Unit Test Marathon for HSC - Maharashtra Board Mathematics

Mathematical Logic, Matrices, Differentiation, and Applications of Derivatives

Statements, Truth Values, and Open Sentences

A statement is a declarative or assertive sentence that is either true or false, but not both. Statements are denoted by symbols such as $P$, $Q$, or $R$. The truth value of a statement is $T$ (true) if the statement is correct, and $F$ (false) if it is incorrect.

• Examples:
  • "2 is an even number" is a statement with truth value $T$.
  • "10 is an odd number" is a statement with truth value $F$.
  • "What is your name?" is not…

Mathematical Logic, Matrices, Differentiation, and Applications of Derivatives

Statements, Truth Values, and Open Sentences

A statement is a declarative or assertive sentence that is either true or false, but not both. Statements are denoted by symbols such as $P$, $Q$, or $R$. The truth value of a statement is $T$ (true) if the statement is correct, and $F$ (false) if it is incorrect.

• Examples:
  • "2 is an even number" is a statement with truth value $T$.
  • "10 is an odd number" is a statement with truth value $F$.
  • "What is your name?" is not a statement; it is an interrogative sentence.
• Open Sentence: An expression like $x + 4 = 5$ is not a statement unless $x$ is specified, as its truth value depends on $x$.

Logical Connectives and Compound Statements

Logical connectives are used to combine two or more statements to form compound statements. There are five main logical connectives: AND, OR, IF-THEN, IF AND ONLY IF, and NOT.

• Connectives and Symbols:
  • AND (Conjunction): $\wedge$
  • OR (Disjunction): $\vee$
  • IF-THEN (Implication): $\rightarrow$
  • IF AND ONLY IF (Biconditional): $\leftrightarrow$
  • NOT (Negation): $\neg$
• Simple Statement: Cannot be split further.
• Compound Statement: Combination of two or more simple statements.

Truth Tables and Logical Equivalence

Truth tables are used to determine the truth value of compound statements for all possible truth values of their components. Logical equivalence occurs when two statements have identical truth tables.

• AND ($P \wedge Q$): True only if both $P$ and $Q$ are true.
• OR ($P \vee Q$): False only if both $P$ and $Q$ are false.
• IF-THEN ($P \rightarrow Q$): False only if $P$ is true and $Q$ is false.
• IF AND ONLY IF ($P \leftrightarrow Q$): True if both $P$ and $Q$ have the same truth value.
• NOT ($\neg P$): True if $P$ is false, and vice versa.
• Negation of Negation: $\neg(\neg P) = P$.

Tautology, Contradiction, and Contingency

A tautology is a statement pattern that is always true, a contradiction is always false, and a contingency is sometimes true and sometimes false.

• Tautology: All truth values in the final column are $T$.
• Contradiction: All truth values are $F$.
• Contingency: Mixture of $T$ and $F$ in the truth table.

Converse, Inverse, Contrapositive, and Dual

For an implication $P \rightarrow Q$:

• Converse: $Q \rightarrow P$
• Inverse: $\neg P \rightarrow \neg Q$
• Contrapositive: $\neg Q \rightarrow \neg P$

The dual of a logical expression is obtained by interchanging AND and OR, $T$ and $F$, and universal and existential quantifiers.

Laws of Algebra of Statements

Several laws govern the manipulation of logical statements:

• De Morgan’s Laws:
  • $\neg(P \wedge Q) = \neg P \vee \neg Q$
  • $\neg(P \vee Q) = \neg P \wedge \neg Q$
• Identity Laws:
  • $P \wedge T = P$
  • $P \vee F = P$
• Complementary Laws:
  • $P \wedge \neg P = F$
  • $P \vee \neg P = T$
• Absorption Laws:
  • $P \vee (P \wedge Q) = P$
  • $P \wedge (P \vee Q) = P$
• Commutative, Associative, and Distributive Laws are also applicable as described.

Switching Circuits and Input-Output Tables

Switching circuits can be represented using logical statements. Series circuits correspond to AND operations, and parallel circuits correspond to OR operations.

• Series (AND): Both switches must be closed for the lamp to light.
• Parallel (OR): Either switch closed will light the lamp.
• Input-Output Table: Replace $T$ with $1$ and $F$ with $0$.

Matrices: Types, Order, and Operations

A matrix is a rectangular array of numbers arranged in rows and columns. The order of a matrix is $m \times n$ (rows $\times$ columns).

• Types:
  • Row Matrix: Single row.
  • Column Matrix: Single column.
  • Square Matrix: Number of rows equals number of columns.
  • Identity Matrix ($I$): Diagonal elements are $1$, others $0$.
• Trace: Sum of diagonal elements.
• Transpose: Rows become columns.
• Determinant: For $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det(A) = ad - bc$.

Invertibility and Finding the Inverse of a Matrix

A matrix is invertible if its determinant is not zero. The inverse is found using the adjoint method or elementary transformations.

• If $\det(A) = 0$: $A$ is singular, not invertible, $A^{-1}$ does not exist.
• If $\det(A) \neq 0$: $A$ is non-singular, invertible, $A^{-1}$ exists.
• Inverse Formula: $A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)$
• Adjoint: Transpose of the cofactor matrix.

Differentiation: Basic Formulas and Rules

Differentiation is the process of finding the derivative of a function. Key formulas and rules include:

• Basic Derivatives:
  • $\frac{d}{dx}(c) = 0$
  • $\frac{d}{dx}(x^n) = n x^{n-1}$
  • $\frac{d}{dx}(a^x) = a^x \log a$
  • $\frac{d}{dx}(e^x) = e^x$
  • $\frac{d}{dx}(\log x) = 1/x$
• Trigonometric Derivatives:
  • $\frac{d}{dx}(\sin x) = \cos x$
  • $\frac{d}{dx}(\cos x) = -\sin x$
  • $\frac{d}{dx}(\tan x) = \sec^2 x$
  • $\frac{d}{dx}(\cot x) = -\csc^2 x$
  • $\frac{d}{dx}(\sec x) = \sec x \tan x$
  • $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• Inverse Trigonometric Derivatives:
  • $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$
  • $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$
  • $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$
  • $\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2}$
  • $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$
  • $\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2-1}}$
• Product Rule: $\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}$
• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
• Chain Rule (Composite Function): If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

Applications of Derivatives: Tangents, Normals, and Rate Problems

The derivative at a point gives the slope of the tangent to the curve at that point. The normal is perpendicular to the tangent.

• Equation of Tangent: $y - y_1 = m(x - x_1)$, where $m = \frac{dy}{dx}$ at $(x_1, y_1)$.
• Equation of Normal: $y - y_1 = m'(x - x_1)$, where $m' = -1/m$.
• Rate Problems: If $y$ changes $k$ times as fast as $x$, then $\frac{dy}{dt} = k \frac{dx}{dt}$.

Maxima, Minima, and Optimization

To find maxima or minima, set the first derivative to zero and use the second derivative test.

• First Derivative Test: Set $f'(x) = 0$ to find critical points.
• Second Derivative Test:
  • If $f''(x) > 0$, minimum at $x$.
  • If $f''(x) < 0$, maximum at $x$.
• Optimization Problems: Formulate the quantity to be maximized/minimized, express in terms of one variable, differentiate, set derivative to zero, and solve.

Mean Value Theorems and Increasing/Decreasing Functions

The Lagrange Mean Value Theorem (LMVT) states that if $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

• Rolle’s Theorem: If $f(a) = f(b)$, then $f'(c) = 0$ for some $c \in (a, b)$.
• Increasing Function: $f'(x) > 0$ (strictly increasing), $f'(x) \geq 0$ (increasing).
• Decreasing Function: $f'(x) < 0$ (strictly decreasing), $f'(x) \leq 0$ (decreasing).

Logarithmic and Parametric Differentiation

Logarithmic differentiation is used when the function is a product, quotient, or power of functions.

• Steps:
  • Take logarithm on both sides.
  • Differentiate using chain rule.
  • Simplify to find $\frac{dy}{dx}$.
• Parametric Differentiation: If $x = f(t)$ and $y = g(t)$, then $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

Approximation and Related Rates

Approximation uses the formula $f(a + h) \approx f(a) + h f'(a)$ for small $h$. Related rates involve differentiating with respect to time.

• Example: If surface area $A$ of a sphere increases at rate $\frac{dA}{dt}$, and $A = 4\pi r^2$, then $\frac{dA}{dt} = 8\pi r \frac{dr}{dt}$.
• Volume of Sphere: $V = \frac{4}{3}\pi r^3$, so $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

Key Formula Table

All formulas and methods above are directly stated or derived in the transcript.