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

• Physics - March 2025 - English Medium View Answer Key
• Physics - March 2025 - Marathi Medium View Answer Key
• Physics - March 2025 - Hindi Medium View Answer Key
• Physics - March 2024 - English Medium View Answer Key Answer Key
• Physics - March 2024 - Marathi Medium View Answer Key
• Physics - March 2024 - Hindi Medium View Answer Key
• Physics - March 2023 - English Medium View Answer Key
• Physics - July 2023 - English Medium View Answer Key
• Physics - March 2022 - English Medium View Answer Key
• Physics - July 2022 - English Medium View Answer Key
• Physics - July 2021 - English Medium View Answer Key
• Physics - February 2020 - English Medium View Answer Key
• Physics - February 2019 - English Medium View Answer Key
• Physics - February 2018 - English Medium View Answer Key
• Physics - March 2013 View
• Physics - October 2013 View Answer Key
• Physics - March 2014 View Answer Key
• Physics - October 2014 View Answer Key
• Physics - March 2015 View Answer Key
• Physics - October 2015 View Answer Key
• Physics - March 2016 View Answer Key
• Physics - July 2016 View Answer Key
• Physics - March 2017 View Answer Key
• Physics - July 2017 View Answer Key

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
1	Solid State	-
2	Solutions	-
3	Ionic Equilibrium	-
4	Chemical Thermodynamics	-
5	Electrochemistry	-
6	Chemical Kinetics	-
7	Group 16, 17 & 18 Elements	-
8	Transition & Inner Transition Elements	☑ Expected
9	Coordination Compounds	☑ Expected
10	Halogen Derivatives	-
11	Alcohols, Phenols & Ethers	-
12	Aldehydes, Ketones & Carboxylic Acids	-
13	Amines	-
14	Biomolecules	-
15	Polymer Chemistry	-
16	Green & 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

• Chemistry - March 2025 - English Medium View Answer Key
• Chemistry - March 2025 - Marathi Medium View Answer Key
• Chemistry - March 2025 - Hindi Medium View Answer Key
• Chemistry - March 2024 - English Medium View Answer Key
• Chemistry - March 2024 - Marathi Medium View Answer Key
• Chemistry - March 2024 - Hindi Medium View Answer Key
• Chemistry - March 2023 - English Medium View Answer Key
• Chemistry - July 2023 - English Medium View Answer Key
• Chemistry - March 2022 - English Medium View Answer Key
• Chemistry - July 2022 - English Medium View Answer Key
• Chemistry - March 2021 - English Medium View Answer Key
• Chemistry - March 2020 - English Medium View Answer Key
• Chemistry - March 2013 View
• Chemistry - October 2013 View
• Chemistry - March 2014 View
• Chemistry - October 2014 View
• Chemistry - March 2015 View
• Chemistry - July 2015 View
• Chemistry - March 2016 View
• Chemistry - July 2016 View
• Chemistry - March 2017 View
• Chemistry - July 2017 View
• Chemistry - March 2019 View

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).

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

• Physics - March 2025 - English Medium View Answer Key
• Physics - March 2025 - Marathi Medium View Answer Key
• Physics - March 2025 - Hindi Medium View Answer Key
• Physics - March 2024 - English Medium View Answer Key Answer Key
• Physics - March 2024 - Marathi Medium View Answer Key
• Physics - March 2024 - Hindi Medium View Answer Key
• Physics - March 2023 - English Medium View Answer Key
• Physics - July 2023 - English Medium View Answer Key
• Physics - March 2022 - English Medium View Answer Key
• Physics - July 2022 - English Medium View Answer Key
• Physics - July 2021 - English Medium View Answer Key
• Physics - February 2020 - English Medium View Answer Key
• Physics - February 2019 - English Medium View Answer Key
• Physics - February 2018 - English Medium View Answer Key
• Physics - March 2013 View
• Physics - October 2013 View Answer Key
• Physics - March 2014 View Answer Key
• Physics - October 2014 View Answer Key
• Physics - March 2015 View Answer Key
• Physics - October 2015 View Answer Key
• Physics - March 2016 View Answer Key
• Physics - July 2016 View Answer Key
• Physics - March 2017 View Answer Key
• Physics - July 2017 View Answer Key

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

गणित और सांख्यिकी (वाणिज्य) - मार्च 2024
सम्पूर्ण हल (हिंदी माध्यम)

समय: 3 घंटे | अधिकतम अंक: 80

विभाग - १ (Section I)

प्र. १. (अ) निम्नलिखित बहुविकल्पीय प्रश्नों के विकल्पों में से सही विकल्प चुनकर लिखिए (प्रत्येक १ अंक):

(i) निम्नलिखित में से कौन सा कथन नहीं है:

• (अ) धूम्रपान स्वास्थ्य के लिए हानिकारक है।
• (ब) \(2+2=4\)
• (क) 2 एकमात्र सम अभाज्य संख्या है।
• (ड) यहाँ आओ।

उत्तर: (ड) यहाँ आओ।

स्पष्टीकरण: यह एक आज्ञार्थक वाक्य (Imperative sentence) है, इसलिए यह तार्किक कथन नहीं है।

(ii) यदि \(x+y+z=3\), \(x+2y+3z=4\), \(x+4y+9z=6\) तब \((y, z) = ...\)

• (अ) (-1, 0)
• (ब) (1, 0)
• (क) (1, -1)
• (ड) (-1, 1)

उत्तर: (ब) (1, 0)

हल:
समीकरण (2) - (1): \(y + 2z = 1\)
विकल्प (ब) में \(y=1, z=0\) रखने पर: \(1 + 0 = 1\) (संतुष्ट करता है)।

(iii) यदि \(y = \log(\frac{e^{x}}{x^{2}})\) तब \(\frac{dy}{dx} = ?\)

• (अ) \(\frac{2-x}{x}\)
• (ब) \(\frac{x-2}{x}\)
• (क) \(\frac{e-x}{ex}\)
• (ड) \(\frac{x-e}{ex}\)

उत्तर: (ब) \(\frac{x-2}{x}\)

\(y = \log e^x - \log x^2 = x - 2\log x\)
\(\frac{dy}{dx} = 1 - \frac{2}{x} = \frac{x-2}{x}\)

(iv) \(\int\frac{dx}{\sqrt{1-x}}\) का मान है:

• (अ) \(2\sqrt{1-x}+c\)
• (ब) \(-2\sqrt{1-x}+c\)
• (क) \(\sqrt{x}+c\)
• (ड) \(x+c\)

उत्तर: (ब) \(-2\sqrt{1-x}+c\)

(v) \(\int\frac{dx}{(x-8)(x+7)} = ...\)

• (अ) \(\frac{1}{15}\log|\frac{x+2}{x+1}|+c\)
• (ब) \(\frac{1}{15}\log|\frac{x+8}{x+7}|+c\)
• (क) \(\frac{1}{15}\log|\frac{x-8}{x+7}|+c\)
• (ड) \((x-8)(x+7)+c\)

उत्तर: (क) \(\frac{1}{15}\log|\frac{x-8}{x+7}|+c\)

(vi) समीकरण \(y=k_{1}e^{x}+k_{2}e^{-x}\) का अवकल समीकरण है:

• (अ) \(\frac{d^{2}y}{dx^{2}}-y=0\)
• (ब) \(\frac{d^{2}y}{dx^{2}}+y\frac{dy}{dx}=0\)
• (क) \(\frac{d^{2}y}{dx^{2}}+\frac{dy}{dx}=0\)
• (ड) \(\frac{d^{2}y}{dx^{2}}+y=0\)

उत्तर: (अ) \(\frac{d^{2}y}{dx^{2}}-y=0\)

प्र. १. (ब) निम्नलिखित कथन सत्य हैं या असत्य, लिखिए (प्रत्येक १ अंक):

• (i) \(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt\)
  सत्य (True)
• (ii) \(\int\frac{x-1}{(x+1)^{3}}e^{x}dx=e^{x}f(x)+c\), के लिए \(f(x)=(x+1)^{2}\)
  असत्य (False) (सही उत्तर \(\frac{1}{(x+1)^2}\) होना चाहिए)
• (iii) अवकल समीकरण के घात (order) और कोटि (degree) सदैव धनात्मक पूर्णांक होते हैं।
  सत्य (True)

प्र. १. (क) निम्नलिखित रिक्त स्थानों की पूर्ति कीजिए (प्रत्येक १ अंक):

• (i) किसी बिंदु (a, b) पर स्पर्श रेखा (tangent) की प्रवणता (slope) प्रवणता / Gradient कहलाती है।
• (ii) यदि \(f'(x)=\frac{1}{x}+x\) तथा \(f(1)=\frac{5}{2}\) तब \(f(x)=\log x + \frac{x^2}{2} + \) 2
• (iii) अवकल समीकरण का हल जिसे सामान्य हल में स्वेच्छ अचरों को विशिष्ट मान देकर प्राप्त किया जाता है, विशिष्ट हल (Particular Solution) कहलाता है।

प्र. २. (अ) निम्नलिखित में से किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ३ अंक):

(i) सत्यता सारणी का उपयोग करके जांचिए कि क्या कथन पुनरुक्ति (tautology), विरोधाभास (contradiction) या आकस्मिकता (contingency) है: \(\sim p\rightarrow(p\rightarrow\sim q)\)

हल:

p	q	~p	~q	p -> ~q	~p -> (p -> ~q)
T	T	F	F	F	T
T	F	F	T	T	T
F	T	T	F	T	T
F	F	T	T	T	T

चूँकि अंतिम स्तंभ में सभी मान 'T' हैं, अतः यह एक पुनरुक्ति (Tautology) है।

(ii) यदि \(x=e^{3t}\), \(y=e^{(4t+5)}\) तब \(\frac{dy}{dx}\) ज्ञात कीजिए।

हल:
\(\frac{dx}{dt} = 3e^{3t}\) और \(\frac{dy}{dt} = 4e^{(4t+5)}\)
\(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{4e^{(4t+5)}}{3e^{3t}}\)
\(= \frac{4}{3} e^{(4t+5-3t)} = \frac{4}{3}e^{t+5}\)

(iii) यदि \(A=[\begin{matrix}7&3&0\\ 0&4&-2\end{matrix}]\) \(B=[\begin{matrix}0&-2&3\\ 2&1&-4\end{matrix}]\) तो \(A^{T}+4B^{T}\) ज्ञात कीजिए।

हल:
\(A^T = [\begin{matrix}7&0\\ 3&4\\ 0&-2\end{matrix}]\), \(B^T = [\begin{matrix}0&2\\ -2&1\\ 3&-4\end{matrix}]\)
\(4B^T = [\begin{matrix}0&8\\ -8&4\\ 12&-16\end{matrix}]\)
\(A^T + 4B^T = [\begin{matrix}7&0\\ 3&4\\ 0&-2\end{matrix}] + [\begin{matrix}0&8\\ -8&4\\ 12&-16\end{matrix}] = [\begin{matrix}7&8\\ -5&8\\ 12&-18\end{matrix}]\)

प्र. २. (ब) निम्नलिखित में से किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ४ अंक):

(i) समान अर्थ वाले कथनों के युग्मों को पहचानिए। (कुत्ता वाले कथन)

हल:
(अ) \(p \to q\) (If D is dog, D is good)
(ब) \(q \to p\) (If D is good, D is dog) - विलोम (Converse)
(क) \(\sim q \to \sim p\) (If D not good, D not dog) - प्रतिधनात्मक (Contrapositive)
(ड) \(\sim p \to \sim q\) (If D not dog, D not good) - प्रतिलोम (Inverse)

तार्किक समतुल्यता: 1. (अ) और (क) समान हैं (\(p \to q \equiv \sim q \to \sim p\))
2. (ब) और (ड) समान हैं (\(q \to p \equiv \sim p \to \sim q\))

(ii) फलन \(f(x)=2x^{3}-21x^{2}+36x-20\) का निम्नतम मान (minimum value) ज्ञात कीजिए।

हल:
\(f'(x) = 6x^2 - 42x + 36\).
\(f'(x) = 0 \Rightarrow x^2 - 7x + 6 = 0 \Rightarrow (x-6)(x-1)=0\).
\(x=1, x=6\).
\(f''(x) = 12x - 42\).
\(x=6\) पर, \(f''(6) = 72 - 42 = 30 > 0\) (निम्नतम)।
निम्नतम मान: \(f(6) = 2(216) - 21(36) + 36(6) - 20 = -128\).

(iii) रेखा \(y=-2x\), X अक्ष एवं रेखाओं \(x=-1\) और \(x=2\) से घिरे क्षेत्र का क्षेत्रफल ज्ञात कीजिए।

हल:
क्षेत्रफल = \(|\int_{-1}^{0} (-2x)dx| + |\int_{0}^{2} (-2x)dx|\)
\(A_1 = [-x^2]_{-1}^{0} = -(0 - 1) = 1\)
\(A_2 = [-x^2]_{0}^{2} = -(4 - 0) = -4 \Rightarrow |A_2| = 4\)
कुल क्षेत्रफल = \(1 + 4 = 5\) वर्ग इकाई।

प्र. ३. (अ) निम्नलिखित में से किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ३ अंक):

(i) यदि \(y=x^{e^{x}}\) तो \(\frac{dy}{dx}\) ज्ञात कीजिए।

हल:
दोनों पक्षों का log लेने पर: \(\log y = e^x \log x\)
अवकलन: \(\frac{1}{y}\frac{dy}{dx} = e^x(\frac{1}{x}) + \log x(e^x)\)
\(\frac{dy}{dx} = y \cdot e^x (\frac{1}{x} + \log x) = x^{e^x} e^x (\frac{1+x\log x}{x})\)

(ii) यदि \(f'(x)=4x^{3}-3x^{2}+2x+k\), \(f(0)=1\) और \(f(1)=4\) तो \(f(x)\) ज्ञात कीजिए।

हल:
समाकलन करने पर: \(f(x) = x^4 - x^3 + x^2 + kx + c\)
\(f(0)=1 \Rightarrow c=1\)
\(f(1)=4 \Rightarrow 1 - 1 + 1 + k + 1 = 4 \Rightarrow k=2\)
अतः \(f(x) = x^4 - x^3 + x^2 + 2x + 1\)

(iii) अवकल समीकरण प्राप्त कीजिए जिसका व्यापक हल \(x^{3}+y^{3}=35ax\) है।

हल:
\(\frac{x^3+y^3}{x} = 35a\). अवकलन करने पर:
\(\frac{x(3x^2+3y^2 y') - (x^3+y^3)(1)}{x^2} = 0\)
\(3x^3 + 3xy^2 \frac{dy}{dx} - x^3 - y^3 = 0\)
\(2x^3 - y^3 + 3xy^2 \frac{dy}{dx} = 0\)

प्र. ३. (ब) निम्नलिखित में से किसी एक उपप्रश्न को हल कीजिए (प्रत्येक ४ अंक):

(i) आव्यूह का सहखण्डज विधि (adjoint method) से व्युत्क्रम ज्ञात कीजिए: \(A = [\begin{matrix}3&1&5\\ 2&7&8\\ 1&2&5\end{matrix}]\)

हल:
\(|A| = 3(35-16) - 1(10-8) + 5(4-7) = 57 - 2 - 15 = 40 \neq 0\).
Cofactors (सहखंड):
\(C_{11}=19, C_{12}=-2, C_{13}=-3\)
\(C_{21}=5, C_{22}=10, C_{23}=-5\)
\(C_{31}=-27, C_{32}=-14, C_{33}=19\)
Adj A (सहखण्डज) = Cofactors आव्यूह का परिवर्तन (Transpose).
\(A^{-1} = \frac{1}{40} [\begin{matrix}19&5&-27\\ -2&10&-14\\ -3&-5&19\end{matrix}]\)

(ii) उपभोग व्यय \(E_{c}=0.0006x^{2}+0.003x\). जब आय ₹ 200 है तब APC, MPC और MPS ज्ञात कीजिए।

हल:
\(APC = \frac{E_c}{x} = 0.0006x + 0.003\). \(x=200\) पर, \(APC = 0.123\).
\(MPC = \frac{dE_c}{dx} = 0.0012x + 0.003\). \(x=200\) पर, \(MPC = 0.243\).
\(MPS = 1 - MPC = 1 - 0.243 = 0.757\).

प्र. ३. (क) निम्नलिखित में से किसी एक कृति (activity) को पूर्ण कीजिए (प्रत्येक ४ अंक):

(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) जनसंख्या वृद्धि (Population Growth)

\(\frac{dP}{dt}=kP \Rightarrow \log P = kt + c\)
(i) \(c = \) \(\log(1,00,000)\)
(ii) जब \(t=25, P=2,00,000\), तो \(k = \) \(\frac{1}{25}\log 2\)
(iii) \(P=4,00,000\) के लिए, \(t = \) 50 वर्ष।

विभाग - २ (Section II)

प्र. ४. (अ) सही विकल्प चुनकर लिखिए (प्रत्येक १ अंक):

(i) अंकित मूल्य और वर्तमान मूल्य के बीच के अंतर को ... कहा जाता है।

उत्तर: (ब) सच्ची छूट (True Discount)

(ii) एक साधारण वार्षिकी में, भुगतान या प्राप्तियाँ ... में होती है।

उत्तर: (ब) प्रत्येक अवधि के अंत

(iii) \(b_{xy}\) और \(b_{yx}\) हैं:

उत्तर: (ब) मूल के परिवर्तन से स्वतंत्र लेकिन पैमाने से नहीं

(iv) डॉरबिश-बावलीस मूल्य सूचकांक संख्या है:

उत्तर: (अ) \(\frac{\frac{\Sigma p_{1}q_{0}}{\Sigma p_{0}q_{1}}+\frac{\Sigma p_{1}q_{1}}{\Sigma p_{0}q_{0}}}{2}\times100\)

(v) L.P.P. का उद्देश्य फलन (objective function) है:

उत्तर: (ब) एक फलन जिसको अधिकतम या न्यूनतम किया जाता है।

(vi) हंगेरियन पद्धति (Assignment Problem) के लिए लाभ अधिकतम समस्या की आवश्यकता है:

उत्तर: (अ) सभी लाभों को अवसर हानियों में परिवर्तित करना।

प्र. ४. (ब) सत्य या असत्य लिखिए (प्रत्येक १ अंक):

• (i) ब्रोकर एक एजेंट है... (गारंटी देता है): असत्य (False) (यह डेल क्रेडर एजेंट होता है)
• (ii) \(\sum\frac{p_{0}q_{0}}{p_{1}q_{1}}\times100\) मूल्य सूचकांक है: असत्य (False)
• (iii) L.P.P. का इष्टतम मूल्य व्यवहार्य क्षेत्र के केंद्र में होता है: असत्य (False) (यह कोने के बिंदुओं पर होता है)

प्र. ४. (क) रिक्त स्थानों की पूर्ति कीजिए (प्रत्येक १ अंक):

• (i) बैंकर की छूट हमेशा सच्ची छूट से अधिक (Greater) होती है।
• (ii) भारित सापेक्ष पद्धति (Weighted Average of Price Relatives) सूत्र: \(\frac{\sum IW}{\sum W}\)
• (iii) पहले कार्य शुरु करने और अंतिम कार्य पूरा करने के बीच का समय: कुल व्यतीत समय (Total Elapsed Time)

प्र. ५. (अ) किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ३ अंक):

(i) दीपक की सैलरी ₹ 4,000 से ₹ 5,000 हुई। कमीशन 3% से 2% हो गया। आय समान है। बिक्री ज्ञात करें।

हल:
माना बिक्री = \(x\)
पुरानी आय = \(4000 + 0.03x\)
नई आय = \(5000 + 0.02x\)
दोनों बराबर हैं: \(4000 + 0.03x = 5000 + 0.02x\)
\(0.01x = 1000 \Rightarrow x = 1,00,000\)
बिक्री = ₹ 1,00,000

(ii) \(b_{yx}=0.4\), \(b_{xy}=0.9\), \(V(X)=9\). \(V(Y)\) ज्ञात करें।

हल:
\(r^2 = b_{yx} \times b_{xy} = 0.4 \times 0.9 = 0.36 \Rightarrow r=0.6\)
\(\sigma_x = \sqrt{9} = 3\)
सूत्र: \(b_{yx} = r \frac{\sigma_y}{\sigma_x} \Rightarrow 0.4 = 0.6 (\frac{\sigma_y}{3})\)
\(0.4 = 0.2 \sigma_y \Rightarrow \sigma_y = 2\)
\(V(Y) = \sigma_y^2 = 4\).

(iii) 4 वार्षिकी केंद्रित गतिमान औसत (4-yearly centered moving averages) निकालें।

हल (Trend Values):
1978: \((0+2+3+3)/4\) और अगले का औसत = 2.25
1979: 2.75
1980: 3.25
1981: 3.875
1982: 4.875
1983: 6.25

प्र. ५. (ब) किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ४ अंक):

(i) वॉल्श की कीमत सूचकांक संख्या 150 है। 'x' का मान ज्ञात कीजिए।

हल:
वॉल्श सूत्र: \(P_{01} = \frac{\sum p_1 \sqrt{q_0 q_1}}{\sum p_0 \sqrt{q_0 q_1}} \times 100\)
भार \(W = \sqrt{q_0 q_1}\): 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) खिलौना निर्माण (Sequencing Problem) A->B->C. कुल उपयोगी समय और मशीन B का निष्क्रिय समय निकालें।

हल:
नियम: Min A (12) \(\ge\) Max B (12) (सत्य)।
काल्पनिक मशीनें G = A+B, H = B+C बनायें।
क्रम (Sequence): 3 - 2 - 5 - 4 - 1
कुल उपयोगी समय (Total Elapsed Time): 102 घंटे।
मशीन B का निष्क्रिय समय (Idle Time): 62 घंटे।

(iii) संभाव्यता वितरण: k, P(X < 3), P(X > 6) ज्ञात करें।

हल:
(अ) \(\sum P(x) = 1 \Rightarrow 10k^2 + 9k - 1 = 0 \Rightarrow k = 0.1\) (k>0)
(ब) \(P(X < 3) = P(1)+P(2) = k+2k = 3k = 0.3\)
(क) \(P(X > 6) = P(7) = 7k^2+k = 7(0.01)+0.1 = 0.17\)

प्र. ६. (अ) किन्हीं दो उपप्रश्नों को हल कीजिए (प्रत्येक ३ अंक):

(i) बीमा दावा (Insurance Claim): पॉलिसी 75%, प्रीमियम 0.70% (₹2625), हानि 60%।

हल:
प्रीमियम = पॉलिसी मूल्य \(\times\) दर \(\Rightarrow 2625 = P.V. \times 0.007 \Rightarrow P.V. = 3,75,000\)
पॉलिसी संपत्ति का 75% है \(\Rightarrow\) संपत्ति मूल्य = \(3,75,000 / 0.75 = 5,00,000\)
हानि = \(5,00,000 \times 0.60 = 3,00,000\)
दावा (Claim) = \(\frac{P.V.}{Property Value} \times Loss = 0.75 \times 3,00,000\) = ₹ 2,25,000

(ii) स्वत्वार्पण समस्या (Assignment Problem): न्यूनतम खर्च ज्ञात करें।

हल:
इष्टतम शेड्यूल:
\(M_1 \rightarrow B\) (10)
\(M_2 \rightarrow C\) (13)
\(M_3 \rightarrow A\) (5)
न्यूनतम खर्च = \(10 + 13 + 5 = 28\) (सौ रुपये में) = ₹ 2800.

(iii) 10% खराब अंडे। 10 अंडों के नमूने में कम-से-कम एक खराब होने की प्रायिकता।

हल:
\(p = 0.1, q = 0.9, n = 10\)
\(P(X \ge 1) = 1 - P(X=0)\)
\(= 1 - {^{10}C_0} (0.1)^0 (0.9)^{10} = 1 - (0.9)^{10}\)

प्र. ६. (ब) किसी एक उपप्रश्न को हल कीजिए (प्रत्येक ४ अंक):

(i) प्रवृत्ति रेखा (Trend Line) - न्यूनतम वर्ग विधि।

हल:
वर्ष (n=7): मध्य वर्ष 1995। \(u = \frac{t-1995}{5}\).
समीकरण \(y = a + bu\).
\(a = \frac{\sum y}{n} = \frac{30}{7} = 4.286\)
\(b = \frac{\sum uy}{\sum u^2} = \frac{-44}{28} = -1.571\)
रेखा: \(y = 4.286 - 1.571(\frac{t-1995}{5})\)

(ii) न्यूनतम कीजिए: \(z=6x+2y\) शर्तें: \(x+2y\ge3, x+4y\ge4, 3x+y\ge3\).

हल:
कोने के बिंदु (Corner Points): A(0, 3), B(0.6, 1.2), C(2, 0.5), D(4, 0).
Z का मान:
A: 6, B: 6, C: 13, D: 24.
न्यूनतम मान 6 है (बिंदु A और B को मिलाने वाले रेखाखंड पर)।

प्र. ६. (क) निम्नलिखित में से किसी एक कृति (Activity) को पूर्ण कीजिए (प्रत्येक ४ अंक):

(i) प्रतिगमन (Regression): \(x=10, \bar{y}=12, V(X)=9, \sigma_y=4, r=0.6\). जब x=5 तो y?

\(Y - 12 = r \cdot \frac{\sigma_y}{\sigma_x} (X-10)\)
\(Y - 12 = 0.6 \times \frac{4}{\boxed{3}} (X-10)\)
जब \(x=5\):
\(Y - 12 = \boxed{0.8} \times (-5)\)
\(Y - 12 = -4 \Rightarrow Y = \boxed{8}\)

(ii) पॉइसन वितरण: \(X \sim P(m)\), \(P(X=1)=P(X=2)\).

\(\frac{e^{-m}m^1}{1!} = \frac{e^{-m}m^2}{\boxed{2!}}\)
\(m = \boxed{2}\)
\(P(X=2) = \frac{e^{-2}2^2}{2!} = \boxed{0.2706}\)

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.