MATHS Final Exam Marathon 1 2026 - Gujarat Board Mathematics

Relations, Functions, Trigonometric Inverse Functions, Matrices, Determinants, Continuity, and Differentiation

Blueprint and Marking Scheme

The first part of the syllabus consists of six chapters, with a focus on 2-mark, 3-mark, and 4-mark subjective questions. The marking scheme for each chapter is explicitly stated, and students should prioritize subjective questions for maximum marks.

• Chapter 1: 3 MCQs, one 3-mark question; total 6 marks.
• Chapter 2: 4 MCQs, two 2-mark questions; total 8 marks.
• Chapter 3: 4 MCQs, one 3-mark, one 4-mark question; total 11 marks.
• Chapter 4: 4 MCQs, one 3-mark, one 4-mark question; total…

Relations, Functions, Trigonometric Inverse Functions, Matrices, Determinants, Continuity, and Differentiation

Blueprint and Marking Scheme

The first part of the syllabus consists of six chapters, with a focus on 2-mark, 3-mark, and 4-mark subjective questions. The marking scheme for each chapter is explicitly stated, and students should prioritize subjective questions for maximum marks.

• Chapter 1: 3 MCQs, one 3-mark question; total 6 marks.
• Chapter 2: 4 MCQs, two 2-mark questions; total 8 marks.
• Chapter 3: 4 MCQs, one 3-mark, one 4-mark question; total 11 marks.
• Chapter 4: 4 MCQs, one 3-mark, one 4-mark question; total 11 marks.
• Chapter 5: 3 MCQs, one 2-mark, one 3-mark, one 4-mark question; total 9 marks.
• Chapter 6: 4 MCQs, one 3-mark, one 4-mark question; total 11 marks.

Relations: Reflexive, Symmetric, Transitive

A relation $R$ on a set $A$ is analyzed for reflexivity, symmetry, and transitivity using explicit definitions and examples.

• Reflexive: For all $x \in A$, $(x, x) \in R$.
• Symmetric: If $(x, y) \in R$, then $(y, x) \in R$.
• Transitive: If $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$.

Example: For $A = \{1,2,3,4,5,6\}$, $R = \{(x, y): y$ is divisible by $x\}$:

• Reflexive: $x$ divides $x$ for all $x$.
• Symmetric: Not always, as $y$ divisible by $x$ does not imply $x$ divisible by $y$.
• Transitive: If $y$ divisible by $x$ and $z$ divisible by $y$, then $z$ divisible by $x$.

Types of Functions: One-One and Onto

Functions are tested for injectivity (one-one) and surjectivity (onto) using explicit examples and definitions.

• One-One (Injective): If $f(x_1) = f(x_2)$ implies $x_1 = x_2$.
• Onto (Surjective): Every element in the codomain is mapped by some element in the domain.

Example: $f: \mathbb{R} \to \mathbb{R}$, $f(x) = [x]$ (greatest integer function):

• Not one-one: $f(1.2) = f(1.4) = 1$ but $1.2 \neq 1.4$.
• Not onto: Range is integers, codomain is reals.

Trigonometric Inverse Functions: Properties and Proofs

Trigonometric inverse functions are analyzed for their domains, ranges, and key identities, with explicit derivations and conditions.

• Key Identity: $3\cos^{-1}x = \cos^{-1}(4x^3 - 3x)$ for $x \in [\frac{1}{2}, 1]$.

  • Let $x = \cos\theta$, then $3\cos^{-1}x = 3\theta$ and $4x^3 - 3x = \cos 3\theta$.
  • The result holds as $3\theta$ remains in the principal value range.

• Domain and Range Table:

• Important Results:
  • $\tan^{-1}\left(\frac{1-x}{1+x}\right) = \frac{\pi}{4} - \tan^{-1}x$ for $x > 0$.
  • $\tan^{-1}\left(\frac{3a x - x^3}{a^3 - 3a x^2}\right) = 3\tan^{-1}\left(\frac{x}{a}\right)$ for $-\frac{a}{3} < x < a$.

Matrices: Multiplication, Inverse, and Properties

Matrix operations are performed with explicit steps for multiplication, finding inverses, and using adjoints.

• Matrix Multiplication: Possible when the number of columns in the first matrix equals the number of rows in the second.
• Inverse of a Matrix: $A^{-1} = \frac{1}{|A|} \text{adj}(A)$, where $|A| \neq 0$.
• Adjoint: For a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

Example: For $A = \begin{pmatrix} 2 & 3 \\ -4 & 6 \end{pmatrix}$,

• $|A| = 2 \times 6 - (-4) \times 3 = 12 + 12 = 24$.
• $\text{adj}(A) = \begin{pmatrix} 6 & -3 \\ 4 & 2 \end{pmatrix}$.
• $A^{-1} = \frac{1}{24} \begin{pmatrix} 6 & -3 \\ 4 & 2 \end{pmatrix}$.

Determinants: Properties and Applications

Determinants are used for solving systems of equations, finding inverses, and checking matrix singularity.

• Determinant of $2 \times 2$ matrix: $|A| = ad - bc$.
• Determinant of $3 \times 3$ matrix: Use Sarrus' rule or cofactor expansion.
• Properties:
  • If $|A| = 0$, $A$ is singular and $A^{-1}$ does not exist.
  • $|kA| = k^n|A|$ for an $n \times n$ matrix.

Example: For $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $|A| = 1 \times 4 - 2 \times 3 = 4 - 6 = -2$.

Continuity: Definitions and Tests

A function $f(x)$ is continuous at $x = a$ if the left-hand limit, right-hand limit, and $f(a)$ are all equal.

• Test for Continuity at $x = a$:
  1. Compute $\lim_{x \to a^-} f(x)$.
  2. Compute $\lim_{x \to a^+} f(x)$.
  3. Compute $f(a)$.
  4. If all three are equal, $f(x)$ is continuous at $x = a$.

Example: For $f(x) = \frac{1}{x-5}$, $f$ is continuous for all $x \neq 5$.

• Piecewise Function: For $f(x) = x^2 - 3$ if $x \leq 2$, $x^2 + 1$ if $x > 2$:
  • Check continuity at $x = 2$ by evaluating both sides and $f(2)$.

Differentiation: Rules and Applications

Differentiation is performed using standard rules, chain rule, product rule, and quotient rule as explicitly described.

• Product Rule: If $y = u \cdot v$, then $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$.
• Quotient Rule: If $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$.
• Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

Examples:

• For $y = x^{\sin x}$, use logarithmic differentiation:

  • $\ln y = \sin x \ln x$
  • $\frac{1}{y} \frac{dy}{dx} = \cos x \ln x + \frac{\sin x}{x}$
  • $\frac{dy}{dx} = x^{\sin x} \left(\cos x \ln x + \frac{\sin x}{x}\right)$

• For $y = \log x^{\cos x}$:

  • $\ln y = \cos x \ln x$
  • $\frac{1}{y} \frac{dy}{dx} = -\sin x \ln x + \frac{\cos x}{x}$
  • $\frac{dy}{dx} = x^{\cos x} \left(-\sin x \ln x + \frac{\cos x}{x}\right)$

Application Problems: Rate of Change and Maxima/Minima

Application-based questions involve related rates and optimization, with explicit stepwise solutions.

• Ladder Problem: For a ladder of length $5$ m, with the bottom moving away from the wall at $2$ cm/s when $x = 4$ m, use $x^2 + y^2 = 25$ and differentiate to find $\frac{dy}{dt}$.

• Sand Cone Problem: For a sand cone with $dv/dt = 12$ cm$^3$/s, $h = 4$ cm, and $r = 6h$, use $V = \frac{1}{3}\pi r^2 h$ and substitute $r$ in terms of $h$ to find $\frac{dh}{dt}$.

• Maxima/Minima: For $f(x) = -2x^3 - 9x^2 - 12x + 1$, find $f'(x)$, set to zero, solve for $x$, and test intervals for increasing/decreasing behavior.

• Absolute Maximum/Minimum: For $f(x)$ on $[a, b]$, evaluate $f(a)$, $f(b)$, and $f$ at critical points in $[a, b]$; the largest is the maximum, the smallest is the minimum.

• Local Maximum/Minimum: Use the second derivative test: if $f''(x) < 0$ at a critical point, it's a maximum; if $f''(x) > 0$, it's a minimum.

• Example: For $f(x) = \sin 2x$ on $[0, 2\pi]$, critical points are found by solving $f'(x) = 0$.

Key Exam Takeaways

• Focus on subjective questions for maximum marks; MCQs are fewer in number.
• For each chapter, prioritize the types of questions and marks distribution as explicitly stated.
• Practice all methods and steps as shown, especially for relations, function types, trigonometric identities, matrix operations, determinants, continuity, and differentiation.
• For application and word problems, always write the explicit steps, formulas, and substitutions as demonstrated.
• Use the provided domain and range table for trigonometric inverse functions in all relevant questions.
• For maxima/minima and continuity, always check all required conditions and show all steps for full marks.