Mathematics Intermediate Part I, Chapter 11

• Mathematics: Intermediate Part I, Chapter 11, Important Multiple Choice Questions.

Contents Overview

• Chapter 1 deals with the Number System
• Chapter 2 covers Sets, Functions, and Groups
• Chapter 3 is about Matrices and Determinants
• Chapter 4 focuses on Quadratic Equations
• Chapter 5 discusses Partial Fractions
• Chapter 6 covers Sequences and Series
• Chapter 7 explores Permutations, Combinations, and Probability
• Chapter 8 is about Mathematical Induction and the Binomial Theorem
• Chapter 9 deals with the Fundamentals of Trigonometry
• Chapter 10 covers Trigonometric Identities
• Chapter 11 discusses Trigonometric Functions and Their Graphs
• Chapter 12 focuses on the Application of Trigonometry
• Chapter 13 is about Inverse Trigonometric Functions
• Chapter 14 covers the Solution of Trigonometric Equations

Chapter 1: Number System

• For any complex number z, the absolute value |z| is always equal to |−z|, |−z̄|, and z̄.
• Numbers that can be written in the form of p/q, where p, q ∈ Z, and q ≠ 0, are rational numbers
• A decimal that has a finite number of digits in its decimal part is called a terminating decimal
• 5.333.... is a rational number
• π is an irrational number
• 22/7 is an irrational number
• The multiplicative inverse of '0' is not defined
• If 'a' is any non-zero real number, then its multiplicative inverse is 1/a
• For all a ∈ R, a = a is a Reflexive Property
• For all a, b ∈ R, a = b ⇒ b = a is called a Symmetric Property
• The golden rule of fractions states that for k ≠ 0, a/b = (ka)/(kb)
• The set {1, -1} possesses a closure property with respect to multiplication
• If a < b, then a/b < 1
• The additive identity in the set of complex numbers is (0, 0)
• The multiplicative identity in the set of complex numbers is (1, 0)
• The modulus of z = a + ib is √(a² + b²)
• i¹³ equals i
• The multiplicative inverse of (4, -7) is (4/65, 7/65)
• (0.3)(0.5) = -0.15
• (-i)² = -1
• √3 is an irrational number
• The product of √-2 × √-2 = -2
• The imaginary part of the complex number (b, a) is 'a'
• If z = -1 - i, then z̄ = -1 + i
• The property 7.8 + (-7.8) = 0 demonstrates the additive inverse
• If x ≠ 0, then the multiplicative inverse of x is 1/x
• (-i)¹⁵ = i
• If z₁ and z₂ are complex numbers, then |z₁ + z₂| ≤ |z₁| + |z₂|
• A recurring decimal represents a rational number
• Every recurring decimal is a rational number
• The imaginary part of (-2 + 3i³) is -3
• The product of two conjugate complex numbers is a real number
• (0, 1)³ is equal to -i
• If n is an even integer, then iⁿ is equal to ±1
• Factors of 3(x² + y²) are √3(x + iy)(x - iy)
• The real part of (2 + i)/i is 1

Chapter 2: Sets, Functions, and Groups

• If x ∈ L ∪ M, then x ∈ L or x ∈ M or both
• The total number of subsets that can be formed from the set {x, y, z} is 8
• If x ∈ B' = U – B, then x ∉ B and x ∈ U
• If L ∪ M = L ∩ M, then L is equal to M
• A set is a collection of objects which are well-defined and distinct
• The set of odd numbers between 1 and 9 are {3, 5, 7}
• Every recurring non-terminating decimal represents Q̄ or irrational numbers
• A diagram which represents a set is called a Venn diagram
• R - {0} i.e real numbers excluding zero, is a group w.r.t the binary operation of multiplication
• In a proposition, if p ↔ q then q → p is called the converse of p → q
• If A and B are disjoint sets, then A ∩ B = φ (empty set)
• ~p → q is called the converse of p → q
• If A ⊆ B then n(A ∩ B) = n(A)
• The inverse of any element of a group is unique
• Every function is a Relation
• The number of all subsets of a set having three elements is 8
• The graph of a linear function is a straight line
• The identity element in the set of complex numbers (C, .) is (1, 1)
• Squaring a number is a unary operation
• For sets, N ⊂ Z ⊂ Q ⊂ R is a true statement.
• The set of integers is a group w.r.t Addition
• If A = φ, then P(A) = {φ}
• If A and B are disjoint sets then A ∩ B = φ
• The set of non-zero real numbers w.r.t multiplication is a group.
• Division is not a binary operation.
• If y = √x, where x ≥ 0, is a function, then its inverse is a parabola.
• An onto function is also called surjective.
• A (1–1) function is also called Injective or one-to-one
• For the propositions p and q, (p ∧ q) → p is a tautology.
• For the propositions p and q, p → (p ∨ q) is a tautology.
• If set A has 2 elements and set B has 4 elements, then the number of elements in A × B is 8
• The inverse of a line is not defined correctly in the options
• The function f = {(x, y), y = x} is an Identity function
• If y = √x, x ≥ 0 is a function, then its inverse is a parabola.
• The truth set of a tautology is a Universal set
• A compound proposition which is always true is called a Tautology
• A compound proposition which is always neither true nor false is called a contingency.
• A compound proposition which is always wrong is called a contradiction.
• If S = {}, then the order of set S is 1.
• The symbol used to denote negation of a proposition is '~'.
• If B ⊆ A, then n(B – A) is equal to 0

Chapter 3: Matrices and Determinants

• A rectangular array of numbers enclosed by square brackets is called a Matrix
• The horizontal lines of numbers in a matrix are called Rows
• The vertical lines of numbers in a matrix are called Columns
• If a matrix A has m rows and n columns, then the order of A is m × n
• The element aij of any matrix A is present in the ith row and jth column
• A matrix A is called real if all aij are Real numbers
• If a matrix A has only one row, it is called a Row vector
• If a matrix A has only one column, it is called a Column vector
• If a matrix A has the same number of rows and columns, it is called a Square matrix
• If a matrix A has different numbers of rows and columns, it is called a Rectangular matrix
• For a square matrix A = [aij]n×n, the elements A₁₁, A₂₂, A₃₃...ann are on the Main diagonal
• For a square matrix A = [aij], if all aij = 0, i ≠ j and all aij = k (non-zero) for i = j, then A is called a Scalar Matrix
• For a square matrix A = [aij], if all aij = 0, i ≠ j and at least aij ≠ 0, i = j, then A is called a Diagonal matrix
• The matrix [7] is a Square matrix
• If A is a matrix of order m × n, then the matrix of order n × m is the Transpose of A
• For any two matrices A and B, (A + B)ᵗ = Aᵗ + Bᵗ
• Let A be a matrix and n is an integer; then A + A + A + ⋯ to n terms = nA
• If the order of A is m × n and the order of B is n × q, then the order of AB is m × q
• If adj A = [[-1, -2], [3, -4]], then matrix A = [[-4, 2], [-3, -1]]
• If A is a non-singular matrix, then A⁻¹ = (1/|A|) adjA
• If AX = B, then X = A⁻¹B
• The inverse of a matrix exists if it is non-singular.
• For any matrix A, it is always true that |A| = |Aᵗ|
• If all entries of a square matrix of order 3 are multiplied by k, then the value of |kA| is equal to k³|A|
• For a non-singular matrix A, it is true that (A⁻¹)⁻¹ = A
• For any non-singular matrices A and B, it is true that (AB)⁻¹ = B⁻¹A⁻¹
• A square matrix A = [aij] for which aij = 0, i > j, then A is called Upper triangular
• A square matrix A = [aij] for which aij = 0, i < j, then A is called Lower triangular
• Any matrix A is called singular if |A| = 0
• R (Real Numbers) is a field
• A square matrix A is symmetric if Aᵗ = A
• A square matrix A is skew-symmetric if Aᵗ = -A
• The main diagonal elements of a skew-symmetric matrix must be 0
• The main diagonal elements of a skew-hermitian matrix must be any complex number
• In echelon form of a matrix, the first non-zero entry is called the Leading entry
• The additive inverse of a matrix exists only if it is any matrix of order m × n
• The multiplicative inverse of a matrix exists only if it is a non-singular matrix
• The number of non-zero rows in echelon form of a matrix is called the Rank of the matrix
• If A is any square matrix, then A + Aᵗ is a symmetric matrix.
• If A is any square matrix, then A² must be symmetric if A is symmetric.
• In a homogenous system of linear equations, the solution (0,0,0) is a Trivial solution
• If AX = 0, then X = 0
• If the system of linear equations has no solution at all, then it is called an Inconsistent system
• If the system x + λy = 4; 2x + 4y = −3 does not possess a unique solution, then λ=2.
• The inverse of a Unit matrix is a Unit matrix
• The transpose of a row matrix is a column matrix

Chapter 4: Quadratic Equation

• The equation ax² + bx + c = 0 will be quadratic if a ≠ 0.
• The solution set of the equation x² − 4x + 4 = 0 is {2}.
• The quadratic formula for solving the equation ax² + bx + c = 0; a ≠ 0 is x = (-b ± √(b² - 4ac)) / (2a).
• To convert ax²ⁿ + bxⁿ + c = 0 (a ≠ 0) into quadratic form, the correct substitution is y = xⁿ.
• The equation in which the variable occurs as an exponent is called an Exponential equation.
• To convert 4^(1+x) + 4^(1-x) = 10 into a quadratic equation, the substitution is y = 4^x.
• The equations involving radical expressions of the variable are called Radical equations.
• The cube roots of unity are 1, (-1+√(-3))/2, (-1-√(-3))/2.
• The sum of all cube roots of 64 is 0.
• The product of the cube roots of -1 is 1.
• 16w⁸ + 16w⁴ = -16
• The sum of all four fourth roots of unity is 0.
• The product of all four fourth roots of unity is -1.
• The sum of all four fourth roots of 16 is 0.
• The complex cube roots of unity are Conjugate to each other.
• The complex cube roots of unity are reciprocals of each other, and additive inverses.
• The complex fourth roots of unity are squares of each other.
• If the sum of all cube roots of unity is equal to x² + 1, then x is equal to ±i
• If the product of all cube roots of unity is equal to p² + 1, then p is equal to 0.
• The expression x² + 1/x - 3 is not a polynomial.
• If f(x) is divided by x - a, then dividend = (Divisor)(Quotient) + Remainder.
• If f(x) is divided by x - a, by the remainder theorem, then the remainder is f(a).
• The polynomial (x – a) is a factor of f(x) if and only if f(a) = 0
• x - 2 is a factor of x² - kx + 4, if k is 4.
• If x = -2 is the root of kx⁴ – 13x² + 36 = 0, then k = -2.
• x + a is a factor of xⁿ + aⁿ when n is any odd integer.
• x - a is a factor of xⁿ – aⁿ when n is any integer.
• The sum of roots of ax² – bx - c = 0 is (a ≠ 0) b/a.
• The sum of roots of ax² + bx - c = 0 is (a ≠ 0) -b/a.
• If 2 and -5 are roots of a quadratic equation, then the equation is x² + 3x − 10 = 0.
• If α and β are the roots of 3x² – 2x + 4 = 0, then the value of α + β is 2/3.
• If roots of ax² + bx + c = 0, (a ≠ 0) are real, then Disc≥ 0
• If roots of ax² + bx + c = 0, (a ≠ 0) are equal, then Disc= 0
• The graph of a quadratic equation is a Parabola
• ω²⁸ + ω²⁹ + 1 = 0
• Synthetic division is a process of division
• The degree of a quadratic equation is 2.
• Basic techniques for solving quadratic equations are finding factors.
• If 16ω⁴ + 16ω⁸=16.

Chapter 5: Partial Fractions

• An open sentence formed by using the sign "=" is called an Equation.
• If an equation is true for all values of the variable, then it is called an identity.
• (x + 3)(x + 4) = x² + 7x + 12 is an identity.
• The quotient of two polynomials P(x)/Q(x) is called a rational fraction
• A fraction P(x)/Q(x), Q(x) ≠ 0 is a proper fraction if the degree of P(x) <the Degree of Q(x).
• A mixed form of a fraction has a polynomial and a proper fraction.
• When a rational fraction is separated into partial fractions, then the result is always an Identity.
• The number of partial fractions x³/((x + 1)(x² − 1)) is 4.
• Conditional equation 2x + 3 = 0 holds when x is equal to -3/2
• x³ - 6x² + 8x is a reducible factor
• A quadratic factor which cannot be written as a product of linear factors with real coefficients is called an irreducible factor.
• 9x²/(x³-1) is a proper fraction.

Chapter 6: Sequence and Series

• An arrangement of numbers according to some definite rule is called a Sequence
• A Sequence is also known as a Progression
• A sequence is a function whose domain is N.
• A sequence whose range is R (set of real numbers) is called a Real Sequence
• If aₙ = {n + (−1)ⁿ}, then a₁₀ = 12
• The last term of an infinite sequence doesn't exist
• The next term of the sequence 1, 2, 12, 40, ... is not defined
• A sequence {aₙ} in which aₙ - Aₙ₋₁ is the same number for all n ∈ N, n > 1 is called an A.P (Arithmetic Progression)
• If the nth term of an A.P is 3n – 1, then the 10th term is 29
• If aₙ₋₁, aₙ, aₙ₊₁ are in A.P, then aₙ is A.M (Arithmetic Mean)
• The arithmetic mean between √2 and 3√2 is 2√2
• The sum of terms of a sequence is called a Series
• No term of a G. P is 0
• A.M between c and d is (c+d)/2
• If A, G, and H are Arithmetic, Geometric and Harmonic means between two +ive numbers then A >=G >=H
• The general term of a G. P is aₙ = arⁿ⁻¹

Chapter 7: Permutation, Combination and Probability

• 20P₃= 6840
• If ⁿP₂ = 30, then n = 6
• The number of diagonals in a 10-sided figure is calculated as ¹⁰C₂ - 10
• ⁿC₀ = 1
• Number of arrangements of the world "MATHEMATICS" can be made ¹¹!/((3,2,1,1,1,1,1) * (2,2,1,1,1,1,1))
• If n is negative, then n! is not defined.
• ⁿ⁻¹Cᵣ + ⁿ⁻¹Cᵣ₋₁ = ⁿCᵣ
• The number of signals that can be given by 5 flags of different colors, using 3 at a time is 60.
• ⁿPₙ = factorial 'n'
• If an event A can occur in p ways and event B can occur by q ways, the, the numbers of ways of both events occur is p.q
• If "ⁿC₁₂ = ⁿC₁₃, then the value of n = 20
• The probability of non-occurrence of an event E is equal to 1-P(E).
• For independent events, P(A ∩ B) = P(A).P(B)
• If A card is drawn from a deck of 52 playing cards, then the probability of getting an ace card is 4/13.
• If four persons want to sit on circular sofa, then the total number of ways are 6
• In the playing of two teams A and B, is team A does not lose the probability of is ½.
• Let S = {1, 2, 3, ..., 10} the probability that a number is divided by 4 is 3/10.
• If rolling a tie, then probability of getting 3 or getting 5 is 1/3
• If a coin is tossed 5 times, then n(S) is equal to 32
• The number of ways for sitting 4 persons in atrain on a straight sofa is 24.
• Sample space for tossing a coin is {H,T}.
• If an event always occurs, then it is called a Certain event
• If E is a certain even, then P(E) =1
• In a permutation, when nPr or P(n,r), then n>=r
• Non occurrence of an event E is denoted by E^C

Chapter 8: Mathematical Induction and Binomial Theorem

• The statement 4ⁿ + 3 + 4 is true when: n ≥ 2
• The number of terms in the expansion of (a + b)ⁿ are: n+1
• Middle term/s in the expansion of (a – 3x)¹⁴ is/are : T₇ & T₈
• The coefficient of the last term in the expansion of (2 – x)⁷ is : (-1)^7 = -7
• (ⁿC₀) + (ⁿC₁) + (ⁿC₂) + ... + (ⁿCₙ) is equal to: 2²ⁿ⁻¹
• 1 + x + x² + x³ + ... = (1-x)⁻¹ if |x| < -1
• The method of induction was given by Francesco from 1494-1575

Chapter 9: Fundamentals of Trigonometry

• Two rays with a common starting point form an Angle
• The common starting point of the two rays is called the Vertex
• If the rotation of an angle is counter clockwise, then the angle is positive.
• If the initial ray OA rotates in an anti-clockwise direction in such a way that it coincides with itself, the angle then formed is 360°
• One rotation in an anti-clock wise direction is equal to 360°
• If the rotation of the angle is clockwise, then the angle is negative.
• A straight line angle is equal to π radian.
• One right angle is equal to π rotation.
• 1° is equal to 60 minutes.
• 1° is equal to 3600 seconds
• 60th part of 1° is equal to minutes.
• 60th part of 1 is equal to 1".
• 3600th part of 1° is equal to 1"
• The system of angular measurement in which the angle is measured in radian (cicular measure) is called Circular System
• The Sexagesimal system is also called the English System
• The radian is a constant angle
• One sign of the angle is that of the quadrantan in which the terminal sides of the angle falls
• The relation between the length of an arc of a circle and the circular measure of it central angle is The point (0,1) lies on the terminal side of the angle is 90°

Chapter 10: Trigonometric Identities

• The distance between the points A(3,8) & B(5,6) is √8
• The fundamental law of trigonometry is, cos(α - β) = cosαcosβ + sinαsinβ
• The set of sine and cosine of the angle lies in the close interval of [-1,1]
• sin(π/2−θ)=cosθ
• tan (π/2−θ)=cotθ
• The simplest form is cos4θ + cos2θ=2cos⁡3θ cos⁡θ

Chapter 11: Trigonometric Function and their graphs

• The period of trigonometric ration lies in [-1,1]
• The smallest+ive number which when added to the original circular measure of the angle gives the same value of the function is called Period
• For Trigonometric series, the value of 2cos2θ sinθ is = sin3θ

Chapter 12: Application of Trigonometry

• To solve an oblique triangle, we use : Law of Sine
• To solve an oblique triangles when three sides are given, we can use :Law of cosine

Chapter 13: Inverse Trigonometric Functions

• Inverse of a function exist only if it is : √(1-1) function
• The following are trigonometric expressions but not identities