression for electric potential due to an electric dipole. (3M)
  • 6) Define equipotential surface. State and explain its properties. (2M)
  • 7) Define capacity of the capacitor. (2M)
  • 8) Energy stored in a capacitor. (2/3M)
  • 9) With the help of neat diagram, explain how non-polar dielectric material is polarised in external electric field? [3M]

CURRENT ELECTRICITY

  • 1) State and Explain Kirchoff's law. (2M)
  • 2) Obtain the balancing condition in case of Wheatstone bridge. (3M)
  • 3) State and explain the concept of potentiometer. (3M)
  • 4) Define Potential Gradient. (1M)
  • 5) Write a note on galvanometer. (2M)
  • 6) Describe kelvin's method to determine the resistance of a galvanometer by using a meter bridge. (3M)
  • 7) Explain how MCG is converted into an ammeter. [3M]

MAGNETIC FIELDS DUE TO ELECTRIC CURRENT

  • 1) Describe the magnetic field near a current in a long, straight wire. State the expression for the magnetic induction near a straight infinitely long current-carrying wire. [3M]
  • 2) State the factors which the magnetic force on a charge depends upon. Hence state the expression for the Lorentz force on a charge due to an electric field as well as a magnetic field. [3M]
  • 3) Define the SI unit of magnetic induction from Lorentz force. [1M]
  • 4) Explain the condition under which a charged particle will travel through a uniform magnetic field in a helical path. [3M]
  • 5) State under what conditions will a charged particle moving through a uniform magnetic field travel in (i) a straight line (ii) a circular path (iii) a helical path. [3M]
  • 6) What is a cyclotron? State its principle of working. [4M]
  • 7) Biot-savarts law. [2M]
  • 8) Current Carrying in parallel wires. [3M]

MAGNETIC MATERIALS

  • 1) Explain the directional characteristic of a bar magnet. [2M]
  • 2) State the expression for the torque acting on a magnetic dipole in a uniform magnetic field. [3M]
  • 3) Explain what is meant by magnetic potential energy of a bar magnet kept in a uniform magnetic field. Discuss the cases when theta = 0, 180, and 90 degrees. [3M]
  • 4) Derive the expression for the time period of angular oscillations of a bar magnet kept in a uniform magnetic field. [3M]
  • 5) What is the gyromagnetic ratio of an orbital electron? State its dimensions and the SI unit. [2M]

ELECTROMAGNETIC INDUCTION

  • 1) Describe Faraday's magnet and coil experiment. What conclusion can be drawn from the experiment? [3M]
  • 2) State the causes of induced current and explain them on the basis of Lenz's law. [2M]
  • 3) State an expression for the magnetic flux through a loop of finite area A inside a uniform magnetic field. Hence discuss Faraday's second law. [3M]
  • 4) State the SI units and dimensions of (i) magnetic induction (ii) magnetic flux. [2M]
  • 5) Determine the motional emf induced in a straight conductor moving in a uniform magnetic field with constant velocity. [3M]
  • 6) What is an ac generator? State the principle of an ac generator. [3M]
  • 7) Explain back emf in a motor. [3M]
  • 8) Explain the concept of self-induction. [3M]
  • 9) Derive an expression for the energy stored in the magnetic field of an inductor. [3M]
  • 10) Obtain an expression for the self-inductance of a solenoid. [3M]
  • 11) Obtain an expression for the energy density of a magnetic field. [3M]
  • 12) Explain the concept/phenomenon of mutual induction. [2M]
  • 13) What is a transformer? State the principle of working of a transformer. [4M]
  • 14) Derive expressions for a transformer for the emf and current in terms of the turn's ratio. [3M]

AC CIRCUITS

  • 1) Write an expression for an alternating emf that varies sinusoidally with time. [4M]
  • 2) Draw a Phasor diagram showing e and i in the case of a purely inductive circuit. [3M]
  • 3) An alternating emf is applied to an LR circuit. Obtain the expressions for the applied emf and the effective resistance. Draw the phasor diagram. [3M]
  • 4) An alternating emf is applied to a CR circuit. Obtain an expression for the phase difference and effective resistance. Draw the phasor diagram. [4M]
  • 5) What is meant by the term impedance? State the formula for it in the case of an LCR series circuit. [3M]
  • 6) State the expression for the average power consumed over one cycle in the case of a series LCR AC circuit. [3M]
  • 7) How are oscillations produced using an inductor and a capacitor. [3M]
  • 8) Explain electrical resonance in an LCR series circuit. Deduce the expression for the resonant frequency of the circuit. [3M]
  • 9) Explain the term sharpness of resonance and Q factor (quality factor). [2M]

DUAL NATURE OF RADIATION AND MATTER

  • 1) What was Hertz's observation regarding emission of electrons from a metal surface? [3M]
  • 2) With a neat diagram, describe the apparatus to study the characteristics of photoelectric effect. [3M]
  • 3) Define (1) threshold frequency (2) threshold wavelength (3) stopping potential. [3M]
  • 4) State the characteristics of photoelectric effect. [2M]
  • 5) Explain how wave theory of light fails to explain the characteristics of photoelectric effect. [3M]
  • 6) Give Einstein's explanation of the photoelectric effect. [4M]
  • 7) Write Einstein's photoelectric equation and explain its various terms. How does the equation explain various features? [4M]
  • 8) What is a photocell? Describe its construction and working with a neat labelled diagram. [3M]
  • 9) Derive an expression for the de Broglie wavelength associated with an electron accelerated from rest through a potential difference V. [3M]

STRUCTURE OF ATOMS AND NUCLEI

  • 1) With the help of a neat labelled diagram, describe the Geiger-Marsden experiment. [3M]
  • 2) Explain Rutherford's model of the atom. [2M]
  • 3) State and explain the formula that gives wavelengths of lines in the hydrogen spectrum. [3M]
  • 4) Derive an expression for the linear speed of an electron in a Bohr orbit. Show it is inversely proportional to principal quantum number. [3M]
  • 5) How is the nuclear size determined? State the relation between nuclear size and mass number. [3M]
  • 6) Define mass defect and state an expression for it. [3M]
  • 7) Explain the term nuclear binding energy and binding energy per nucleon. [3M]
  • 8) State the law of radioactive decay and express it in the exponential form. [3M]
  • 9) Define half-life of a radioactive element and obtain the relation between half-life and decay constant. [3M]
  • 10) Postulates of Bohr atomic model. [2M]

SEMICONDUCTOR DEVICES

  • 1) What is a PN-junction diode? What is a depletion region? What is barrier potential? [3M]
  • 2) Explain the forward bias and reverse bias conditions of a diode. [3M]
  • 3) What is rectification? How does a pn-junction diode act as a rectifier? [3M]
  • 4) Distinguish between a half-wave rectifier and full-wave rectifier. [2M]
  • 5) Explain ripple in the output of a rectifier. What is ripple factor? [2M]
  • 6) Explain Zener breakdown. [2M]
  • 7) Explain the I-V characteristics of a photodiode. [2M]
  • 8) What is a light-emitting diode (LED)? [3M]
  • 9) Describe with a neat diagram the construction of an LED. [4M]
  • 10) What are the different transistor configurations in a circuit? Show them schematically. [3M]
  • 11) Define AND, OR, and NOT logic gates. Give logic symbol, Boolean expression and truth table of each. [3M]
  • 12) Obtain the relation between alpha_DC and beta_DC. [2/3M]
Note: All questions listed above are important for the 2025-2026 HSC examinations. Ensure you focus particularly on the questions with higher mark allocations.

HSC Physics Board Papers with Solution

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Maharashtra HSC Class 12 Chemistry Chapter-wise Blueprint & Exam Pattern 2025-26

Maharashtra HSC Class 12 Chemistry Blueprint (2025-26)

Below is the detailed exam pattern and chapter-wise analysis for the Maharashtra HSC Class 12 Chemistry Board Exam. The theory paper consists of 70 Marks, and the Practical exam holds 30 Marks.

1. Theory Exam Pattern (70 Marks)

Marks Type Question Type No. of Questions Total Marks
1 mark MCQ (10) + VSA (8) 18 18
2 marks Short Answer I (Attempt 8 of 12) 8 16
3 marks Short Answer II (Attempt 8 of 12) 8 24
4 marks Long Answer (Attempt 3 of 5) 3 12
Total 70

2. Chapter-wise List & Question Distribution

The following list details the chapters included in the syllabus. Based on the 2025-26 blueprint, specific chapters are targeted for Long Answer (4 Marks) questions.

Chapter No. Chapter Name Long Answer (4 Marks) Focus
1Solid State-
2Solutions-
3Ionic Equilibrium-
4Chemical Thermodynamics-
5Electrochemistry-
6Chemical Kinetics-
7Group 16, 17 & 18 Elements-
8Transition & Inner Transition Elements☑ Expected
9Coordination Compounds☑ Expected
10Halogen Derivatives-
11Alcohols, Phenols & Ethers-
12Aldehydes, Ketones & Carboxylic Acids-
13Amines-
14Biomolecules-
15Polymer Chemistry-
16Green & Nano Chemistry-

3. Practical Exam

Total Marks: 30 Marks

Students must maintain their journals and prepare for viva voce and experiments as per college guidelines.

4. Preparation Strategy for 2025-26

  • High Weightage Focus: Focus on chapters that cover all mark types, specifically Chemical Thermodynamics, Coordination Compounds, and Group 16-18 Elements.
  • Daily Revision: Start your daily study routine by revising 1 & 2 mark MCQs/VSA questions from every chapter to secure the base 18 marks.
  • Long Answers: specifically practice long answer questions from chapters marked within the 4 Marks category (Transition Elements & Coordination Compounds).
  • Practice: Reinforce your learning by solving Previous Year Questions (PYQs) and board sample sets.

12th Chemistry with Solution

HSC Chemistry

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Most Important Chapters for HSC 12th Physics: Weightage & Topic List

Chapter-Wise Weightage (Approximate)

The following table outlines the approximate weightage for the most important chapters in the HSC 12th Physics syllabus. Focusing on these can significantly boost your exam score.

Chapter Name Weightage (%)
Magnetic Effects of Current & Magnetism 10%
Electrostatics 8-10%
Electromagnetic Induction & AC 8-10%
Ray Optics & Optical Instruments 8%
Current Electricity 6-8%
Dual Nature of Matter & Radiation 6-8%
Semiconductor Electronics 7-8%
Wave Optics 5-7%
Atoms & Nuclei 6%

Detailed Chapter Analysis & Important Topics

Below is a detailed breakdown of the frequently asked topics and the nature of questions (numerical vs. conceptual) for each high-weightage chapter.

1. Electrostatics High Weightage

Important Topics:

  • Coulomb's Law
  • Electric Field & Potential
  • Gauss's Theorem
  • Capacitors & Dielectrics
  • Energy Stored in Capacitor
Why Important?
  • Conceptual + Numerical Problems are frequently asked.
  • Direct theory-based questions appear in Long Answers.

2. Current Electricity

Important Topics:

  • Ohm's Law & Resistance Combination
  • Kirchhoff's Law & Wheatstone Bridge
  • Meter Bridge & Potentiometer
  • Colour Coding of Resistors
Why Important?
  • Numerical Problems on Ohm's Law, Resistances, and Meter Bridge.
  • Conceptual Questions on Kirchhoff's Law & Potentiometer.

3. Magnetic Effects of Current & Magnetism

Important Topics:

  • Biot-Savart Law
  • Ampere's Circuital Law
  • Moving Coil Galvanometer
  • Torque on a Magnetic Dipole
  • Earth's Magnetism
Why Important?
  • Theory + Derivations are important for long answers.
  • Numericals on Biot-Savart Law & Galvanometer Conversion often appear.

4. Electromagnetic Induction & Alternating Current

Important Topics:

  • Faraday's Laws of Electromagnetic Induction
  • Lenz's Law & Eddy Currents
  • Self & Mutual Inductance
  • AC Circuits (LCR Circuits, Resonance)
  • Transformers & Power in AC Circuits
Why Important?
  • Concept-based numericals on AC Circuits & LCR Circuit.
  • Theory questions on Faraday's Laws & Eddy Currents.

5. Ray Optics & Optical Instruments

Important Topics:

  • Reflection & Refraction Laws
  • Lens Formula & Mirror Formula
  • Total Internal Reflection & Critical Angle
  • Microscope & Telescope Working
Why Important?
  • Diagram-based questions on lenses & optical instruments.
  • Numericals on Lens & Mirror Formula.

6. Wave Optics

Important Topics:

  • Huygens Principle
  • Young's Double-Slit Experiment (YDSE)
  • Diffraction & Polarization
Why Important?
  • Conceptual & Derivation-Based Questions.
  • Numericals on YDSE (Fringe Width Calculation).
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7. Dual Nature of Matter & Radiation

Important Topics:

  • Photoelectric Effect & Einstein's Equation
  • Work Function & Threshold Frequency
  • de Broglie Wavelength
Why Important?
  • Frequently asked numerical problems on Photoelectric Effect & de Broglie Wavelength.
  • Direct Conceptual Questions in Theory.

8. Atoms & Nuclei

Important Topics:

  • Bohr's Model of Hydrogen Atom
  • Energy Levels & Spectral Series
  • Nuclear Fission & Fusion
  • Radioactivity (Half-Life & Decay Constant)
Why Important?
  • Derivations & Numericals on Bohr's Model & Half-Life.
  • Short Answer Questions on Radioactivity.

9. Semiconductor Electronics

Important Topics:

  • PN Junction Diode & Its Characteristics
  • Zener Diode & Voltage Regulation
  • Logic Gates (AND, OR, NOT, NAND, NOR)
Why Important?
  • Circuit Diagram-Based Questions.
  • Boolean Algebra & Logic Gate Problems.

12th Physics Previous Years Papers with Solution

HSC Physics Board Papers with Solution

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HSC Commerce Mathematics & Statistics March 2024 Exam Question Paper with Solutions HIndi Medium

рдЧрдгिрдд рдФрд░ рд╕ांрдЦ्рдпिрдХी (рд╡ाрдгिрдЬ्рдп) - рдоाрд░्рдЪ 2024
рд╕рдо्рдкूрд░्рдг рд╣рд▓ (рд╣िंрджी рдоाрдз्рдпрдо)

рд╕рдордп: 3 рдШंрдЯे | рдЕрдзिрдХрддрдо рдЕंрдХ: 80

рд╡िрднाрдЧ - рез (Section I)

рдк्рд░. рез. (рдЕ) рдиिрдо्рдирд▓िрдЦिрдд рдмрд╣ुрд╡िрдХрд▓्рдкीрдп рдк्рд░рд╢्рдиों рдХे рд╡िрдХрд▓्рдкों рдоें рд╕े рд╕рд╣ी рд╡िрдХрд▓्рдк рдЪुрдирдХрд░ рд▓िрдЦिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

(i) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХौрди рд╕ा рдХрдерди рдирд╣ीं рд╣ै:
  • (рдЕ) рдзूрдо्рд░рдкाрди рд╕्рд╡ाрд╕्рде्рдп рдХे рд▓िрдП рд╣ाрдиिрдХाрд░рдХ рд╣ै।
  • (рдм) \(2+2=4\)
  • (рдХ) 2 рдПрдХрдоाрдд्рд░ рд╕рдо рдЕрднाрдЬ्рдп рд╕ंрдЦ्рдпा рд╣ै।
  • (рдб) рдпрд╣ाँ рдЖрдУ।
рдЙрдд्рддрд░: (рдб) рдпрд╣ाँ рдЖрдУ।
рд╕्рдкрд╖्рдЯीрдХрд░рдг: рдпрд╣ рдПрдХ рдЖрдЬ्рдЮाрд░्рдердХ рд╡ाрдХ्рдп (Imperative sentence) рд╣ै, рдЗрд╕рд▓िрдП рдпрд╣ рддाрд░्рдХिрдХ рдХрдерди рдирд╣ीं рд╣ै।
(ii) рдпрджि \(x+y+z=3\), \(x+2y+3z=4\), \(x+4y+9z=6\) рддрдм \((y, z) = ...\)
  • (рдЕ) (-1, 0)
  • (рдм) (1, 0)
  • (рдХ) (1, -1)
  • (рдб) (-1, 1)
рдЙрдд्рддрд░: (рдм) (1, 0)
рд╣рд▓:
рд╕рдоीрдХрд░рдг (2) - (1): \(y + 2z = 1\)
рд╡िрдХрд▓्рдк (рдм) рдоें \(y=1, z=0\) рд░рдЦрдиे рдкрд░: \(1 + 0 = 1\) (рд╕ंрддुрд╖्рдЯ рдХрд░рддा рд╣ै)।
(iii) рдпрджि \(y = \log(\frac{e^{x}}{x^{2}})\) рддрдм \(\frac{dy}{dx} = ?\)
  • (рдЕ) \(\frac{2-x}{x}\)
  • (рдм) \(\frac{x-2}{x}\)
  • (рдХ) \(\frac{e-x}{ex}\)
  • (рдб) \(\frac{x-e}{ex}\)
рдЙрдд्рддрд░: (рдм) \(\frac{x-2}{x}\)
\(y = \log e^x - \log x^2 = x - 2\log x\)
\(\frac{dy}{dx} = 1 - \frac{2}{x} = \frac{x-2}{x}\)
(iv) \(\int\frac{dx}{\sqrt{1-x}}\) рдХा рдоाрди рд╣ै:
  • (рдЕ) \(2\sqrt{1-x}+c\)
  • (рдм) \(-2\sqrt{1-x}+c\)
  • (рдХ) \(\sqrt{x}+c\)
  • (рдб) \(x+c\)
рдЙрдд्рддрд░: (рдм) \(-2\sqrt{1-x}+c\)
(v) \(\int\frac{dx}{(x-8)(x+7)} = ...\)
  • (рдЕ) \(\frac{1}{15}\log|\frac{x+2}{x+1}|+c\)
  • (рдм) \(\frac{1}{15}\log|\frac{x+8}{x+7}|+c\)
  • (рдХ) \(\frac{1}{15}\log|\frac{x-8}{x+7}|+c\)
  • (рдб) \((x-8)(x+7)+c\)
рдЙрдд्рддрд░: (рдХ) \(\frac{1}{15}\log|\frac{x-8}{x+7}|+c\)
(vi) рд╕рдоीрдХрд░рдг \(y=k_{1}e^{x}+k_{2}e^{-x}\) рдХा рдЕрд╡рдХрд▓ рд╕рдоीрдХрд░рдг рд╣ै:
  • (рдЕ) \(\frac{d^{2}y}{dx^{2}}-y=0\)
  • (рдм) \(\frac{d^{2}y}{dx^{2}}+y\frac{dy}{dx}=0\)
  • (рдХ) \(\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}=0\)
  • (рдб) \(\frac{d^{2}y}{dx^{2}}+y=0\)
рдЙрдд्рддрд░: (рдЕ) \(\frac{d^{2}y}{dx^{2}}-y=0\)

рдк्рд░. рез. (рдм) рдиिрдо्рдирд▓िрдЦिрдд рдХрдерди рд╕рдд्рдп рд╣ैं рдпा рдЕрд╕рдд्рдп, рд▓िрдЦिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

  • (i) \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt\)
    рд╕рдд्рдп (True)
  • (ii) \(\int\frac{x-1}{(x+1)^{3}}e^{x}dx=e^{x}f(x)+c\), рдХे рд▓िрдП \(f(x)=(x+1)^{2}\)
    рдЕрд╕рдд्рдп (False) (рд╕рд╣ी рдЙрдд्рддрд░ \(\frac{1}{(x+1)^2}\) рд╣ोрдиा рдЪाрд╣िрдП)
  • (iii) рдЕрд╡рдХрд▓ рд╕рдоीрдХрд░рдг рдХे рдШाрдд (order) рдФрд░ рдХोрдЯि (degree) рд╕рджैрд╡ рдзрдиाрдд्рдордХ рдкूрд░्рдгांрдХ рд╣ोрддे рд╣ैं।
    рд╕рдд्рдп (True)

рдк्рд░. рез. (рдХ) рдиिрдо्рдирд▓िрдЦिрдд рд░िрдХ्рдд рд╕्рдеाрдиों рдХी рдкूрд░्рддि рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

  • (i) рдХिрд╕ी рдмिंрджु (a, b) рдкрд░ рд╕्рдкрд░्рд╢ рд░ेрдЦा (tangent) рдХी рдк्рд░рд╡рдгрддा (slope) рдк्рд░рд╡рдгрддा / Gradient рдХрд╣рд▓ाрддी рд╣ै।
  • (ii) рдпрджि \(f'(x)=\frac{1}{x}+x\) рддрдеा \(f(1)=\frac{5}{2}\) рддрдм \(f(x)=\log x + \frac{x^2}{2} + \) 2
  • (iii) рдЕрд╡рдХрд▓ рд╕рдоीрдХрд░рдг рдХा рд╣рд▓ рдЬिрд╕े рд╕ाрдоाрди्рдп рд╣рд▓ рдоें рд╕्рд╡ेрдЪ्рдЫ рдЕрдЪрд░ों рдХो рд╡िрд╢िрд╖्рдЯ рдоाрди рджेрдХрд░ рдк्рд░ाрдк्рдд рдХिрдпा рдЬाрддा рд╣ै, рд╡िрд╢िрд╖्рдЯ рд╣рд▓ (Particular Solution) рдХрд╣рд▓ाрддा рд╣ै।

рдк्рд░. реи. (рдЕ) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рей рдЕंрдХ):

(i) рд╕рдд्рдпрддा рд╕ाрд░рдгी рдХा рдЙрдкрдпोрдЧ рдХрд░рдХे рдЬांрдЪिрдП рдХि рдХ्рдпा рдХрдерди рдкुрдирд░ुрдХ्рддि (tautology), рд╡िрд░ोрдзाрднाрд╕ (contradiction) рдпा рдЖрдХрд╕्рдоिрдХрддा (contingency) рд╣ै: \(\sim p\rightarrow(p\rightarrow\sim q)\)
рд╣рд▓:
pq~p~qp -> ~q~p -> (p -> ~q)
TTFFFT
TFFTTT
FTTFTT
FFTTTT
рдЪूँрдХि рдЕंрддिрдо рд╕्рддंрдн рдоें рд╕рднी рдоाрди 'T' рд╣ैं, рдЕрддः рдпрд╣ рдПрдХ рдкुрдирд░ुрдХ्рддि (Tautology) рд╣ै।
(ii) рдпрджि \(x=e^{3t}\), \(y=e^{(4t+5)}\) рддрдм \(\frac{dy}{dx}\) рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
\(\frac{dx}{dt} = 3e^{3t}\) рдФрд░ \(\frac{dy}{dt} = 4e^{(4t+5)}\)
\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{4e^{(4t+5)}}{3e^{3t}}\)
\(= \frac{4}{3} e^{(4t+5-3t)} = \frac{4}{3}e^{t+5}\)
(iii) рдпрджि \(A=[\begin{matrix}7&3&0\\ 0&4&-2\end{matrix}]\) \(B=[\begin{matrix}0&-2&3\\ 2&1&-4\end{matrix}]\) рддो \(A^{T}+4B^{T}\) рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
\(A^T = [\begin{matrix}7&0\\ 3&4\\ 0&-2\end{matrix}]\), \(B^T = [\begin{matrix}0&2\\ -2&1\\ 3&-4\end{matrix}]\)
\(4B^T = [\begin{matrix}0&8\\ -8&4\\ 12&-16\end{matrix}]\)
\(A^T + 4B^T = [\begin{matrix}7&0\\ 3&4\\ 0&-2\end{matrix}] + [\begin{matrix}0&8\\ -8&4\\ 12&-16\end{matrix}] = [\begin{matrix}7&8\\ -5&8\\ 12&-18\end{matrix}]\)

рдк्рд░. реи. (рдм) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) рд╕рдоाрди рдЕрд░्рде рд╡ाрд▓े рдХрдердиों рдХे рдпुрдЧ्рдоों рдХो рдкрд╣рдЪाрдиिрдП। (рдХुрдд्рддा рд╡ाрд▓े рдХрдерди)
рд╣рд▓:
(рдЕ) \(p \to q\) (If D is dog, D is good)
(рдм) \(q \to p\) (If D is good, D is dog) - рд╡िрд▓ोрдо (Converse)
(рдХ) \(\sim q \to \sim p\) (If D not good, D not dog) - рдк्рд░рддिрдзрдиाрдд्рдордХ (Contrapositive)
(рдб) \(\sim p \to \sim q\) (If D not dog, D not good) - рдк्рд░рддिрд▓ोрдо (Inverse)

рддाрд░्рдХिрдХ рд╕рдорддुрд▓्рдпрддा: 1. (рдЕ) рдФрд░ (рдХ) рд╕рдоाрди рд╣ैं (\(p \to q \equiv \sim q \to \sim p\))
2. (рдм) рдФрд░ (рдб) рд╕рдоाрди рд╣ैं (\(q \to p \equiv \sim p \to \sim q\))
(ii) рдлрд▓рди \(f(x)=2x^{3}-21x^{2}+36x-20\) рдХा рдиिрдо्рдирддрдо рдоाрди (minimum value) рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
\(f'(x) = 6x^2 - 42x + 36\).
\(f'(x) = 0 \Rightarrow x^2 - 7x + 6 = 0 \Rightarrow (x-6)(x-1)=0\).
\(x=1, x=6\).
\(f''(x) = 12x - 42\).
\(x=6\) рдкрд░, \(f''(6) = 72 - 42 = 30 > 0\) (рдиिрдо्рдирддрдо)।
рдиिрдо्рдирддрдо рдоाрди: \(f(6) = 2(216) - 21(36) + 36(6) - 20 = -128\).
(iii) рд░ेрдЦा \(y=-2x\), X рдЕрдХ्рд╖ рдПрд╡ं рд░ेрдЦाрдУं \(x=-1\) рдФрд░ \(x=2\) рд╕े рдШिрд░े рдХ्рд╖ेрдд्рд░ рдХा рдХ्рд╖ेрдд्рд░рдлрд▓ рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
рдХ्рд╖ेрдд्рд░рдлрд▓ = \(|\int_{-1}^{0} (-2x)dx| + |\int_{0}^{2} (-2x)dx|\)
\(A_1 = [-x^2]_{-1}^{0} = -(0 - 1) = 1\)
\(A_2 = [-x^2]_{0}^{2} = -(4 - 0) = -4 \Rightarrow |A_2| = 4\)
рдХुрд▓ рдХ्рд╖ेрдд्рд░рдлрд▓ = \(1 + 4 = 5\) рд╡рд░्рдЧ рдЗрдХाрдИ।

рдк्рд░. рей. (рдЕ) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рей рдЕंрдХ):

(i) рдпрджि \(y=x^{e^{x}}\) рддो \(\frac{dy}{dx}\) рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
рджोрдиों рдкрдХ्рд╖ों рдХा log рд▓ेрдиे рдкрд░: \(\log y = e^x \log x\)
рдЕрд╡рдХрд▓рди: \(\frac{1}{y}\frac{dy}{dx} = e^x(\frac{1}{x}) + \log x(e^x)\)
\(\frac{dy}{dx} = y \cdot e^x (\frac{1}{x} + \log x) = x^{e^x} e^x (\frac{1+x\log x}{x})\)
(ii) рдпрджि \(f'(x)=4x^{3}-3x^{2}+2x+k\), \(f(0)=1\) рдФрд░ \(f(1)=4\) рддो \(f(x)\) рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
рд╕рдоाрдХрд▓рди рдХрд░рдиे рдкрд░: \(f(x) = x^4 - x^3 + x^2 + kx + c\)
\(f(0)=1 \Rightarrow c=1\)
\(f(1)=4 \Rightarrow 1 - 1 + 1 + k + 1 = 4 \Rightarrow k=2\)
рдЕрддः \(f(x) = x^4 - x^3 + x^2 + 2x + 1\)
(iii) рдЕрд╡рдХрд▓ рд╕рдоीрдХрд░рдг рдк्рд░ाрдк्рдд рдХीрдЬिрдП рдЬिрд╕рдХा рд╡्рдпाрдкрдХ рд╣рд▓ \(x^{3}+y^{3}=35ax\) рд╣ै।
рд╣рд▓:
\(\frac{x^3+y^3}{x} = 35a\). рдЕрд╡рдХрд▓рди рдХрд░рдиे рдкрд░:
\(\frac{x(3x^2+3y^2 y') - (x^3+y^3)(1)}{x^2} = 0\)
\(3x^3 + 3xy^2 \frac{dy}{dx} - x^3 - y^3 = 0\)
\(2x^3 - y^3 + 3xy^2 \frac{dy}{dx} = 0\)

рдк्рд░. рей. (рдм) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрд╕ी рдПрдХ рдЙрдкрдк्рд░рд╢्рди рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) рдЖрд╡्рдпूрд╣ рдХा рд╕рд╣рдЦрдг्рдбрдЬ рд╡िрдзि (adjoint method) рд╕े рд╡्рдпुрдд्рдХ्рд░рдо рдЬ्рдЮाрдд рдХीрдЬिрдП: \(A = [\begin{matrix}3&1&5\\ 2&7&8\\ 1&2&5\end{matrix}]\)
рд╣рд▓:
\(|A| = 3(35-16) - 1(10-8) + 5(4-7) = 57 - 2 - 15 = 40 \neq 0\).
Cofactors (рд╕рд╣рдЦंрдб):
\(C_{11}=19, C_{12}=-2, C_{13}=-3\)
\(C_{21}=5, C_{22}=10, C_{23}=-5\)
\(C_{31}=-27, C_{32}=-14, C_{33}=19\)
Adj A (рд╕рд╣рдЦрдг्рдбрдЬ) = Cofactors рдЖрд╡्рдпूрд╣ рдХा рдкрд░िрд╡рд░्рддрди (Transpose).
\(A^{-1} = \frac{1}{40} [\begin{matrix}19&5&-27\\ -2&10&-14\\ -3&-5&19\end{matrix}]\)
(ii) рдЙрдкрднोрдЧ рд╡्рдпрдп \(E_{c}=0.0006x^{2}+0.003x\). рдЬрдм рдЖрдп ₹ 200 рд╣ै рддрдм APC, MPC рдФрд░ MPS рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
\(APC = \frac{E_c}{x} = 0.0006x + 0.003\). \(x=200\) рдкрд░, \(APC = 0.123\).
\(MPC = \frac{dE_c}{dx} = 0.0012x + 0.003\). \(x=200\) рдкрд░, \(MPC = 0.243\).
\(MPS = 1 - MPC = 1 - 0.243 = 0.757\).

рдк्рд░. рей. (рдХ) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрд╕ी рдПрдХ рдХृрддि (activity) рдХो рдкूрд░्рдг рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) \(\int_{0}^{2}\frac{dx}{4+x-x^{2}}\)
\(=\int_{0}^{2}\frac{dx}{-x^{2}+\boxed{x}+\boxed{4}}\)
\(=\int_{0}^{2}\frac{dx}{-x^{2}+x+\frac{1}{4}-\boxed{1/4}+4}\)
\(=-\int_{0}^{2}\frac{dx}{(x-\frac{1}{2})^{2}-(\boxed{\sqrt{17}/2})^{2}}\)
\(=\frac{1}{\sqrt{17}}\log(\frac{20+4\sqrt{17}}{20-4\sqrt{17}})\)
(ii) рдЬрдирд╕ंрдЦ्рдпा рд╡ृрдж्рдзि (Population Growth)
\(\frac{dP}{dt}=kP \Rightarrow \log P = kt + c\)
(i) \(c = \) \(\log(1,00,000)\)
(ii) рдЬрдм \(t=25, P=2,00,000\), рддो \(k = \) \(\frac{1}{25}\log 2\)
(iii) \(P=4,00,000\) рдХे рд▓िрдП, \(t = \) 50 рд╡рд░्рд╖।

рд╡िрднाрдЧ - реи (Section II)

рдк्рд░. рек. (рдЕ) рд╕рд╣ी рд╡िрдХрд▓्рдк рдЪुрдирдХрд░ рд▓िрдЦिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

(i) рдЕंрдХिрдд рдоूрд▓्рдп рдФрд░ рд╡рд░्рддрдоाрди рдоूрд▓्рдп рдХे рдмीрдЪ рдХे рдЕंрддрд░ рдХो ... рдХрд╣ा рдЬाрддा рд╣ै।
рдЙрдд्рддрд░: (рдм) рд╕рдЪ्рдЪी рдЫूрдЯ (True Discount)
(ii) рдПрдХ рд╕ाрдзाрд░рдг рд╡ाрд░्рд╖िрдХी рдоें, рднुрдЧрддाрди рдпा рдк्рд░ाрдк्рддिрдпाँ ... рдоें рд╣ोрддी рд╣ै।
рдЙрдд्рддрд░: (рдм) рдк्рд░рдд्рдпेрдХ рдЕрд╡рдзि рдХे рдЕंрдд
(iii) \(b_{xy}\) рдФрд░ \(b_{yx}\) рд╣ैं:
рдЙрдд्рддрд░: (рдм) рдоूрд▓ рдХे рдкрд░िрд╡рд░्рддрди рд╕े рд╕्рд╡рддंрдд्рд░ рд▓ेрдХिрди рдкैрдоाрдиे рд╕े рдирд╣ीं
(iv) рдбॉрд░рдмिрд╢-рдмाрд╡рд▓ीрд╕ рдоूрд▓्рдп рд╕ूрдЪрдХांрдХ рд╕ंрдЦ्рдпा рд╣ै:
рдЙрдд्рддрд░: (рдЕ) \(\frac{\frac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{1}}+\frac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{0}}}{2}\times100\)
(v) L.P.P. рдХा рдЙрдж्рджेрд╢्рдп рдлрд▓рди (objective function) рд╣ै:
рдЙрдд्рддрд░: (рдм) рдПрдХ рдлрд▓рди рдЬिрд╕рдХो рдЕрдзिрдХрддрдо рдпा рди्рдпूрдирддрдо рдХिрдпा рдЬाрддा рд╣ै।
(vi) рд╣ंрдЧेрд░िрдпрди рдкрдж्рдзрддि (Assignment Problem) рдХे рд▓िрдП рд▓ाрдн рдЕрдзिрдХрддрдо рд╕рдорд╕्рдпा рдХी рдЖрд╡рд╢्рдпрдХрддा рд╣ै:
рдЙрдд्рддрд░: (рдЕ) рд╕рднी рд▓ाрднों рдХो рдЕрд╡рд╕рд░ рд╣ाрдиिрдпों рдоें рдкрд░िрд╡рд░्рддिрдд рдХрд░рдиा।

рдк्рд░. рек. (рдм) рд╕рдд्рдп рдпा рдЕрд╕рдд्рдп рд▓िрдЦिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

  • (i) рдм्рд░ोрдХрд░ рдПрдХ рдПрдЬेंрдЯ рд╣ै... (рдЧाрд░ंрдЯी рджेрддा рд╣ै): рдЕрд╕рдд्рдп (False) (рдпрд╣ рдбेрд▓ рдХ्рд░ेрдбрд░ рдПрдЬेंрдЯ рд╣ोрддा рд╣ै)
  • (ii) \(\sum\frac{p_{0}q_{0}}{p_{1}q_{1}}\times100\) рдоूрд▓्рдп рд╕ूрдЪрдХांрдХ рд╣ै: рдЕрд╕рдд्рдп (False)
  • (iii) L.P.P. рдХा рдЗрд╖्рдЯрддрдо рдоूрд▓्рдп рд╡्рдпрд╡рд╣ाрд░्рдп рдХ्рд╖ेрдд्рд░ рдХे рдХेंрдж्рд░ рдоें рд╣ोрддा рд╣ै: рдЕрд╕рдд्рдп (False) (рдпрд╣ рдХोрдиे рдХे рдмिंрджुрдУं рдкрд░ рд╣ोрддा рд╣ै)

рдк्рд░. рек. (рдХ) рд░िрдХ्рдд рд╕्рдеाрдиों рдХी рдкूрд░्рддि рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рез рдЕंрдХ):

  • (i) рдмैंрдХрд░ рдХी рдЫूрдЯ рд╣рдоेрд╢ा рд╕рдЪ्рдЪी рдЫूрдЯ рд╕े рдЕрдзिрдХ (Greater) рд╣ोрддी рд╣ै।
  • (ii) рднाрд░िрдд рд╕ाрдкेрдХ्рд╖ рдкрдж्рдзрддि (Weighted Average of Price Relatives) рд╕ूрдд्рд░: \(\frac{\sum IW}{\sum W}\)
  • (iii) рдкрд╣рд▓े рдХाрд░्рдп рд╢ुрд░ु рдХрд░рдиे рдФрд░ рдЕंрддिрдо рдХाрд░्рдп рдкूрд░ा рдХрд░рдиे рдХे рдмीрдЪ рдХा рд╕рдордп: рдХुрд▓ рд╡्рдпрддीрдд рд╕рдордп (Total Elapsed Time)

рдк्рд░. рел. (рдЕ) рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рей рдЕंрдХ):

(i) рджीрдкрдХ рдХी рд╕ैрд▓рд░ी ₹ 4,000 рд╕े ₹ 5,000 рд╣ुрдИ। рдХрдоीрд╢рди 3% рд╕े 2% рд╣ो рдЧрдпा। рдЖрдп рд╕рдоाрди рд╣ै। рдмिрдХ्рд░ी рдЬ्рдЮाрдд рдХрд░ें।
рд╣рд▓:
рдоाрдиा рдмिрдХ्рд░ी = \(x\)
рдкुрд░ाрдиी рдЖрдп = \(4000 + 0.03x\)
рдирдИ рдЖрдп = \(5000 + 0.02x\)
рджोрдиों рдмрд░ाрдмрд░ рд╣ैं: \(4000 + 0.03x = 5000 + 0.02x\)
\(0.01x = 1000 \Rightarrow x = 1,00,000\)
рдмिрдХ्рд░ी = ₹ 1,00,000
(ii) \(b_{yx}=0.4\), \(b_{xy}=0.9\), \(V(X)=9\). \(V(Y)\) рдЬ्рдЮाрдд рдХрд░ें।
рд╣рд▓:
\(r^2 = b_{yx} \times b_{xy} = 0.4 \times 0.9 = 0.36 \Rightarrow r=0.6\)
\(\sigma_x = \sqrt{9} = 3\)
рд╕ूрдд्рд░: \(b_{yx} = r \frac{\sigma_y}{\sigma_x} \Rightarrow 0.4 = 0.6 (\frac{\sigma_y}{3})\)
\(0.4 = 0.2 \sigma_y \Rightarrow \sigma_y = 2\)
\(V(Y) = \sigma_y^2 = 4\).
(iii) 4 рд╡ाрд░्рд╖िрдХी рдХेंрдж्рд░िрдд рдЧрддिрдоाрди рдФрд╕рдд (4-yearly centered moving averages) рдиिрдХाрд▓ें।
рд╣рд▓ (Trend Values):
1978: \((0+2+3+3)/4\) рдФрд░ рдЕрдЧрд▓े рдХा рдФрд╕рдд = 2.25
1979: 2.75
1980: 3.25
1981: 3.875
1982: 4.875
1983: 6.25

рдк्рд░. рел. (рдм) рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) рд╡ॉрд▓्рд╢ рдХी рдХीрдордд рд╕ूрдЪрдХांрдХ рд╕ंрдЦ्рдпा 150 рд╣ै। 'x' рдХा рдоाрди рдЬ्рдЮाрдд рдХीрдЬिрдП।
рд╣рд▓:
рд╡ॉрд▓्рд╢ рд╕ूрдд्рд░: \(P_{01} = \frac{\sum p_1 \sqrt{q_0 q_1}}{\sum p_0 \sqrt{q_0 q_1}} \times 100\)
рднाрд░ \(W = \sqrt{q_0 q_1}\): A(3), B(6), C(5), D(4)
\(\sum p_1 W = 30 + 96 + 115 + 104 = 345\)
\(\sum p_0 W = 15 + 6x + 75 + 40 = 130 + 6x\)
\(150 = \frac{345}{130+6x} \times 100 \Rightarrow 1.5(130+6x) = 345\)
\(195 + 9x = 345 \Rightarrow 9x = 150 \Rightarrow x = 16.67\)
(ii) рдЦिрд▓ौрдиा рдиिрд░्рдоाрдг (Sequencing Problem) A->B->C. рдХुрд▓ рдЙрдкрдпोрдЧी рд╕рдордп рдФрд░ рдорд╢ीрди B рдХा рдиिрд╖्рдХ्рд░िрдп рд╕рдордп рдиिрдХाрд▓ें।
рд╣рд▓:
рдиिрдпрдо: Min A (12) \(\ge\) Max B (12) (рд╕рдд्рдп)।
рдХाрд▓्рдкрдиिрдХ рдорд╢ीрдиें G = A+B, H = B+C рдмрдиाрдпें।
рдХ्рд░рдо (Sequence): 3 - 2 - 5 - 4 - 1
рдХुрд▓ рдЙрдкрдпोрдЧी рд╕рдордп (Total Elapsed Time): 102 рдШंрдЯे।
рдорд╢ीрди B рдХा рдиिрд╖्рдХ्рд░िрдп рд╕рдордп (Idle Time): 62 рдШंрдЯे।
(iii) рд╕ंрднाрд╡्рдпрддा рд╡िрддрд░рдг: k, P(X < 3), P(X > 6) рдЬ्рдЮाрдд рдХрд░ें।
рд╣рд▓:
(рдЕ) \(\sum P(x) = 1 \Rightarrow 10k^2 + 9k - 1 = 0 \Rightarrow k = 0.1\) (k>0)
(рдм) \(P(X < 3) = P(1)+P(2) = k+2k = 3k = 0.3\)
(рдХ) \(P(X > 6) = P(7) = 7k^2+k = 7(0.01)+0.1 = 0.17\)

рдк्рд░. рем. (рдЕ) рдХिрди्рд╣ीं рджो рдЙрдкрдк्рд░рд╢्рдиों рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рей рдЕंрдХ):

(i) рдмीрдоा рджाрд╡ा (Insurance Claim): рдкॉрд▓िрд╕ी 75%, рдк्рд░ीрдоिрдпрдо 0.70% (₹2625), рд╣ाрдиि 60%।
рд╣рд▓:
рдк्рд░ीрдоिрдпрдо = рдкॉрд▓िрд╕ी рдоूрд▓्рдп \(\times\) рджрд░ \(\Rightarrow 2625 = P.V. \times 0.007 \Rightarrow P.V. = 3,75,000\)
рдкॉрд▓िрд╕ी рд╕ंрдкрдд्рддि рдХा 75% рд╣ै \(\Rightarrow\) рд╕ंрдкрдд्рддि рдоूрд▓्рдп = \(3,75,000 / 0.75 = 5,00,000\)
рд╣ाрдиि = \(5,00,000 \times 0.60 = 3,00,000\)
рджाрд╡ा (Claim) = \(\frac{P.V.}{Property Value} \times Loss = 0.75 \times 3,00,000\) = ₹ 2,25,000
(ii) рд╕्рд╡рдд्рд╡ाрд░्рдкрдг рд╕рдорд╕्рдпा (Assignment Problem): рди्рдпूрдирддрдо рдЦрд░्рдЪ рдЬ्рдЮाрдд рдХрд░ें।
рд╣рд▓:
рдЗрд╖्рдЯрддрдо рд╢ेрдб्рдпूрд▓:
\(M_1 \rightarrow B\) (10)
\(M_2 \rightarrow C\) (13)
\(M_3 \rightarrow A\) (5)
рди्рдпूрдирддрдо рдЦрд░्рдЪ = \(10 + 13 + 5 = 28\) (рд╕ौ рд░ुрдкрдпे рдоें) = ₹ 2800.
(iii) 10% рдЦрд░ाрдм рдЕंрдбे। 10 рдЕंрдбों рдХे рдирдоूрдиे рдоें рдХрдо-рд╕े-рдХрдо рдПрдХ рдЦрд░ाрдм рд╣ोрдиे рдХी рдк्рд░ाрдпिрдХрддा।
рд╣рд▓:
\(p = 0.1, q = 0.9, n = 10\)
\(P(X \ge 1) = 1 - P(X=0)\)
\(= 1 - {^{10}C_0} (0.1)^0 (0.9)^{10} = 1 - (0.9)^{10}\)

рдк्рд░. рем. (рдм) рдХिрд╕ी рдПрдХ рдЙрдкрдк्рд░рд╢्рди рдХो рд╣рд▓ рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) рдк्рд░рд╡ृрдд्рддि рд░ेрдЦा (Trend Line) - рди्рдпूрдирддрдо рд╡рд░्рдЧ рд╡िрдзि।
рд╣рд▓:
рд╡рд░्рд╖ (n=7): рдордз्рдп рд╡рд░्рд╖ 1995। \(u = \frac{t-1995}{5}\).
рд╕рдоीрдХрд░рдг \(y = a + bu\).
\(a = \frac{\sum y}{n} = \frac{30}{7} = 4.286\)
\(b = \frac{\sum uy}{\sum u^2} = \frac{-44}{28} = -1.571\)
рд░ेрдЦा: \(y = 4.286 - 1.571(\frac{t-1995}{5})\)
(ii) рди्рдпूрдирддрдо рдХीрдЬिрдП: \(z=6x+2y\) рд╢рд░्рддें: \(x+2y\ge3, x+4y\ge4, 3x+y\ge3\).
рд╣рд▓:
рдХोрдиे рдХे рдмिंрджु (Corner Points): A(0, 3), B(0.6, 1.2), C(2, 0.5), D(4, 0).
Z рдХा рдоाрди:
A: 6, B: 6, C: 13, D: 24.
рди्рдпूрдирддрдо рдоाрди 6 рд╣ै (рдмिंрджु A рдФрд░ B рдХो рдоिрд▓ाрдиे рд╡ाрд▓े рд░ेрдЦाрдЦंрдб рдкрд░)।

рдк्рд░. рем. (рдХ) рдиिрдо्рдирд▓िрдЦिрдд рдоें рд╕े рдХिрд╕ी рдПрдХ рдХृрддि (Activity) рдХो рдкूрд░्рдг рдХीрдЬिрдП (рдк्рд░рдд्рдпेрдХ рек рдЕंрдХ):

(i) рдк्рд░рддिрдЧрдорди (Regression): \(x=10, \bar{y}=12, V(X)=9, \sigma_y=4, r=0.6\). рдЬрдм x=5 рддो y?
\(Y - 12 = r \cdot \frac{\sigma_y}{\sigma_x} (X-10)\)
\(Y - 12 = 0.6 \times \frac{4}{\boxed{3}} (X-10)\)
рдЬрдм \(x=5\):
\(Y - 12 = \boxed{0.8} \times (-5)\)
\(Y - 12 = -4 \Rightarrow Y = \boxed{8}\)
(ii) рдкॉрдЗрд╕рди рд╡िрддрд░рдг: \(X \sim P(m)\), \(P(X=1)=P(X=2)\).
\(\frac{e^{-m}m^1}{1!} = \frac{e^{-m}m^2}{\boxed{2!}}\)
\(m = \boxed{2}\)
\(P(X=2) = \frac{e^{-2}2^2}{2!} = \boxed{0.2706}\)
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Title: Maths & Stats (Commerce) Board Paper Solution March 2024 (Hindi Medium) Labels: Maths Commerce, HSC Board 2024, Hindi Medium Solution, Solved Paper Permanent Link: maths-stats-commerce-march-2024-solution-hindi Search Description: Complete solved paper for HSC Commerce Maths & Statistics March 2024 (Hindi Medium) with step-by-step explanations.
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