Rbse Solutions Class 12 Maths Chapter 4 Miscellaneous Exercises
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
Rbse Solutions Class 12 Maths: Exercise 4.5
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
Rbse Solutions Class 12 Maths: Exercise 4.4
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. $A = \begin{bmatrix} 1 & 2 … Read more
Rbse Solutions Class 12 Maths: Exercise 4.3
Here are the solutions for Exercise 4.3, focusing on finding Minors and Cofactors and using them to evaluate determinants. The Minor ($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
Rbse Solutions Class 12 Maths: Exercise 4.2
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
Rbse Solutions Class 12 Maths: Exercise 4.1
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
Rbse Solutions Class 12 Maths Chapter 3 Miscellaneous : Matrices
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
Rbse Solutions Class 12 Maths: Exercise 3.3
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
Rbse Solutions Class 12 Maths: Exercise 3.2 (Matrix Operations)
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 Chapter 3, Exercise 3.1(Matrices).
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
Rbse Solutions Class 12 Maths Chapter 2 Miscellaneous Exercise
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
Rbse Solutions Class 12 Maths: Exercise 2.2 (Properties and Identities)
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
Rbse Solutions Class 12 Maths Exercise 2.1 (Principal Values)
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
Rbse Solutions Class 12 Maths :Chapter 1 Miscellaneous Exercise
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
Rbse Solutions Class 12 Maths: Exercise 1.2 (Functions)
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
Rbse Solutions Class 12 Maths Chapter 1: Exercise 1.1 (Relations and Functions)
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
Class 12 Physics Books, Solutions, Syllabus & Notes PDF (RBSE/CBSE)
Download RBSE Class 12 Physics Books PDF for 2026-27. Get chapter-wise NCERT solutions, latest marking scheme (56 marks theory), and 50-district updated notes. Quick Overview: This guide provides the complete RBSE Class 12 Physics curriculum for 2026. It includes PDF downloads for both Hindi and English mediums, a detailed blueprint of the 56-mark theory paper, … Read more
NCERT Solutions for Class 10 Social Science in Hindi medium updated 2021-22
ncert कक्षा 10 सामाजिक विज्ञान हल Get here all NCERT Solutions for Class 10 History in Hindi Chapter 1: यूरोप में राष्ट्रवाद का उदय Chapter 2: भारत में राष्ट्रवाद Chapter 3: भूमंडलीकृत विश्व का बनना Chapter 4: औद्योगिकीकरण की युग Chapter 2 The Nationalist Movement in Indo-China (इंडो-चाइना में राष्ट्रवादी आंदोलन) Chapter 5: मुद्रण संस्कृति … Read more
Rbse Class 12 Chemistry Chapter 3: Chemical Kinetics Pyqs
kinetics 🚀 $\text{RBSE}$ Class 12 Chemical Kinetics $\text{PYQs}$: Your Complete Guide (2013-2023) Mastering Chemical Kinetics is essential for high scores in the $\text{RBSE}$ Chemistry Board Exam. This guide covers the most critical Previous Year Questions ($\text{PYQs}$) on reaction order, rate laws, half-life, and activation energy, with detailed solutions. 1. Reaction Order and Rate Law $\text{PYQs}$ … Read more
RBSE Class 12 Chemistry Chapter 1 Solutions: Solutions (विलयन)
The study of Solutions (विलयन) now forms the foundation of Physical Chemistry in the latest RBSE Class 12 Syllabus. This crucial chapter (Unit 1) has a significant mark weightage (typically 6-7 marks) in the board exams and is heavily focused on numerical problem-solving and the application of four major laws. Mastering the step-by-step NCERT/RBSE Solutions … Read more
Rbse Solutions for Class 9 Maths Chapter 12 Exercise 12.1 | Statistics
Rbse Solutions for Class 9 Maths Chapter 12 Exercise 12.1 | STATISTICS 1. A survey conducted by an organisation for the cause of illness and death among the women between the ages 15 – 44 (in years) worldwide found the following figures (in %): S.No. Causes Female fatality rate (%) 1. Reproductive health conditions 31.8 … Read more
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.4 | Volume of Sphare
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.4 | Volume of Sphare 1. Find the surface area of a sphere of radius: (i) $10.5 \text{ cm}$ (ii) $5.6 \text{ cm}$ (iii) $14 \text{ cm}$ Solution The formula for the surface area of a sphere is $\mathbf{A = 4\pi r^2}$. (i) Radius $r = … Read more
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.3 | Volume of Cone
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.3 | Volume of Cone 1. Volume of the Right Circular Cone (i) Radius $r = 6 \text{ cm}$, Height $h = 7 \text{ cm}$ $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times \frac{22}{7} \times (6)^2 \times 7$$ $$V = \frac{1}{3} \times 22 \times 36 … Read more
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.2 | Surface Area of a Sphere
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.2 | Surface Area of a Sphere 1. Find the surface area of a sphere of radius (i) 10.5cm (ii) 5.6cm (iii) 14cm (Assume π=22/7) Solution: Formula: Surface area of a sphere (SA) = 4πr2 (i) Radius of a sphere, r = 10.5 cm SA = … Read more
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.1 | Surface Area of a Right Circular Cone
Rbse Solutions for Class 9 Maths Chapter 11 Exercise 11.1 | Surface Area of a Right Circular Cone 1. Diameter of the base of a cone is 10.5 cm, and its slant height is 10 cm. Find its curved surface area. (Assume π=22/7) Solution: Radius of the base of cone = diameter/ 2 = (10.5/2)cm … Read more
Rbse Solutions for Class 9 Maths Chapter 10 Exercise 10.1 | HERON’S FORMULA
Rbse Solutions for Class 9 Maths Chapter 10 Exercise 10.1 Master Class 9 Heron’s Formula with step-by-step NCERT solutions for Exercise 10.1. Learn to calculate triangle area using side lengths and the semi-perimeter $s$. Essential for exams! 1. A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area … Read more
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.3
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.3 | Cyclic Quadrilaterals | Angle Subtended by an Arc of a Circle 1. In Fig. 9.23, A, B and C are three points on a circle with centre O, such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other … Read more
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.2
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.2 | Perpendicular from the Centre to a Chord ,Equal Chords and their Distances from the Centre 1. Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centres is 4 cm. Find the length of the common … Read more
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.1 | Circles
Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.1 | Circles Master Rbse Solutions for Class 9 Maths Chapter 9 Exercise 9.1 and in-depth proofs of key theorems. Learn how equal chords subtend equal angles at the centre and vice-versa. Perfect for exam preparation and concept clarity! 1. Recall that two circles are congruent … Read more
Rbse Solutions for Class 9 Maths Chapter 8 Exercise 8.2 | The Mid-point Theorem
Rbse Solutions for Class 9 Maths Chapter 8 Exercise 8.2 | The Mid-point Theorem 1. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.20). AC is a diagonal. Show that:(i) SR || AC and SR = 1/2 AC(ii) PQ = … Read more
Rbse Solutions for Class 9 Maths Chapter 8 Exercise 8.1 | Quadrilaterals
Rbse Solutions for Class 9 Maths Chapter 8 Exercise 8.1 | Quadrilaterals Master Class 9 Maths Chapter 8 Exercise 8.1 with step-by-step solutions. Learn how to prove properties of parallelograms, rhombuses, and rectangles with clear explanations and diagrams. Perfect for CBSE exam preparation. 1. If the diagonals of a parallelogram are equal, then show that … Read more