Common Questions in Bihar Board Class 12 Math (2019-2024)

 

Common Questions in Bihar Board Class 12 Math (2019-2024)

1. Relations and Functions

   • Question: Define and explain types of relations and functions. Give examples of one-to-one, onto, and bijective functions. (Appeared in 2020, 2022, 2024)

2. Inverse Trigonometric Functions

   • Question: Solve for x, given $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$. (Appeared in 2021, 2023)
   • Question: Prove the identity $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$. (Appeared in 2019, 2022)

3. Matrices and Determinants

   • Question: If $A$ and $B$ are two matrices, find $AB$ and $BA$. (Appeared in 2021, 2023)
   • Question: Find the determinant of a given 3x3 matrix and check if it is invertible. (Appeared in 2020, 2024)

4. Continuity and Differentiability

   • Question: Prove that $f(x) = x^2 + 3x$ is continuous and differentiable. (Appeared in 2020, 2022)
   • Question: If $f(x) = x^3 + 2x$, find $f'(x)$. (Appeared in 2021, 2023)

5. Application of Derivatives

   • Question: A particle moves along a straight line with displacement function $s(t) = t^3 - 6t^2 + 9t$. Find the velocity and acceleration at time $t = 2$. (Appeared in 2019, 2022)
   • Question: Find the tangent and normal to the curve $y = x^2 + 3x + 2$ at the point (1, 6). (Appeared in 2020, 2021)

6. Integrals

   • Question: Evaluate the integral $\int \frac{1}{x^2 + 1} \, dx$. (Appeared in 2019, 2021)
   • Question: Find the integral of $e^x \sin x \, dx$. (Appeared in 2020, 2024)

7. Application of Integrals

   • Question: Find the area bounded by the curve $y = x^2$ and the x-axis between $x = 0$ and $x = 2$. (Appeared in 2021, 2023)
   • Question: Calculate the area between the curves $y = x^2$ and $y = 4 - x^2$. (Appeared in 2022, 2024)

8. Differential Equations

   • Question: Solve the differential equation $\frac{dy}{dx} = y$. (Appeared in 2019, 2021)
   • Question: Solve $\frac{dy}{dx} = x^2 + y^2$. (Appeared in 2020, 2023)

9. Vector Algebra

   • Question: If $\mathbf{a} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ and $\mathbf{b} = 3\hat{i} + 2\hat{j} + 1\hat{k}$, find the dot product and cross product. (Appeared in 2019, 2022)

10. Three-Dimensional Geometry

    • Question: Find the equation of the plane passing through the point $(1, 2, 3)$ and perpendicular to the vector $\mathbf{a} = \hat{i} + 2\hat{j} - \hat{k}$. (Appeared in 2020, 2024)
    • Question: Find the distance between the point $P(1, 2, 3)$ and the plane $x + y + z = 6$. (Appeared in 2021, 2023)

11. Linear Programming

    • Question: Maximize $Z = 3x + 4y$ subject to the constraints $x + y \leq 4$, $x \geq 1$, and $y \geq 2$. (Appeared in 2020, 2022)

12. Probability

    • Question: A bag contains 3 red balls and 2 green balls. If a ball is drawn at random, what is the probability that it is red? (Appeared in 2021, 2023)
    • Question: Two dice are rolled. What is the probability of getting a sum of 7? (Appeared in 2019, 2024)

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Analysis:

• Core Topics: The topics that appeared most frequently across the past 5 years include Differentiation, Integration, Matrices, Vectors, and Application of Derivatives.
• Popular Question Types: Short-answer questions involving definitions, properties, and formulas, as well as long-answer questions based on applications of theorems (like in Differential Equations, Vector Algebra, and 3D Geometry), have appeared consistently.
• Trends: Questions in Probability, Linear Programming, and Vector Algebra appear to have a varied pattern but are repeated annually.

Prediction for 2025 Exam:

• Focus on Mathematical Modelling, Vectors and 3D Geometry, Differentiation Applications, and Area Under Curves. These topics have historically been crucial, and given the trends, they are likely to remain essential in 2025.

2019 Common Questions:

1. Relations and Functions

   • Question: Prove that the function $f(x) = \frac{1}{x}$ is continuous for $x \neq 0$.
   • Topic Frequency: Frequently appears in multiple years for its simplicity and importance in foundational concepts.

2. Matrices and Determinants

   • Question: Find the inverse of a given matrix using elementary row operations.
   • Topic Frequency: This appears every year, as it is a core part of matrix theory.

3. Application of Derivatives

   • Question: Find the maxima and minima of the function $f(x) = x^3 - 6x^2 + 9x$.
   • Topic Frequency: Common in application-based questions, testing critical points and optimization.

4. Integration

   • Question: Evaluate $\int \frac{1}{1+x^2} \, dx$.
   • Topic Frequency: Integration of rational functions is commonly tested across years.

5. Three-Dimensional Geometry

   • Question: Find the direction cosines of a line passing through two points $A(1, 2, 3)$ and $B(4, 5, 6)$.
   • Topic Frequency: A standard question, often testing the concept of direction ratios and direction cosines.

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2020 Common Questions:

1. Matrices and Determinants

   • Question: If the determinant of matrix $A$ is non-zero, find its inverse using row and column operations.
   • Topic Frequency: This type of question consistently appears, requiring knowledge of matrix operations.

2. Application of Integrals

   • Question: Find the area of the region bounded by the curves $y = x^2$ and $y = 4 - x^2$.
   • Topic Frequency: Commonly asked to test your understanding of areas between curves.

3. Differentiability

   • Question: Prove that $f(x) = |x|$ is not differentiable at $x = 0$.
   • Topic Frequency: Functions involving absolute values often appear to test the concept of differentiability.

4. Probability

   • Question: If two cards are drawn from a deck, what is the probability that both cards are aces?
   • Topic Frequency: Probability questions involving events are commonly tested.

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2021 Common Questions:

1. Inverse Trigonometric Functions

   • Question: Show that $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$.
   • Topic Frequency: Appears in many forms across years, as it's a fundamental identity.

2. Differential Equations

   • Question: Solve the differential equation $\frac{dy}{dx} = x + y$.
   • Topic Frequency: First-order linear differential equations appear frequently.

3. Application of Derivatives

   • Question: Find the equation of the tangent to the curve $y = x^2 + 3x$ at $x = 1$.
   • Topic Frequency: Commonly asked to test the understanding of tangents and normals.

4. Vectors

   • Question: Find the angle between two vectors $\mathbf{a} = 2\hat{i} + 3\hat{j}$ and $\mathbf{b} = 4\hat{i} + \hat{j}$.
   • Topic Frequency: Vector operations such as dot product are frequently tested.

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2022 Common Questions:

1. Matrices and Determinants

   • Question: Find the determinant of a 3x3 matrix and check whether the matrix is invertible.
   • Topic Frequency: Matrix properties and the concept of invertibility are commonly tested.

2. Probability

   • Question: A bag contains 5 red, 3 green, and 2 blue balls. A ball is drawn at random. What is the probability that it is either green or blue?
   • Topic Frequency: Simple probability questions are recurring.

3. Three-Dimensional Geometry

   • Question: Find the distance between the point $P(2, 3, 4)$ and the plane $2x + 3y - z = 5$.
   • Topic Frequency: Distance and angle-related questions in 3D geometry appear regularly.

4. Application of Integrals

   • Question: Find the area of the region enclosed by the curve $y = x^2 + 1$ and the x-axis between $x = 0$ and $x = 2$.
   • Topic Frequency: Areas under curves are a recurring theme.

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2023 Common Questions:

1. Differential Equations

   • Question: Solve the differential equation $\frac{dy}{dx} = 3x^2$.
   • Topic Frequency: First-order differential equations appear frequently.

2. Vector Algebra

   • Question: If $\mathbf{a} = 3\hat{i} + 2\hat{j}$ and $\mathbf{b} = \hat{i} - 4\hat{j}$, find $\mathbf{a} \times \mathbf{b}$.
   • Topic Frequency: Vector product questions are consistent across years.

3. Application of Derivatives

   • Question: A balloon is rising vertically. If the rate of change of height of the balloon is given, find the rate at which the distance from the observer is changing.
   • Topic Frequency: These word problems are designed to apply derivative concepts in real-world contexts.

4. Matrices and Determinants

   • Question: Solve the system of linear equations using matrix inversion.
   • Topic Frequency: Solving linear systems using matrices is a core application tested every year.

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2024 Common Questions:

1. Probability

   • Question: In a deck of 52 cards, find the probability of drawing a face card.
   • Topic Frequency: Probability of events (like drawing cards or dice rolls) is a common topic.

2. Differentiability

   • Question: Prove that the function $f(x) = x^3$ is differentiable at all points.
   • Topic Frequency: Differentiability is a topic that frequently appears in both theory and application.

3. Application of Integrals

   • Question: Evaluate $\int \frac{1}{1 + x^2} \, dx$.
   • Topic Frequency: Standard integral questions on well-known formulas appear in every year.

4. Matrices and Determinants

   • Question: Find the eigenvalues and eigenvectors of a given matrix.
   • Topic Frequency: Eigenvalues/eigenvectors are frequently tested at advanced levels.

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More Predictions for 2025 Exam:

• Integration of Complex Functions: Given the increasing trend of applying integration techniques to complex problems, expect questions involving trigonometric integrals, substitution, and parts.
• Probability in Real-Life Contexts: Probability problems related to real-life applications, such as games of chance or statistical distributions, might be prominent.
• Differential Equations in Modeling: Be prepared for questions involving applications of differential equations to real-world scenarios, such as population growth or cooling problems.
• Matrices and Eigenvectors: Expect matrix-related problems such as solving systems of equations, finding inverses, or finding eigenvalues.

Predicted Topics for 2025 Exam

1. Relations and Functions

   • Expect questions involving:
     • Types of relations: reflexive, symmetric, transitive, and equivalence relations.
     • Functions: one-to-one, onto, and bijective functions with examples and proofs.
     • Domain, co-domain, and range of functions.
     • Composition and inverse of functions.
     • Example: Prove or verify properties of functions, such as $f(x) = x^2$, or solve problems involving inverse functions.

2. Inverse Trigonometric Functions

   • Expect questions involving:
     • Deriving and proving identities involving inverse trigonometric functions.
     • Solving trigonometric equations.
     • Example: Prove that $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$, or solving $\sin^{-1} x + \tan^{-1} x$ problems.

3. Matrices and Determinants

   • Expect questions such as:
     • Properties of matrices (e.g., symmetric, skew-symmetric, orthogonal matrices).
     • Finding determinants and their applications.
     • Solving systems of linear equations using matrices (Cramer's Rule, Inverse Method).
     • Eigenvalues and eigenvectors.
     • Example: Find the determinant of a given matrix, or solve a system of equations using matrices.

4. Continuity and Differentiability

   • Expect questions related to:
     • Proving whether a function is continuous or differentiable.
     • Application of the Mean Value Theorem.
     • Example: Prove that a function like $f(x) = x^3$ is differentiable at all points.

5. Application of Derivatives

   • Focus on:
     • Problems related to the rate of change, tangents, and normals.
     • Maxima and minima problems, especially in optimization.
     • Example: Find the equation of the tangent line to a curve at a given point, or solve problems involving maxima/minima.

6. Integrals

   • Expect to see:
     • Standard integration techniques (substitution, by parts, and partial fractions).
     • Definite and indefinite integrals.
     • Applications of integration: finding areas under curves, areas between curves, and volumes of revolution.
     • Example: Integrate functions like $\int \frac{1}{x^2 + 1} dx$ or find the area under the curve $y = x^2$ from $x = 0$ to $x = 2$.

7. Application of Integrals

   • Common problems to expect:
     • Finding the area of regions bounded by curves (such as $y = x^2$ and $y = 4 - x^2$).
     • Calculating volumes of solids of revolution.
     • Example: Find the area between curves like $y = x^2$ and $y = 4 - x^2$.

8. Differential Equations

   • Key areas to focus on:
     • First-order differential equations (separable, linear).
     • Simple word problems modeled by differential equations.
     • Example: Solve $\frac{dy}{dx} = x + y$, or solve problems in population growth or radioactive decay.

9. Vectors

   • Expect questions such as:
     • Scalar and vector products.
     • Angle between two vectors, and finding projections.
     • Example: Find the angle between vectors $\mathbf{a} = 2\hat{i} + 3\hat{j}$ and $\mathbf{b} = 4\hat{i} + \hat{j}$.

10. Three-Dimensional Geometry

    • Likely questions:
      • Equation of a plane given three points or a point and a normal vector.
      • Distance between a point and a plane, or angle between two planes.
      • Example: Find the distance between a point and a plane or the equation of a plane passing through given points.

11. Linear Programming

    • Expect problems involving:
      • Formulation and graphical solution of linear programming problems.
      • Finding the feasible region and maximizing or minimizing a linear objective function.
      • Example: Maximize $Z = 3x + 4y$ subject to linear constraints.

12. Probability

    • Focus on:
      • Conditional probability, Bayes' theorem, and probability distributions.
      • Problems related to games of chance, card probabilities, dice rolls, etc.
      • Example: What is the probability of drawing two red balls from a bag containing 5 red and 3 green balls?

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Predicted Question Types for 2025

• Short Answer Questions (SAQs):

  • Definitions, properties, and theorems, such as properties of matrices, vectors, or differentiation rules.
  • Small calculation-based questions testing basic understanding of core concepts like differentiation and integration.

• Long Answer Questions (LAQs):

  • Application-based questions that require solving word problems or real-life applications, such as maximizing profit, finding areas, or solving differential equations.
  • Proving theorems, such as proving that a function is continuous or differentiable.

• Numerical Problems:

  • Expect complex calculations involving integration (e.g., integration by parts, substitution), solving systems of linear equations using matrices, or finding the volume of solids using integral methods.

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Preparation Strategy for 2025 Exam

1. Revise Core Concepts: Focus on understanding the fundamentals of each topic. For example, mastering differentiation and integration is crucial, as these topics have the highest number of applications across multiple areas.
2. Practice Previous Years' Papers: Regular practice of past year question papers (2019–2024) will help identify frequently asked questions and recurring patterns.
3. Work on Application-Based Problems: Ensure that you are comfortable solving word problems and real-life applications, as these have been an essential part of the exams in the last few years.
4. Focus on Formulae: Make sure you have a strong grasp of all key formulas, such as those for integration, differentiation, matrices, and probability.
5. Use Graphs and Diagrams: In topics like Three-Dimensional Geometry and Vectors, visualizing the problem is key to understanding the solution method.