Here are the solutions for the Miscellaneous Exercises on Chapter 4, covering determinants, matrix inverse, and systems of equations. 1. Prove that the determinant $\begin{vmatrix} x & \sin \theta & \cos \theta \\ – \sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{vmatrix}$ is independent of $\theta$. Let $D$ be … Read more
Here are the solutions for Exercise 4.5, covering the consistency of linear equations and solving systems using the matrix method. A system of linear equations $AX = B$ is consistent if it has one or infinitely many solutions. Examine the Consistency of the System of Equations (1 to 6) 1. $x + 2y = 2$ … Read more
Here are the detailed solutions for Exercise 4.4, covering the adjoint, inverse of matrices, and related properties. Find adjoint of each of the matrices in Exercises 1 and 2. The adjoint of a matrix $A$ is the transpose of the matrix formed by its cofactors. $\text{adj}(A) = [\text{Cofactors}]^T$. 1. $A = \begin{bmatrix} 1 & 2 … Read more
Here are the solutions for Exercise 4.3, focusing on finding Minors and Cofactors and using them to evaluate determinants. The Minor ($M_{ij}$) of an element $a_{ij}$ is the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column. The Cofactor ($A_{ij}$) of an element $a_{ij}$ is given by $A_{ij} = (-1)^{i+j} M_{ij}$. … Read more
Here are the detailed solutions for Exercise 4.2, which covers finding the area of a triangle, checking for collinearity, and finding the equation of a line using the determinant method. 1. Find the area of the triangle with vertices at the points given: The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and … Read more
Here are the solutions for Exercise 4.1, which involves evaluating determinants of $2 \times 2$ and $3 \times 3$ matrices and verifying determinant properties. 1. Evaluate the determinant: $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$ The determinant of a $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is … Read more
This section covers proofs of matrix properties, solving matrix equations, and applying matrix algebra to real-world problems. Question 1. If $A$ and $B$ are symmetric matrices, prove that $AB – BA$ is a skew symmetric matrix. Solution: Given that $A$ and $B$ are symmetric matrices, by definition: $$A’ = A \quad \text{and} \quad B’ = … Read more
This exercise focuses on the properties of the Transpose of a Matrix ($A’$ or $A^T$), and the definition and application of Symmetric and Skew Symmetric Matrices. Question 1. Find the transpose of each of the following matrices: (i) $A = \begin{bmatrix} 5 \\ 1/2 \\ -1 \end{bmatrix}$ (ii) $B = \begin{bmatrix} 1 & -1 \\ … Read more
This exercise covers fundamental matrix operations: addition, subtraction, scalar multiplication, and matrix multiplication. Question 1. Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find (i) $A+B$, (ii) $A-B$, (iii) … Read more
Rbse Solutions Class 12 Maths Exercise 3.1 Question 1. In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & 5/2 & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements … Read more
Here are the complete solutions for the first 14 questions of the Miscellaneous Exercise on NCERT Class 12 Maths Chapter 2 (Inverse Trigonometric Functions). Find the value of the following: Question 1. $\cos^{-1} \left(\cos \frac{13\pi}{6}\right)$ Solution: We know that the Principal Value Branch (PVB) of $\cos^{-1} x$ is $[0, \pi]$. Since $\frac{13\pi}{6}$ is outside this … Read more
Complete NCERT Solutions for Class 12 Maths Chapter 2, Exercise 2.2. Master identities like 3sin⁻¹x and 2tan⁻¹x, and solve inverse trig equations with step-by-step proofs. Question 1. Prove $3 \sin^{-1} x = \sin^{-1} (3x – 4x^3)$, $x \in [-1/2, 1/2]$ Solution: We use the substitution method. Let $x = \sin \theta$. Then $\theta = \sin^{-1} … Read more
Get complete NCERT Solutions for Class 12 Maths Chapter 2, Exercise 2.1 (Inverse Trigonometric Functions). Master finding the Principal Value of sin⁻¹, cos⁻¹, and tan⁻¹. The Principal Value Branch (PVB) is crucial for these solutions. The PVB for each function is listed below: Function Principal Value Branch (PVB) $\sin^{-1}(x)$ $[-\pi/2, \pi/2]$ $\cos^{-1}(x)$ $[0, \pi]$ $\tan^{-1}(x)$ … Read more
Complete solutions for NCERT Class 12 Maths Chapter 1 Miscellaneous Exercise. Detailed steps for bijection, equivalence relations, and functions like f(x)=x/(1+|x|). Question 1. Show that the function $f: \mathbb{R} \to \{x \in \mathbb{R}: -1 < x < 1\}$ defined by $f(x) = \frac{x}{1 + |x|}$, $x \in \mathbb{R}$ is one-one and onto function. Solution: The … Read more
Are you struggling to prove if a function is Injective or Surjective? Exercise 1.2 is one of the most important sections of the RBSE Class 12 Maths syllabus. In the 2026 board exam, you can expect at least one 3-mark or 4-mark question from this specific exercise. In this guide, we provide the NCERT-based solutions … Read more
If you’ve just opened your NCERT Maths book, Chapter 1 might look a bit intimidating with all those $x, y$ and set brackets. But here’s a secret from someone who has analyzed the RBSE Ajmer Board patterns for years: Exercise 1.1 is actually a scoring goldmine if you understand the logic instead of memorizing the … Read more
Namaste, RBSE Class 12 students! Are you ready to ace your Mathematics board exams? RBSE Solutions for Class 12 Maths Chapter 1: Relations and Functions is a foundational unit that lays the groundwork for many advanced topics in calculus and beyond. Understanding this chapter thoroughly is key to scoring well, and with the updated syllabus … Read more
Get complete, step-by-step solutions for NCERT Class 10 Maths Chapter 9 Exercise 9.1 on Some Applications of Trigonometry (Heights and Distances). Learn to solve problems involving Angle of Elevation and Angle of Depression using trigonometric ratios ($\sin, \cos, \tan$) and specific angle values ($30^\circ, 45^\circ, 60^\circ$). Solutions cover scenarios like finding the height of a … Read more
Get comprehensive solutions for the NCERT Class 12 Maths Chapter 13 Miscellaneous Exercise. Practice advanced problems covering Bayes’ Theorem This exercise reviews key concepts from Probability, including Conditional Probability, Independence, Bayes’ Theorem, and Binomial Distribution. 1. Conditional Probability $P(B|A)$ where $P(A) \ne 0$ The formula for conditional probability is $P(B|A) = \frac{P(A \cap B)}{P(A)}$. (i) … Read more
Find complete, step-by-step solutions for NCERT Class 12 Maths Exercise 13.3 (Probability). Master problems using the Theorem of Total Probability and Bayes’ Theorem This exercise primarily involves the Theorem of Total Probability and Bayes’ Theorem. Theorem of Total Probability $P(E) = P(E|E_1)P(E_1) + P(E|E_2)P(E_2) + \dots + P(E|E_n)P(E_n)$ Bayes’ Theorem $$P(E_i|E) = \frac{P(E|E_i)P(E_i)}{\sum_{j=1}^{n} P(E|E_j)P(E_j)}$$ 1. … Read more
Get detailed solutions for NCERT Class 12 Maths Exercise 13.2 (Probability). Learn to identify and solve problems based on Independent Events using the formula $P(A \cap B) = P(A) \cdot P(B)$. Covers finding probabilities for card draws (with and without replacement), system reliability, and checking for independence using the Multiplication Rule. Essential for Chapter 13. … Read more
Get step-by-step solutions for NCERT Class 12 Maths Exercise 13.1 (Probability). Master the Conditional Probability formula, This exercise focuses on calculating conditional probability, defined as: $$P(E|F) = \frac{P(E \cap F)}{P(F)}, \quad \text{where } P(F) > 0$$ 1. Given $P(E) = 0.6$, $P(F) = 0.3$, and $P(E \cap F) = 0.2$. Find $P(E|F)$ and $P(F|E)$. 2. … Read more
Get detailed, step-by-step solutions for NCERT Class 12 Maths Exercise 12.1 (Linear Programming). Learn to solve LPPs by the Graphical Method, including steps to identify the feasible region (bounded and unbounded), determine corner points, and find the maximum or minimum value of the objective function ($Z$). Includes cases for unique and multiple optimal solutions and … Read more
Find complete solutions for the NCERT Class 12 Maths Miscellaneous Exercise on Three Dimensional Geometry (Chapter 11). Review advanced topics including finding the angle between lines using direction ratios, determining the equation of a line parallel to an axis, calculating the value of $k$ for perpendicular lines, and solving complex problems on Shortest Distance (SD) … Read more
Get detailed solutions for NCERT Class 12 Maths Exercise 11.2 (Three Dimensional Geometry). Learn to find vector and Cartesian equations of a line passing through a point and parallel to a vector. Master conditions for parallel and perpendicular lines, calculating the angle between two lines, and finding the Shortest Distance (SD) between skew lines using … Read more
Get step-by-step solutions for NCERT Class 12 Maths Exercise 11.1 (Three Dimensional Geometry). Master calculating Direction Cosines ($l, m, n$) from angles, converting Direction Ratios ($a, b, c$) to Direction Cosines, and using them to prove the collinearity of three points. Includes finding the direction cosines of the sides of a triangle. Essential for the … Read more
Get detailed solutions for the NCERT Class 12 Maths Miscellaneous Exercise on Vector Algebra (Chapter 10). This exercise covers finding unit vectors, displacement, direction cosines, section formula, collinearity, area of a parallelogram, and applications of the dot and cross product. Ideal for comprehensive revision and securing top marks in vector concepts. This exercise reviews all … Read more
Get detailed solutions for NCERT Class 12 Maths Exercise 10.3 focusing on the Scalar (Dot) Product of vectors. Learn to calculate the angle between two vectors ($\theta$), find the projection of one vector onto another, determine conditions for perpendicularity ($\vec{a} \cdot \vec{b} = 0$), and prove vector properties involving magnitudes and orthogonality. Essential for mastering … Read more
Find complete solutions for NCERT Class 12 Maths Exercise 10.4 covering the Vector (Cross) Product. Learn to calculate This exercise focuses on the vector product (or cross product) of two vectors, defined by $\mathbf{\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \hat{n}}$ and its applications in geometry (Area of Triangle and Parallelogram). 1. Find $|\vec{a} … Read more
Get detailed solutions for NCERT Class 12 Maths Exercise 10.2 (Vector Algebra). Master computing vector magnitude, finding scalar and vector components, calculating unit vectors ($\hat{a}$), and determining direction cosines. Includes solving problems on collinearity, vector addition, section formula (internal and external division), and geometric applications (midpoint, right-angled triangles). Essential for securing marks in Chapter 10. … Read more
Complete solutions for NCERT Class 12 Maths Exercise 10.1 (Vectors). Learn to represent vectors graphically (like displacement) and classify quantities as scalars (e.g., mass, time, power) or vectors (e.g., velocity, acceleration, force). Identify and distinguish between different types of vectors (coinitial, equal, collinear). Essential foundation for the Vectors chapter. This exercise introduces the basic concepts … Read more
Find complete solutions for the NCERT Class 12 Maths Miscellaneous Exercise on Chapter 9 (Differential Equations). This comprehensive set of problems tests mastery across all solution methods: Variable Separable, Homogeneous, and Linear DE forms. Includes identifying Order and Degree, verifying solutions, and solving complex particular solution and application problems. Essential review for final exam preparation. … Read more
Get comprehensive solutions for NCERT Class 12 Maths Exercise 9.5. Learn to solve first-order Linear Differential Equations using the Integrating Factor (IF) method. Practice finding the IF, calculating general solutions, and determining particular solutions by applying initial conditions. Includes solving equations linear in $x$ and the Bernoulli’s form. Essential for complete mastery of Chapter 9. … Read more