Zahid Notes 1st year math mcqs with answers.pdf

1|Page MATHEMATICS MULTIPLE CHOICE QUESTIONS WITH ANSWERS 11 M.SALMAN SHERAZI 03337727666/03067856232 2|Page Contents UNIT # Title Page # 1 Number System Sets, Functions and Groups Matrices and Determinants Quadratic Equations Partial Fractions Sequence and Series Permutation, Combinations and Proba...

1|Page MATHEMATICS MULTIPLE CHOICE QUESTIONS WITH ANSWERS 11 M.SALMAN SHERAZI 03337727666/03067856232 2|Page Contents UNIT # Title Page # 1 Number System Sets, Functions and Groups Matrices and Determinants Quadratic Equations Partial Fractions Sequence and Series Permutation, Combinations and Probability Mathematical Induction and Binomial Theorem Fundamentals of Trigonometry Trigonometric Identities Trigonometric Functions and their Graphs Application of Trigonometry Inverse Trigonometric Functions Solution of Trigonometric Equations 3 2 3 4 5 6 7 8 9 10 11 12 13 14 4 6 8 9 10 11 13 13 15 17 18 19 21 3|Page UNIT # 01 Number Systems Each question has four possible answer. Tick the correct answer. 1. For any complex number 𝒛, it is always true that |𝒛| is equal to: (a) |𝑧| (b) | − 𝑧| (c) | − 𝑧| (d) ✔ all of these 𝒑 2. The numbers which can be written in the form of 𝒒 , 𝒑, 𝒒 ∈ 𝒁 , 𝒒 ≠ 𝟎 are : (a) 3. (a) 4. (a) 5. (a) 6. ✔Rational number (b) Irrational number (c) Complex number (d) Whole number A decimal which has a finite numbers of digits in its decimal part is called______ decimal. ✔Terminating (b) Non-Terminating (c) Recurring (d) Non recurring 𝟓. 𝟑𝟑𝟑 …. Is Rational (b) Irrational (c) an integer (d) a prime number 𝝅 is Rational (b) ✔ Irrational (c) Natural number (d) None 𝟐𝟐 𝟕 is (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. ✔Rational (a) 12. (a) 13. (a) 14. (a) 15. (a) 16. ✔𝑘𝑏 (a) 17. (a) 18. (b) ✔√𝑎2 + 𝑏2 √𝑎 + 𝑏 𝒊𝟏𝟑 equals: ✔𝑖 (b) – 𝑖 The multiplicative inverse of (𝟒, −𝟕) is: (b) Irrational (c) an integer Multiplicative inverse of ′𝟎′ is 0 (b) any real number (c) ✔ not defined If 𝒂 is any non-zero real number, then multiplicative inverse is 1 1 –𝑎 (b) ✔ 𝑎 (c) − 𝑎 For all 𝒂 ∈ 𝑹 , 𝒂 = 𝒂 is …. Property. ✔Reflexive (b) Symmetric (c) Transitive For all 𝒂, 𝒃 ∈ 𝑹 , 𝒂 = 𝒃 ⟹ 𝒃 = 𝒂 is called….. property. Reflexive (b) ✔ Symmetric (c) Transitive 𝒂 Golden rule of fraction is that for 𝒌 ≠ 𝟎, = 𝑘𝑎 (b) 𝑙 (c) 𝑏 The set {𝟏, −𝟏} possesses closure property 𝒘. 𝒓. 𝒕 ′+′ (b) ✔ ′ × ′ (c) ′ ÷ ′ If 𝒂 < 𝑏 then 1 1 1 1 𝑎 𝑏 The additive identity in set of complex number is ✔(0,0) (b) (0,1) (c) (1,0) The multiplicative identity of complex number is (0,0) (b) (0,1) (c) ✔ (1,0) The modulus of 𝒛 = 𝒂 + 𝒊𝒃 is 4 7 (a) (− , − ) 65 65 19. (𝟎, 𝟑)(𝟎, 𝟓) = (a) 15 20. (a) 21. (a) 22. (a) 23. (a) 24. (a) 𝒃 𝑎𝑏 𝑘𝑎 𝟐𝟏 − 𝟐 (b) (− 4 7 , ) 65 65 (b) ✔-15 (c) 𝑎 − 𝑏 (c) 1 4 7 ,− ) 65 65 (c) ( (c) −8𝑖 (−𝟏) = 𝑖 (b) ✔ – 𝑖 (c) 1 √𝟑 is __________ Rational (b) ✔ Irrational (c) Integer Product √−𝟐 × √−𝟐 is equal to _____ -2 (b) ✔2 (c) 0 The imaginary part of the complex number (𝒃 , 𝒂) is _________ ✔𝑎 (b) 𝑏 (c) 𝑖𝑎 If 𝒛 = −𝟏 − 𝒊 then 𝒛 =______ (−1, −1) (b) ✔(−1,1) (c) (1, −1) (d) a whole number (d) 1 (d) not defined (d) Trichotomy (d) Trichotomy (d) 𝑘𝑏 𝑏 (d) ′ − ′ (d) 𝑎 − 𝑏 > 0 (d) (1,1) (d) (1,1) (d) √𝑎2 − 𝑏 2 (d) -1 4 7 (d) ✔( , ) (d) 8𝑖 65 65 (d) -1 (d) Prime (d) 4 (d) None of these (d) (1,1) 4|Page 25. The property 𝟕. 𝟖 + (−𝟕. 𝟖) = 𝟎 is______ (a) Commutative (b)‘.’ inverse (c) ✔ ‘ + ’ inverse (d) Associative 26. If 𝒙 = 𝟎 then multiplicative inverse of 𝒙 is 1 (a) 0 (b) 1 (c) 𝑥 (d) ✔ None of these 𝟏𝟓 27. (−𝒊) =_______ (a) 1 (b) -1 (c) ✔ 𝑖 (d) – 𝑖 28. If 𝒛1 and 𝒛2 are complex numbers then |𝒛1+𝒛2| is _______ (a) 𝑗 then A is called: ✔Upper triangular (b) Lower triangular (c) Symmetric (d) Hermitian A square matrix 𝑨 = [𝒂𝒊𝒋 ] for which 𝒂𝒊𝒋 = 𝟎, 𝒊 < 𝑗 then A is called: Upper triangular (b) ✔Lower triangular (c) Symmetric (d) Hermitian Any matrix A is called singular if: ✔|𝐴| = 0 (b) |𝐴| ≠ 0 (c) 𝐴𝑡 = 𝐴 (d) 𝐴𝐴−1 = 𝐼 Which of the following Sets is a field. R (b) Q (c) C (d) ✔all of these Which of the following Sets is not a field. R (b) Q (c) C (d) ✔Z A square matrix A is symmetric if: (a) 𝐴𝑡 = 𝐴 (b) ✔ 𝐴𝑡 = −𝐴 35. A square matrix A is Hermitian if: (a) 𝐴𝑡 = 𝐴 (b) 𝐴𝑡 = −𝐴 36. A square matrix A is skew- Hermitian if: (a) 37. (a) 38. (a) 39. (a) 40. (a) 41. (a) 42. (a) 43. (a) 44. (a) −4 3 ] −2 −1 𝑡 𝑡 (c) (𝐴) = 𝐴 𝑡 (c) ✔(𝐴) = 𝐴 𝑡 𝑡 (d) (𝐴) = −𝐴 𝑡 (d) (𝐴) = −𝐴 𝑡 (d) (𝐴) = −𝐴 𝑡 𝐴𝑡 = 𝐴 (b) 𝐴𝑡 = −𝐴 (c) (𝐴) = 𝐴 (d) ✔(𝐴) = −𝐴 The main diagonal elements of a skew symmetric matrix must be: 1 (b) ✔ 0 (c) any non-zero number (d) any complex number The main diagonal elements of a skew hermitian matrix must be: 1 (b) ✔ 0 (c) any non-zero number (d) any complex number In echelon form of matrix, the first non zero entry is called: ✔Leading entry (b) first entry (c) preceding entry (d) Diagonal entry The additive inverse of a matrix exist only if it is: Singular (b) non singular (c) null matrix (d) ✔ any matrix of order 𝑚 × 𝑛 The multiplicative inverse of a matrix exist only if it is: Singular (b) ✔ non singular (c) null matrix (d) any matrix of order 𝑚 × 𝑛 The number of non zero rows in echelon form of a matrix is called: Order of matrix (b) Rank of matrix (c) leading (d) leading row If A is any square matrix then 𝑨 + 𝑨𝒕 is a ✔Symmetric (b) skew symmetric (c) hermitian (d) skew hermitian If A is any square matrix then 𝑨 − 𝑨𝒕 is a Symmetric (b) ✔skew symmetric (c) hermitian (d) skew hermitian 8|Page 𝒕 45. If A is any square matrix then 𝑨 + (𝑨) is a (a) Symmetric (b) skew symmetric 46. (a) 47. (a) 48. (a) 49. (a) 50. (a) 51. (a) 52. (a) 53. (a) 54. (a) 55. (a) 𝒕 (c) ✔ hermitian (d) skew hermitian If A is any square matrix then 𝑨 + (𝑨) is a Symmetric (b) skew symmetric (c) hermitian (d) ✔ skew hermitian If A is symmetric (Skew symmetric), then 𝑨𝟐 must be Singular (b) non singular (c) ✔symmetric (d) non trivial solution In a homogeneous system of linear equations , the solution (0,0,0) is: ✔Trivial solution (b) non trivial solution (c) exact solution (d) anti symmetric If 𝑨𝑿 = 𝑶 then 𝑿 = 𝐼 (b) ✔ 𝑂 (c) 𝐴−1 (d) Not possible If the system of linear equations have no solution at all, then it is called a/an Consistent system (b)✔ Inconsistent system (c) Trivial System (d) Non Trivial System The value of 𝝀 for which the system 𝒙 + 𝟐𝒚 = 𝟒; 𝟐𝒙 + 𝝀𝒚 = −𝟑 does not possess the unique solution ✔4 (b) -4 (c) ±4 (d) any real number If the system 𝒙 + 𝟐𝒚 = 𝟎; 𝟐𝒙 + 𝝀𝒚 = 𝟎 has non-trivial solution, then 𝝀 is: ✔4 (b) -4 (c) ±4 (d) any real number The inverse of unit matrix is: ✔Unit (b) Singular (c) Skew Symmetric (d) rectangular Transpose of a row matrix is: Diagonal matrix (b) zero matrix (c) ✔ column matrix (d) scalar matrix 𝒙 𝟒 If | | = 𝟎 ⇒ 𝒙 equals 𝟓 𝟏𝟎 ✔2 (b) 4 (c) 6 (d) 8 UNIT # 04 Quadratic Equations Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. The equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 will be quadratic if: 𝑎 = 0, 𝑏 ≠ 0 (b) ✔ 𝑎 ≠ 0 (c) 𝑎 = 𝑏 = 0 (d) 𝑏 = any real number 𝟐 Solution set of the equation 𝒙 − 𝟒𝒙 + 𝟒 = 𝟎 is: {2, −2} (b) ✔ {2} (c) {−2} (d) {4, −4} The quadratic formula for solving the equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎; 𝒂 ≠ 𝟎 is (a) ✔𝑥 = 4. (a) 5. (a) 6. (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. (a) −𝑏±√𝑏2 −4𝑎𝑐 2𝑎 𝟐𝒏 𝒏 (b) 𝑥 = −𝑏±√𝑎2 −4𝑎𝑐 2𝑏 (c) 𝑥 = −𝑎±√𝑏2 −4𝑎𝑐 2 (c) −1, −1+√3𝑖 −1+√3𝑖 , 2 2 (d) None of these To convert 𝒂𝒙 + 𝒃𝒙 + 𝒄 = 𝟎(𝒂 ≠ 𝟎) into quadratic form , the correct substitution is: 1 ✔𝑦 = 𝑥 𝑛 (b) 𝑥 = 𝑦 𝑛 (c) 𝑦 = 𝑥 −𝑛 (d) 𝑦 = 𝑥 The equation in which variable occurs in exponent , called: Exponential function (b) Quadratic equation (c) Reciprocal equation (d) ✔Exponential equation To convert 𝟒𝟏+𝒙 + 𝟒𝟏−𝒙 = 𝟏𝟎 into quadratic , the substitution is: 𝑦 = 𝑥 1−𝑥 (b) 𝑦 = 41+𝑥 (c) 𝑦 = 4𝑥 (d) 𝑦 = 4−𝑥 The equations involving redical expressions of the variable are called: Reciprocal equations(b) ✔ Redical equations (c) Quadratic functions (d) exponential equations The cube roots of unity are : ✔1, −1+√3𝑖 −1+√3𝑖 1+√3𝑖 1+√3𝑖 2 , (b) 1, 2 , 2 Sum of all cube roots of 64 is : ✔0 (b) 1 Product of cube roots of -1 is: 0 (b) -1 𝟖 𝟒 𝟏𝟔𝝎 + 𝟏𝟔𝒘 = 0 (b) ✔ -16 2 (c) 64 (d) −1, (d) -64 1+√3𝑖 1+√3𝑖 , 2 2 (c) ✔1 (d) None (c) 16 (d) -1 9|Page 12. (a) 13. (a) 14. (a) 15. (a) 16. (a) 17. (a) 18. (a) 19. (a) The sum of all four fourth roots of unity is: Unity (b) ✔0 (c) -1 (d) None The product of all four fourth roots of unity is: Unity (b) 0 (c) ✔-1 (d) None The sum of all four fourth roots of 16 is: 16 (b) -16 (c) ✔ 0 (d) 1 The complex cube roots of unity are………………. each other. Additive inverse (b) Equal to (c) ✔Conjugate (d) None of these The complex cube roots of unity are………………. each other. Multiplicative inverse (b) reciprocal (c) square (d) ✔None of these The complex fourth roots of unity are ……. of each other. ✔Additive inverse (b) equal to (c) square of (d) None of these 𝟐 If sum of all cube roots of unity is equal to 𝒙 + 𝟏 , then 𝒙 is equal to: −1 (b) 0 (c) ✔±𝑖 (d) 1 𝟐 If product of all cube roots of unity is equal to 𝒑 + 𝟏 , then 𝒑 is equal to: −1 (b) ✔ 0 (c) ±𝑖 (d) 1 𝟏 20. The expression 𝒙𝟐 + 𝒙 − 𝟑 is polynomial of degree: (a) 21. (a) 22. (a) 23. (a) 24. (a) 25. (a) 26. (a) 27. (a) 28. 2 (b) 3 (c) 1 (d) ✔ not a polynomial If 𝒇(𝒙) is divided by 𝒙 − 𝒂 , then dividend = (Divisor)(…….)+ Remainder. Divisor (b) Dividend (c) ✔ Quotient (d) 𝑓(𝑎) If 𝒇(𝒙) is divided by 𝒙 − 𝒂 by remainder theorem then remainder is: ✔ 𝑓(𝑎) (b) 𝑓(−𝑎) (c) 𝑓(𝑎) + 𝑅 (d) 𝑥 − 𝑎 = 𝑅 The polynomial (𝒙 − 𝒂) is a factor of 𝒇(𝒙) if and only if ✔ 𝑓(𝑎) = 0 (b) 𝑓(𝑎) = 𝑅 (c) Quotient = 𝑅 (d) 𝑥 = −𝑎 𝟐 𝒙 − 𝟐 is a factor of 𝒙 − 𝒌𝒙 + 𝟒, if 𝒌 is: 2 (b) ✔ 4 (c) 8 (d) -4 If 𝒙 = −𝟐 is the root of 𝒌𝒙𝟒 − 𝟏𝟑𝒙𝟐 + 𝟑𝟔 = 𝟎, then 𝒌 = 2 (b) -2 (c) 1 (d) ✔ -1 𝒏 𝒏 𝒙 + 𝒂 is a factor of 𝒙 + 𝒂 when 𝒏 is Any integer (b) any positive integer (c) ✔ any odd integer (d) any real number 𝒏 𝒙 − 𝒂 is a factor of 𝒙 − 𝒂𝒏 when 𝒏 is ✔ Any integer (b) any positive integer (c) any odd integer (d) any real number Sum of roots of 𝒂𝒙𝟐 − 𝒃𝒙 − 𝒄 = 𝟎 is (𝒂 ≠ 𝟎) (a) ✔ 29. (a) 30. (a) 31. (a) 32. (a) 33. (a) 34. (a) 35. (a) 36. (a) 37. (a) 𝑏 𝑎 (b) – 𝑏 𝑎 (c) 𝑐 𝑎 (d) – 𝑐 𝑎 Sum of roots of 𝒂𝒙𝟐 − 𝒃𝒙 − 𝒄 = 𝟎 is (𝒂 ≠ 𝟎) 𝑏 𝑐 𝑐 𝑏 (b) – 𝑎 (c) 𝑎 (d) ✔ – 𝑎 𝑎 If 2 and -5 are roots of a quadratic equation , then equation is: 𝑥 2 − 3𝑥 − 10 = 0 (b) 𝑥 2 − 3𝑥 + 10 = 0 (c) ✔ 𝑥 2 + 3𝑥 − 10 = 0 (d) 𝑥 2 + 3𝑥 + 10 = 0 If 𝜶 and 𝜷 are the roots of 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎, then the value of 𝜶 + 𝜷 is: 2 2 4 4 ✔3 (b) − 3 (c) 3 (d) − 3 If roots of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are real , then Disc≥ 0 (b) Disc< 0 (c) Disc≠ 0 If roots of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are complex , then ✔Disc≥ 0 (b) ✔ Disc< 0 (c) Disc≠ 0 𝟐 If roots of 𝒂𝒙 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are equal , then ✔Disc= 0 (b) Disc< 0 (c) Disc≠ 0 Graph of quadratic equation is: Straight line (b) Circle (c) square 𝝎𝟐𝟖 + 𝝎𝟐𝟗 + 𝟏 = ✔0 (b) 1 (c) -1 Synthetic division is a process of: Addition (b) multiplication (c) subtraction (d) Disc≤ 0 (d) Disc≤ 0 (d) None of these (d) ✔Parabola (d) 𝜔 (d) ✔ division 10 | P a g e 38. (a) 39. (a) 40. (a) Degree of quadratic equation is: 0 (b)1 (c) ✔2 Basic techniques for solving quadratic equations are 1 (b) 2 (c) ✔3 𝟒 𝟖 𝟏𝟔𝝎 + 𝟏𝟔𝝎 = 0 (b) ✔ -16 (c) 16 (d) 3 (d) 4 (d) -1 UNIT # 05 Partial Fractions Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) An open sentence formed by using sign of “ = ” is called a/an ✔Equation (b) Formula (c) Rational fraction If an equation is true for all values of the variable, then it is called: a conditional equation (b) ✔ an identity (c) proper rational fraction (𝒙 + 𝟑)(𝒙 + 𝟒) = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟐 is a/an: Conditional equation (b) ✔an identity (c) proper rational fraction 4. The quotient of two (a) ✔Rational fraction 5. A 𝑷(𝒙) polynomials 𝑸(𝒙) , 𝑸(𝒙) ≠ 𝟎 is called : (b) Irrational fraction 𝑷(𝒙) fraction 𝑸(𝒙) , 𝑸(𝒙) (c) Partial fraction (d) Theorem (d) All of these (d) a formula (d) Proper fraction ≠ 𝟎 is called proper fraction if : (a) ✔Degree of 𝑃(𝑥) < Degree of 𝑄(𝑥) (b) Degree of 𝑃(𝑥) = Degree of 𝑄(𝑥) (c) Degree of 𝑃(𝑥) > Degree of 𝑄(𝑥) (d) Degree of 𝑃(𝑥) ≥ Degree of 𝑄(𝑥) 𝑷(𝒙) 6. A fraction 𝑸(𝒙) , 𝑸(𝒙) ≠ 𝟎 is called proper fraction if : (a) Degree of 𝑃(𝑥) < Degree of 𝑄(𝑥) (b) Degree of 𝑃(𝑥) = Degree of 𝑄(𝑥) (c) Degree of 𝑃(𝑥) > Degree of 𝑄(𝑥) (d) ✔ Degree of 𝑃(𝑥) ≥ Degree of 𝑄(𝑥) 7. A mixed form of fraction is : (a) An integer+ improper fraction (b) a polynomial+improper fraction (c) ✔a polynomial+proper fraction (d) a polynomial+rational fraction 8. When a rational fraction is separated into partial fractions, then result is always : (a) A conditional equations (b) ✔an identity (c) a partial fraction (d) an improper fraction 𝒙𝟑 9. The number of Partial fraction of 𝒙(𝒙+𝟏)(𝒙𝟐 −𝟏)are: (a) 2 (b) 3 10. The number of Partial fraction of (a) 2 (b) 3 11. Partial fractions of (a) 1 2(𝑥−1) + 1 2(𝑥+1) 𝟏 are: 𝒙𝟐 −𝟏 (b) ✔ (c) ✔4 𝒙𝟓 are: 𝒙(𝒙+𝟏)(𝒙𝟐 −𝟒) (d) ✔ 6 (c) 4 1 2(𝑥−1) − 1 2(𝑥+1) (c) − (d) None of these 1 1 + 2(𝑥−1) 2(𝑥+1) (d) − 1 1 − 2(𝑥−1) 2(𝑥+1) Conditional equation 𝟐𝒙 + 𝟑 = 𝟎 holds when 𝒙 is equal to: 3 3 1 ✔− 2 (b) 2 (c) 3 (d) 1 Which is a reducible factor: 𝑥 3 − 6𝑥 2 + 8𝑥 (b) 𝑥 2 + 16𝑥 (c) 𝑥 2 + 5𝑥 − 6 (d) ✔all of these A quadratic factor which cannot written as a product of linear factors with real coefficients is called: (a) ✔An irreducible factor (b) reducible factor (c) an irrational factor (d) an improper factor 12. (a) 13. (a) 14. 15. 𝟗𝒙𝟐 is 𝒙𝟑 −𝟏 an (a) Improper fraction (b) ✔ Proper fraction (c) Polynomial (d) equation 11 | P a g e UNIT # 06 Sequence and Series Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) 4. (a) 5. (a) 6. (a) 7. (a) 8. (a) 9. (a) An arrangement of numbers according to some definite rule is called: ✔Sequence (b) Combination (c) Series (d) Permutation A sequence is also known as: Real sequence (b) ✔Progression (c) Arrangement (d) Complex sequence A sequence is function whose domain is 𝑍 (b) 𝑁 (c) 𝑄 (d) ✔ 𝑅 As sequence whose range is 𝑹 𝒊. 𝒆., set of real numbers is called: ✔Real sequence (b) Imaginary sequence (c) Natural sequence (d) Complex sequence If 𝒂𝒏 = {𝒏 + (−𝟏)𝒏 }, then 𝒂𝟏𝟎 = 10 (b) ✔ 11 (c) 12 (d) 13 The last term of an infinite sequence is called : 𝑛𝑡ℎ term (b) 𝑎𝑛 (c) last term (d) ✔ does not exist The next term of the sequence 𝟏, 𝟐, 𝟏𝟐, 𝟒𝟎, … is ✔112 (b) 120 (c) 124 (d) None of these A sequence {𝒂𝒏 } in which 𝒂𝒏 − 𝒂𝒏−𝟏 is the same number for all 𝒏 ∈ 𝑵,𝒏 > 1 is called: ✔A.P (b) G.P (c) H.P (d) None of these 𝒏𝒕𝒉 term of an A.P is 𝟑𝒏 − 𝟏 then 10th term is : 9 (b) ✔ 29 (c) 12 (d) cannot determined 𝟏 𝟐 𝟓 𝟐 𝟕 𝟐 10. 𝒏𝒕𝒉 term of the series (𝟑) + ( ) + ( ) + ⋯ 𝟐 𝟑 2𝑛−1 2 (a) ✔ ( 11. (a) 12. (a) 3 ) 2𝑛−1 2 ) 3 (b) ( If 𝒂𝒏−𝟏 , 𝒂𝒏 , 𝒂𝒏+𝟏 are in A.P, then 𝒂𝒏 is ✔A.M (b) G.M Arithmetic mean between 𝒄 and 𝒅 is: 𝑐+𝑑 𝑐+𝑑 (b) 2𝑐𝑑 ✔ 2 2𝑛 2 (c) ( 3 ) (c) H.M (d) Mid point 2𝑐𝑑 2 (c) 𝑐+𝑑 (d) 𝑐+𝑑 13. The arithmetic mean between √𝟐 and 𝟑√𝟐 is: 4 (a) 4√2 (b) ✔ 2 (c) √2 14. (a) 15. (a) 16. (a) 17. (a) 18. (a) 19. (a) 20. (a) 21. (a) 22. (a) 23. (a) 24. (a) (d) cannot determined (d) none of these √ The sum of terms of a sequence is called: Partial sum (b) ✔ Series (c) Finite sum (d) none of these 𝟐 Forth partial sum of the sequence {𝒏 } is called: 16 (b) ✔ 1+4+9+16 (c) 8 (d) 1+2+3+4 Sum of 𝒏 −term of an Arithmetic series 𝑺𝒏 is equal to: 𝑛 𝑛 𝑛 𝑛 ✔2 [2𝑎 + (𝑛 − 1)𝑑] (b) 2 [𝑎 + (𝑛 − 1)𝑑] (c) 2 [2𝑎 + (𝑛 + 1)𝑑] (d) 2 [2𝑎 + 𝑙] For any 𝑮. 𝑷., the common ratio 𝒓 is equal to: 𝑎𝑛 𝑎 𝑎 (b) 𝑎𝑛−1 (c) ✔ 𝑎 𝑛 (d) 𝑎𝑛+1 − 𝑎𝑛 , 𝑛 ∈ 𝑁, 𝑛 > 1 𝑎 𝑛+1 𝑛 𝑛−1 No term of a 𝑮. 𝑷., is: ✔0 (b) 1 (c) negative The general term of a 𝑮. 𝑷., is : ✔𝑎𝑛 = 𝑎𝑟𝑛−1 (b) 𝑎𝑛 = 𝑎𝑟 𝑛 (c) 𝑎𝑛 = 𝑎𝑟 𝑛+1 The sum of infinite geometric series is valid if |𝑟| > 1 (b) |𝑟| = 1 (c) |𝑟| ≥ 1 For the series 𝟏 + 𝟓 + 𝟐𝟓 + 𝟏𝟐𝟓 + ⋯ + ∞ , the sum is -4 (b) 4 An infinite geometric series is convergent if |𝑟| > 1 (b) |𝑟| = 1 An infinite geometric series is divergent if |𝑟| < 1 (b) |𝑟| ≠ 1 If sum of series is defined then it is called: ✔Convergent series (b) Divergent series (c) 1−5𝑛 −4 (d) imaginary number (d) None of these (d) ✔ |𝑟| < 1 (d) ✔ not defined (c) |𝑟| ≥ 1 (d) ✔ |𝑟| < 1 (c) finite series (d) Geometric series (c) 𝑟 = 0 (d✔) |𝑟| > 1 12 | P a g e 25. If sum of series is not defined then it is called: (a) Convergent series (b) ✔ Divergent series (c) finite series (d) Geometric series 26. The interval in which series 𝟏 + 𝟐𝒙 + 𝟒𝒙𝟐 + 𝟖𝒙𝟑 + ⋯ is convergent if : 1 1 (b) ✔ − < 𝑥 < (c) |2𝑥| > 1 (d) |𝑥| < 1 2 2 27. If the reciprocal of the terms a sequence form an 𝑨. 𝑷., then it is called: (a) ✔𝐻. 𝑃 (b) 𝐺. 𝑃 (c) 𝐴. 𝑃 (d) sequence 𝟏 𝟏 𝟏 28. The 𝒏𝒕𝒉 term of 𝟐 , 𝟓 , 𝟖 , … is (a) −2 < 𝑥 < 2 (a) 29. (a) 30. (a) 31. (a) 32. (a) 33. (a) 1 34. If 3𝑛−1 𝒂𝒏+𝟏 +𝒃𝒏+𝟏 𝒂𝒏 +𝒃𝒏 (a) ✔0 35. If (a) 0 36. If (a) 37. (a) 38. (a) 1 (b) 3𝑛 − 1 (c) 2𝑛 + 1 (d) 3𝑛+1 Harmonic mean between 𝟐 and 𝟖 is: 16 5 ✔5 (b) 5 (c) ±4 (d) 16 If 𝑨, 𝑮 and 𝑯 are Arithmetic , Geometric and Harmonic means between two positive numbers then 𝐺 2 = 𝐴𝐻 (b) ✔𝐴, 𝐺, 𝐻 𝑎𝑟𝑒 𝑖𝑛 𝐺. 𝑃 (c) 𝐴 > 𝐺 > 𝐻 (d) all of these If 𝑨, 𝑮 and 𝑯 are Arithmetic , Geometric and Harmonic means between two negative numbers then 𝐺 2 = 𝐴𝐻 (b) 𝐴, 𝐺, 𝐻 𝑎𝑟𝑒 𝑖𝑛 𝐺. 𝑃 (c) 𝐴 < 𝐺 < 𝐻 (d) ✔all of these If 𝒂 and 𝒃 are two positive number then 𝐴 𝐻 (c) 𝐴 = 𝐺 = 𝐻 (d) 𝐴 ≥ 𝐺 ≥ 𝐻 If 𝒂 and 𝒃 are two negative number then ✔𝐴 < 𝐺 < 𝐻 (b) 𝐴 > 𝐺 > 𝐻 (c) 𝐴 = 𝐺 = 𝐻 (d) 𝐴 ≥ 𝐺 ≥ 𝐻 ✔ 𝒂𝒏 +𝒃𝒏 𝒂𝒏−𝟏 +𝒃𝒏−𝟏 𝒂𝒏+𝟏 +𝒃𝒏+𝟏 𝒂𝒏 +𝒃𝒏 is A.M between 𝒂 and 𝒃 then 𝒏 is equal to: (b) -1 (d) 2 is G.M between 𝒂 and 𝒃 then 𝒏 is equal to: (b) -1 is H.M between 𝒂 and 𝒃 then 𝒏 is equal to: (b) 𝑛(𝑛+1)(𝑛+2) 6 (c) 1 (c) (2𝑛 − 1)2 (c) ✔ 1 (d) ✔ 2 (c) 1 0 (b) ✔ -1 𝟐 If 𝑺𝒏 = (𝒏 + 𝟏) , then 𝑺𝟐𝒏 is equal to: 2𝑛 + 1 (b) ✔ 4𝑛2 + 4𝑛 + 1 ∑ 𝒏𝟑 = 𝑛(𝑛+1) 2 1 (c) 1 (d) 1 2 (d) cannot be determined 𝑛2 (𝑛+1)2 4 (d) 𝑛(𝑛+1)2 2 UNIT # 07 Permutation, Combination and Probability Each question has four possible answer. Tick the correct answer. 1. 𝟐𝟎𝑷𝟑= (a) 6890 2. If 𝒏𝑷𝟐= 30 then 𝒏 = (a) 4 (b) 6810 (c) ✔6840 (d) 6880 (b) 5 (c) ✔6 (d) 10 3. The number of diagonals in 10-sided figure is (a) 10 (b) 10𝐶2 (c) ✔10𝐶2 − 10 (d) 45 4. 𝒏𝑪𝟎= (a) 0 (b) ✔ 1 (c) 𝑛 (d) 𝑛! 5. How many arrangement of the word “MATHEMATICS” can be made 11 11 (a) 11! (b) ( ) (c) ✔ ( ) (d) None 3,2,1,1,1,1,1 2,2,2,1,1,1,1,1 6. If 𝒏 is negative then 𝒏! is (a) 1 (b) 0 (c) unique (d) ✔Not defined 13 | P a g e 7. 𝒏 − 𝟏𝑪𝒓 + 𝒏 − 𝟏𝑪𝒓−𝟏 equals (a) ✔𝑛𝐶𝑟 (b) 𝑛𝐶𝑟−1 (c) 𝑛 − 1𝐶𝑟 (d) 𝑛 + 1𝐶𝑟 8. How many signals can be given by 5 flags of different colors , using 3 at a time (a) 120 (b) ✔60 (c) 24 (d) 15 9. 𝒏𝑷𝒏 = (a) 1 (b) 𝑛 (c) ✔ 𝑛! (d) None 10. 𝟖! 𝟕! = 8 (a) ✔8 (b) 7 (c) 56 (d) 7 11. If an event 𝑨 can occur in 𝒑 ways and 𝑩 can occur 𝒒 ways , then number of ways that both events occur is: (a) 𝑝 + 𝑞 (b) ✔ 𝑝. 𝑞 (c) (𝑝𝑞)! (d) (𝑝 + 𝑞)! 𝒏 𝒏 12. If (𝟏𝟐) = (𝟖) then the value of 𝒏, (a) 15 (b) 16 (c) 18 (d) ✔20 13. Probability of non-occurrence of an event E is equal to : (a) ✔1 − 𝑃(𝐸) 14. For independent events 𝑷(𝑨 ∩ 𝑩) = (a) 𝑃(𝐴) + 𝑃(𝐵) 𝑛(𝑆) (b) 𝑃(𝐸) + 𝑛(𝐸) (b) 𝑃(𝐴) − 𝑃(𝐵) 𝑛(𝑆) (c) 𝑛(𝐸) (c) ✔ 𝑃(𝐴). 𝑃(𝐵) (d) 1 + 𝑃(𝐸) (d) 𝑃(𝐴) 𝑃(𝐵) 15. A card is drawn from a deck of 52 playing cards. The probability of card that it is an ace card is: 2 4 1 17 (a) 13 (b) 13 (c) ✔ 13 (d) 13 16. Four persons wants to sit in a circular sofa, the total ways are: (a) 24 (b) ✔6 (c) 4 (d) None of these 17. Two teams 𝑨 and 𝑩 are playing a match, the probability that team 𝑨 does not lose is: 2 1 (b) 3 (c) 1 (d) 0 (a) 3 18. Let 𝑺 = {𝟏, 𝟐, 𝟑, … , 𝟏𝟎} the probability that a number is divided by 4 is : 19. 20. 21. 22. 1 2 1 1 (a) 5 (b) ✔ (c) 10 (d) 2 5 A die is rolled , the probability of getting 3 or 5 is: 2 15 15 1 (a) 3 (b) ✔ 36 (c) 36 (d) 36 A coin is tossed 5 times , then 𝒏(𝑺) is equal to: (a) ✔32 (b) 25 (c) 10 (d) 20 The number of ways for sitting 4 persons in a train on a straight sofa is: (a) ✔24 (b) 6 (c) 4 (d) None of these Sample space for tossing a coin is: (a) {𝐻} (b) {𝑇} (c) {𝐻, 𝐻} (d) ✔ {𝐻, 𝑇} 𝟕 23. If 𝑷(𝑬) = 𝟏𝟐, 𝒏(𝑺) = 𝟖𝟒𝟎𝟎 , 𝒏(𝑬) = (a) 108 (b) ✔4900 (c) 144 24. In a permutation 𝒏𝑷𝒓 𝒐𝒓 𝑷(𝒏, 𝒓) , it is always true that (a) ✔𝑛 ≥ 𝑟 (b) 𝑛 < 𝑟 (c) 𝑛 ≤ 𝑟 (d) 14400 (d) 𝑛 < 0, 𝑟 < 0 25. If an event always occurs , then it is called: (a) Null Event (b) Possible Event (c) ✔Certain event (d) Independent Event 26. If 𝑬 is a certain event , then (a) 𝑃(𝐸) = 0 (b) ✔𝑃(𝐸) = 1 (c) 0 < 𝑃(𝐸) < 1 (d) 𝑃(𝐸) > 1 27. If 𝑬 is an impossible event ,then (a) ✔𝑃(𝐸) = 0 (b) 𝑃(𝐸) = 1 (c) 𝑃(𝐸) ≠ 0 (d) 0 < 𝑃(𝐸) < 1 28. Non occurrence of an event E is denoted by: (a) ∼ 𝐸 (b) ✔ 𝐸 (c) 𝐸 𝑐 (d) All of these 14 | P a g e UNIT # 08 Mathematical Induction and Binomial Theorem Each question has four possible answer. Tick the correct answer. 1. The statement 𝟒𝒏 + 𝟑𝒏 + 𝟒 is true when : (a) 𝑛 = 0 (b) 𝑛 = 1 (c) ✔ 𝑛 ≥ 2 integer 2. The number of terms in the expansion of (𝒂 + 𝒃)𝒏 are: (a) 𝑛 (b) ✔ 𝑛 + 1 (c) 2𝑛 3. Middle term/s in the expansion of (𝒂 − 𝟑𝒙)𝟏𝟒 is/are : (a) 𝑇7 (b) ✔ 𝑇8 (c) 𝑇6 &𝑇7 4. The coefficient of the last term in the expansion of (𝟐 − 𝒙)𝟕 is : (a) 1 (b) ✔ −1 (c) 7 𝟐𝒏 𝟐𝒏 𝟐𝒏 𝟐𝒏 5. ( 𝟎 ) + ( 𝟏 ) + ( 𝟐 ) + ⋯ + (𝟐𝒏) is equal to: (a) 2𝑛 (b) ✔22𝑛 (c) 22𝑛−1 𝟐 𝟑 6. 𝟏 + 𝒙 + 𝒙 + 𝒙 + ⋯ (b) ✔(1 − 𝑥)−1 (c) (1 + 𝑥)−2 (a) (1 + 𝑥)−1 𝒏 7. The middle term in the expansion of (𝒂 + 𝒃)𝒏 is ( + 𝟏) ; then 𝒏 is (a) 8. (a) 9. (a) 10. (a) 11. (a) 12. (a) 𝟐 Odd (b) ✔ even (c) prime The number of terms in the expansion of (𝒂 + 𝒃)𝟐𝟎 is: 18 (b) 20 (c) ✔ 21 The expansion (𝟏 − 𝟒𝒙)−𝟐 is valid if: 1 1 ✔|𝑥| < 4 (b) |𝑥| > (c) −1 < 𝑥 < 1 4 𝒏 The statement 𝟑 < 𝑛! is true, when 𝑛=2 (b) 𝑛 = 4 (c) 𝑛 = 6 General term in the expansion of (𝒂 + 𝒃)𝒏 is: 𝑛 𝑛 (b)✔(𝑟−1 )𝑎𝑛−𝑟 𝑥𝑟 (c) (𝑟+1 (𝑛+1 )𝑎𝑛−𝑟 𝑥^𝑟 )𝑎𝑛−𝑟 𝑥 𝑟 𝑟 The method of induction was given by Francesco who lived from: ✔1494-1575 (b) 1500-1575 (c) 1498-1575 (d) 𝑛 is any +iv (d) 2𝑛−1 (d) 𝑇7 &𝑇8 (d) −7 (d) 22𝑛+1 (d) (1 − 𝑥)−2 (d) none of these (d) 19 (d) |𝑥| < −1 (d) ✔𝑛 > 6 (d) (𝑛𝑟)𝑎𝑛−𝑟 𝑥 𝑟 (d) 1494-1570 UNIT # 09 Fundamentals of Trigonometry Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) 4. (a) 5. (a) 6. (a) 7. (a) 8. (a) Two rays with a common starting point form: Triangle (b) ✔ Angle (c) Radian (d) Minute The common starting point of two rays is called: Origin (b) Initial Point (c) ✔ Vertex (d)All of these If the rotation of the angle is counter clock wise, then angle is: Negative (b) ✔ Positive (c) Non-Negative (d) None of these If the initial ray ⃗⃗⃗⃗⃗⃗ 𝑶𝑨 rates in anti-clockwise direction in such a way that it coincides with itself, the angle then formed is: 180° (b) 270° (c) 300° (d) ✔ 360° One rotation in anti-clock wise direction is equal to: 180° (b) 270° (c) ✔360° (d) 90° Straight line angle is equal to 1 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 (b) 𝜋 radian (c) 180° (d) ✔ All of these 2 One right angle is equal to 𝜋 1 𝑟𝑎𝑑𝑖𝑎𝑛 (b) 90° (c) 4 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 (d) ✔ All of these 2 𝟏° is equal to 1 1 30 minutes (b) ✔ 60 minutes (c) 60 minutes (d) 2 minutes 15 | P a g e 9. (a) 10. (a) 11. (a) 12. (a) 13. (a) 14. (a) 15. (a) 16. (a) 17. (a) 18. (a) 19. (a) 20. (a) 21. (a) 22. (a) 23. (a) 24. (a) 25. (a) 26. (a) 27. (a) 28. (a) 29. (a) 30. (a) 31. (a) 32. (a) 33. (a) 𝟏° is equal to 1 360′′ (b) ✔ 3600′′ (c) (360)′ (d) 60′′ th 60 part of 𝟏° is equal to One second (b) ✔ One minute (c) 1 Radian (d) 𝜋 radian 60th part of 𝟏′ is equal to 1’ (b) ✔ 1’’ (c) 60’’ (d) 3600’’ th 3600 part of 𝟏° is equal to 1’ (b) ✔ 1’’ (c) 60’’ (d) 3600’’ Sexagesimal system is also called German System (b) ✔ English System (c) C.G.S System (d) SI System 𝟏𝟔°𝟑𝟎′ equal to 32° (c) 16.05° (d) 16.2° ✔16.5° (b) 2 ′ ′ Conversion of 𝟐𝟏. 𝟐𝟓𝟔° to 𝑫°𝑴 𝑺 ′ form is: 21°25’6’’ (b) 21°40′27′′ (c) ✔ 21°15′22′′ (d) 21°30′2′′ The angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle is called: 1 Degree (b) 1’ (c) ✔ 1 Radian (d) 1’’ The system of angular measurement in which the angle is measured in radian is called: Sexagesimal System (b) ✔ Circular System (c) English System (d) Gradient System Relation between the length of arc of a circle and the circular measure of it central angle is: 𝑟 𝑙 1 𝑙=𝜃 (b) 𝜃 = 𝑙𝑟 (c) ✔ 𝜃 = 𝑟 (d) 𝑙 = 2 𝑟 2 𝜃 With usual notation , if 𝒍 = 𝟔𝒄𝒎, 𝒓 = 𝟐𝒄𝒎, then unit of 𝜽 is: 𝑐𝑚 (b) 𝑐𝑚2 (c) ✔ No unit (d) 𝑐𝑚3 𝟏° is equal to: 𝜋 (180) ° 𝟏° is equal to: 0.175 𝑟𝑎𝑑 1 radian is equal to 𝜋 (b) 180 𝜋 𝑟𝑎𝑑 (b) ✔0.0175𝑟𝑎𝑑 180 (180) ° (b) 𝜋 𝑟𝑎𝑑 1 radian is equal to: ✔57.296° (b) 5.7296° 3 radian is: ✔171.888° (b) 120° 𝟏𝟎𝟓° = __________𝒓𝒂𝒅𝒊𝒂𝒏 7𝜋 2𝜋 ✔ 12 (b) 3 3’’= ___________ 𝒓𝒂𝒅𝒊𝒂𝒏 53𝜋 𝜋 (b) ✔ 216000 270 𝝅 𝒓𝒂𝒅𝒊𝒂𝒏 𝟒 ✔45° = ________𝒅𝒆𝒈𝒓𝒆𝒆 180 ) 𝑟𝑎𝑑 𝜋 (c) ( (c) 1.75 𝑟𝑎𝑑 180 ° ) 𝜋 𝜋 (d) ✔ 180 𝑟𝑎𝑑 (d) 0.00175𝑟𝑎𝑑 𝜋 𝑟𝑎𝑑 180 (c) ✔ ( (d) (c) 175.27° (d) 17.5276 (c) 300° (d) 270° (c) 5𝜋 12 41𝜋 (c) 720 (d) (d) 5𝜋 6 27721𝜋 32400 (b) 30° (c) 60° (d) 75° Circular measure of angle between the hands of a watch at 𝟒′ 𝑶 𝒄𝒍𝒐𝒄𝒌 is 3𝜋 45° (b) ✔ 120° (c) 2 (d) 270° If 𝒍 = 𝟏. 𝟓 𝒄𝒎 & 𝒓 = 𝟐. 𝟓 𝒄𝒎 then 𝜽 is equal to: 3 5 ✔5 (b) 3 (c) 3.75 (d) 𝑁𝑜𝑛𝑒 If 𝜽 = 𝟒𝟓° , 𝒓 = 𝟏𝟖𝒎𝒎 , then 𝒍 = 9 2 ✔2 𝜋 (b) 𝜋 (c) 812mm (d) 810mm 9 Area of sector of circle of radius 𝒓 is: 1 1 1 1 ✔2 𝑟 2 𝜃 (b) 2 𝑟𝜃 2 (c) 2 (𝑟𝜃)2 (d) 2𝑟2 𝜃 Angles with same initial and terminal sides are called: Acute angles (b) Allied Angles (c) ✔Coterminal angles (d) Quadrentel angles If angle 𝜽 is in degree, then the angle coterminal with 𝜽 is: 𝜃 + 180°𝑘, 𝑘 ∈ 𝑍 (b) ✔ 𝜃 + 360°𝑘, 𝑘 ∈ 𝑍 (c) 𝜃 + 90°𝑘, 𝑘 ∈ 𝑍 (d) 𝜃 + 60°𝑘, 𝑘 ∈ 𝑍 16 | P a g e 34. (a) 35. (a) 36. (a) 37. (a) 38. (a) 39. (a) 40. (a) 41. (a) 42. (a) 43. (a) 44. (a) 45. (a) 46. (a) 47. (a) 48. (a) If angle 𝜽 is in degree, then the angle coterminal with 𝜽 is: 𝜋 𝜋 ✔𝜃 + 2𝜋𝑘, 𝑘 ∈ 𝑍 (b) 𝜃 + 𝜋𝑘, 𝑘 ∈ 𝑍 (c) 𝜃 + 𝑘, 𝑘 ∈ 𝑍 (d) 𝜃 + 3 𝑘, 𝑘 ∈ 𝑍 2 An angle is in standard position , if its vertex is ✔At origin (b) at 𝑥 − 𝑎𝑥𝑖𝑠 (c) 𝑎𝑡𝑦 − 𝑎𝑥𝑖𝑠 (d) in 1st Quad Only If initial and the terminal side of an angle falls on 𝒙 − 𝒂𝒙𝒊𝒔 𝒐𝒓 𝒚 − 𝒂𝒙𝒊𝒔 then it is called: Coterminal angle (b) ✔ Quadrantal angl (c) Allied angle (d) None of these 0°, 90°, 180°, 270° and 360° are called Coterminal angle (b) ✔ Quadrantal angl (c) Allied angle (d) None of these 𝟐 𝟐 𝒔𝒊𝒏 𝜽 + 𝒄𝒐𝒔 𝜽 is equal to: 0 (b) -1 (c) 2 (d) ✔ 1 𝟏 + 𝒕𝒂𝒏𝟐 𝜽 is equal to: csc 2 𝜃 (b) sin2 𝜃 (c) ✔sec 2 𝜃 (d) tan2 𝜃 𝐜𝐬𝐜 𝟐 𝜽 − 𝐜𝐨𝐭 𝟐 𝜽 is equal to: 0 (b) ✔ 1 (c) -1 (d) 2 If 𝒔𝒊𝒏𝜽 < 0 and 𝒄𝒐𝒔𝜽 > 0 then the terminal arm of angle lies in ………….. Quad. I (b) II (c) III (d) ✔ IV If 𝒄𝒐𝒕𝜽 > 0 and 𝒄𝒐𝒔𝒆𝒄𝜽 > 0 then the terminal arm of angle lies in ………….. Quad. ✔I (b) II (c) III (d) IV If 𝒕𝒂𝒏𝜽 < 0 and 𝒄𝒐𝒔𝒆𝒄𝜽 > 0 then the terminal arm of angle lies in ………….. Quad. I (b) ✔ II (c) III (d) IV If 𝒔𝒆𝒄𝜽 < 0 and 𝒔𝒊𝒏𝜽 < 0 then the terminal arm of angle lies in ………….. Quad. I (b) II (c) ✔ III (d) IV In right angle triangle, the measure of the side opposite to 𝟑𝟎° is: ✔Half of Hypotenuse (b) Half of Base (c) Double of base (d) None of these The point (𝟎, 𝟏) lies on the terminal side of angle: 0° (b) ✔ 90° (c) 180° (d) 270° The point (−𝟏, 𝟎) lies on the terminal side of angle: 0° (b) 90° (c) ✔ 180° (d) 270° The point (𝟎, −𝟏) lies on the terminal side of angle: 0° (b) 90° (c) 180° (d) ✔ 270° 𝟏 49. 𝟐𝑺𝒊𝒏𝟒𝟓° + 𝟐 𝑪𝒐𝒔𝒆𝒄𝟒𝟓° = 2 (a) √3 3 √2 (c) −1 (b) ✔ 50. Domain of 𝒔𝒊𝒏𝜽 is: (2𝑛+1)𝜋 (a) ✔𝑅 (b) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 51. Domain of 𝒄𝒐𝒔𝜽 is: (c) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ (a) 𝑅 (b) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 53. Domain of 𝒔𝒆𝒄𝜽 is: (c) ✔ 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ (a) ✔𝑅 (b) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 52. Domain of 𝒕𝒂𝒏𝜽 is: (a) 54. (a) 55. (a) 56. (a) 𝑅 (b) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 𝒄𝒐𝒔𝒆𝒄𝜽𝒔𝒆𝒄𝜽𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽 = ✔1 (b) 0 (𝒔𝒆𝒄𝜽 + 𝒕𝒂𝒏𝜽)(𝒔𝒆𝒄𝜽 − 𝒕𝒂𝒏𝜽) = ✔1 (b) 0 𝟏−𝒔𝒊𝒏𝜽 𝒄𝒐𝒔𝜽 𝑐𝑜𝑠 1−𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃 (b) ✔ 1+𝑠𝑖𝑛𝜃 (c) 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ 2 (2𝑛+1)𝜋 2 (c) ✔ 𝜃 ∈ 𝑅 𝑏𝑢𝑡 𝜃 ≠ (c) 𝑠𝑖𝑛𝜃 𝑠𝑒𝑐𝜃 𝑠𝑖𝑛𝜃 (c) 1−𝑐𝑜𝑠𝜃 (d) 1 , 𝑛 ∈ 𝑍 (d) None of these , 𝑛 ∈ 𝑍 (d) None of these (2𝑛+1)𝜋 2 (2𝑛+1)𝜋 2 , 𝑛 ∈ 𝑍 (d) None of these , 𝑛 ∈ 𝑍 (d) None of these (d) 𝑐𝑜𝑠𝜃 (d) 𝑡𝑎𝑛𝜃 𝑠𝑖𝑛𝜃 (d) 1+𝑐𝑜𝑠𝜃 17 | P a g e UNIT # 10 Trigonometric Identities Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) 4. (a) 5. (a) Distance between the points 𝑨(𝟑, 𝟖) & 𝐵(5,6) is: ✔2√2 (b) 3 (c) 4 (d) √2 Fundamental law of trigonometry is , 𝒄𝒐𝒔(𝜶 − 𝜷) ✔𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (b) 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (c) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 (d) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 𝒄𝒐𝒔(𝜶 + 𝜷) is equal to: 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (b) ✔ 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (c) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 (d) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 𝒔𝒊𝒏(𝜶 + 𝜷) is equal to: 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (b) 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (c) ✔ 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 (d) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 𝒔𝒊𝒏(𝜶 − 𝜷) is equal to: 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (b) 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 (c) 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 (d) ✔ 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽 𝝅 6. 𝒄𝒐𝒔 ( 𝟐 − 𝜷) = (a) 𝑐𝑜𝑠𝛽 7. 𝒄𝒐𝒔 (𝜷 + (a) 𝑐𝑜𝑠𝛽 8. 𝒔𝒊𝒏 (𝜷 − (a) 9. (a) 10. (a) 11. (a) 𝝅 ) 𝟐 𝝅 ) 𝟐 = = 𝑐𝑜𝑠𝛽 𝒄𝒐𝒔(𝟐𝝅 − 𝜽) = ✔𝑐𝑜𝑠𝜃 𝒔𝒊𝒏(𝟐𝝅 − 𝜽) = 𝑐𝑜𝑠𝜃 𝒕𝒂𝒏(𝜶 + 𝜷) = 𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽 1+𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (b) – 𝑐𝑜𝑠𝛽 (c) ✔ 𝑠𝑖𝑛𝛽 (b) ✔ – 𝑐𝑜𝑠𝛽 (c) 𝑠𝑖𝑛𝛽 (b) – 𝑐𝑜𝑠𝛽 (b) – 𝑐𝑜𝑠𝜃 (b) – 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽 (d) – 𝑠𝑖𝑛𝛽 (c) 𝑠𝑖𝑛𝛽 (d) ✔ – 𝑠𝑖𝑛𝛽 (c) 𝑠𝑖𝑛𝜃 (d) – 𝑠𝑖𝑛𝜃 (c) 𝑠𝑖𝑛𝜃 𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽 (d) ✔ – 𝑠𝑖𝑛𝛽 (d) ✔ – 𝑠𝑖𝑛𝜃 𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽 (b) ✔1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (c) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (d) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (b) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (c) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (d) 1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 (b) 𝑡𝑎𝑛𝜃 (c) – 𝑐𝑜𝑡𝜃 (d) – 𝑡𝑎𝑛𝜃 (a) 𝑠𝑖𝑛𝜃 (b) 𝑐𝑜𝑠𝜃 17. 𝒄𝒐𝒔 𝟑𝟏𝟓° is equal to: (c) −𝑠𝑖𝑛𝜃 12. 𝒕𝒂𝒏(𝜶 − 𝜷) = 𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽 (a) ✔1+𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽 𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽 𝑡𝑎𝑛𝛼−𝑡𝑎𝑛𝛽 𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽 13. Angles associated with basic angles of measure 𝜽 to a right angle or its multiple are called: (a) Coterminal angle (b) angle in standard position (c) ✔ Allied angle (d) obtuse angle 𝝅 14. 𝒕𝒂𝒏 ( 𝟐 − 𝜽) = (a) ✔𝑐𝑜𝑡𝜃 15. 𝝅 𝒕𝒂𝒏 ( 𝟐 + (a) 𝑐𝑜𝑡𝜃 16. 𝜽) = 𝟑𝝅 𝒔𝒊𝒏 ( 𝟐 + (a) 1 𝜽) = (b) 𝑡𝑎𝑛𝜃 (b) 0 18. 𝒕𝒂𝒏(−𝟏𝟑𝟓°) is equal to: (a) ✔ 1 (b) 0 19. (a) 20. (a) 21. (a) (c) ✔– 𝑐𝑜𝑡𝜃 (c) ✔ (c) 1 √3 1 √2 𝒔𝒆𝒄(−𝟑𝟎𝟎°) = 1 (b) ✔ 0 (c) 2 𝒔𝒊𝒏(𝟏𝟖𝟎° + 𝜶)𝒔𝒊𝒏(𝟗𝟎° − 𝜶) = ✔ 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 (b) – 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 (c) 𝑐𝑜𝑠𝛾 If 𝜶, 𝜷 and 𝜸 are the angles of a triangle ABC then 𝒔𝒊𝒏(𝜶 + 𝜷) = ✔ 𝑠𝑖𝑛𝛾 (b) – 𝑠𝑖𝑛𝛾 (c) 𝑐𝑜𝑠𝛾 (d) – 𝑡𝑎𝑛𝜃 (d) ✔ – 𝑐𝑜𝑠𝜃 (d) √3 2 (d) -1 (d) -1 (d) – 𝑐𝑜𝑠𝛾 (d) – 𝑐𝑜𝑠𝛾 18 | P a g e 𝜶+𝜷 ) 𝟐 22. If 𝜶, 𝜷 and 𝜸 are the angles of a triangle ABC then 𝒄𝒐𝒔 ( 𝛾 𝛾 (b) – sin 2 𝛾 (c) cos 2 = (a) ✔ sin 2 23. If 𝜶, 𝜷 and 𝜸 are the angles of a triangle ABC then 𝒄𝒐𝒔(𝜶 + 𝜷) = (a) 𝑠𝑖𝑛𝛾 (b) – 𝑠𝑖𝑛𝛾 (c) 𝑐𝑜𝑠𝛾 𝒄𝒐𝒔𝟏𝟏°+𝒔𝒊𝒏𝟏𝟏° 24. 𝒄𝒐𝒔𝟏𝟏°−𝒔𝒊𝒏𝟏𝟏° = (a) 25. (a) 26. (a) 27. ✔ 𝑡𝑎𝑛56° (b) 𝑡𝑎𝑛34° 𝒔𝒊𝒏𝟐𝜶 is equal to: cos2 𝛼 − sin2 𝛼 (b) 1 + 𝑐𝑜𝑠2𝛼 𝒄𝒐𝒔𝟐𝜶 = cos2 𝛼 − sin2 𝛼 (b) 1 − 2 sin2 𝛼 𝒕𝒂𝒏𝟐𝜶 = 2𝑡𝑎𝑛𝛼 (b) ✔ (a) 1+𝑡𝑎𝑛2 𝛼 28. 𝒔𝒊𝒏𝜶 + 𝒔𝒊𝒏𝜷 is equal to: 𝛼+𝛽 (a) ✔ 2 sin ( 2 ) cos ( 𝛼−𝛽 2 ) 2𝑡𝑎𝑛𝛼 1−tan2 𝛼 𝛼+𝛽 𝛼−𝛽 ) sin ( 2 ) 2 (c) −2 sin ( 29. 𝒔𝒊𝒏𝜶 − 𝒔𝒊𝒏𝜷 is equal to: 𝛼+𝛽 𝛼−𝛽 ) cos ( 2 ) 2 𝛼−𝛽 𝛼+𝛽 (c) −2 sin ( 2 ) sin ( 2 ) 𝛼+𝛽 𝛼−𝛽 ) cos ( 2 ) 2 𝛼−𝛽 𝛼+𝛽 ) sin ( ) (c) -2 sin ( 2 2 32. (a) 33. (a) 34. (b) (c) ✔ − 2 sin ( 2 (d) 𝑐𝑜𝑡34° (c) 2 cos2 𝛼 − 1 (d) ✔ All of these (c) ✔ 2𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 2 tan2 𝛼 (c) 1−tan2 𝛼 (d) 2𝑠𝑖𝑛2𝛼𝑐𝑜𝑠2𝛼 tan2 𝛼 (d) 1−tan2 𝛼 𝛼+𝛽 𝛼−𝛽 ) sin ( 2 ) 2 𝛼+𝛽 𝛼−𝛽 (d) 2 cos ( 2 ) cos ( 2 ) (b) 2 cos ( 𝛼+𝛽 2 𝛼−𝛽 ) sin ( 𝛼+𝛽 𝛼−𝛽 ) cos ( 2 ) 2 (c) 2 sin ( 𝛼+𝛽 𝛼−𝛽 ) cos ( 2 ) 2 𝛼+𝛽 (c) 𝑐𝑜𝑡56° (d) 2 cos ( 30. 𝒄𝒐𝒔𝜶 + 𝒄𝒐𝒔𝜷 is equal to: (d) 2 sin ( (d) ✔ – 𝑐𝑜𝑠𝛾 (b) ✔ 2 cos ( (b) 2 sin ( 31. 𝒄𝒐𝒔𝜶 − 𝒄𝒐𝒔𝜷 is equal to: 𝛾 (d) – cos 2 𝛼+𝛽 𝛼−𝛽 ) sin ( 2 ) 2 𝛼+𝛽 𝛼−𝛽 (b) 2 cos ( (d) ✔ 2 cos ( 𝛼−𝛽 ) sin ( 2 ) 2 2 ) cos ( 𝛼+𝛽 𝛼−𝛽 ) sin ( 2 ) 2 𝛼+𝛽 𝛼−𝛽 (d) 2 cos ( 2 ) cos ( 2 ) (b) 2 cos ( 2 ) ) Which is the allied angle ✔ 90° + 𝜃 (b) 60° + 𝜃 (c) 45° + 𝜃 (d) 30° + 𝜃 𝟐𝒔𝒊𝒏𝟕𝜽𝒄𝒐𝒔𝟑𝜽 = ✔ 𝑠𝑖𝑛10𝜃 + 𝑠𝑖𝑛4𝜃 (b) 𝑠𝑖𝑛5𝜃 − 𝑠𝑖𝑛2𝜃 (c) 𝑐𝑜𝑠10𝜃 + 𝑐𝑜𝑠4𝜃 (d) 𝑐𝑜𝑠5𝜃 − 𝑐𝑜𝑠2𝜃 𝟐𝒄𝒐𝒔𝟓𝜽𝒔𝒊𝒏𝟑𝜽 = ✔ 𝑠𝑖𝑛8𝜃 − 𝑠𝑖𝑛2𝜃 (b) 𝑠𝑖𝑛8𝜃 + 𝑠𝑖𝑛2𝜃 (c) 𝑐𝑜𝑠8𝜃 + 𝑐𝑜𝑠2𝜃 (d) 𝑐𝑜𝑠8𝜃 − 𝑐𝑜𝑠2𝜃 UNIT # 11 Trigonometric Functions and their Graphs Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) 4. (a) 5. (a) Domain of 𝒚 = 𝒔𝒊𝒏𝒙 is ✔−∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c) −∞ < 𝑥 < ∞ , 𝑥 ≠ 𝑛𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 Domain of 𝒚 = 𝒄𝒐𝒔𝒙 is ✔−∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c) −∞ < 𝑥 < ∞ , 𝑥 ≠ 𝑛𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 Domain of 𝒚 = 𝒕𝒂𝒏𝒙 is 2𝑛+1 −∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c) ✔ −∞ < 𝑥 < ∞ , 𝑥 ≠ 2 𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 Domain of 𝒚 = 𝒔𝒆𝒄𝒙 is 2𝑛+1 −∞ < 𝑥 < ∞ (b) −1 ≤ 𝑥 ≤ 1 (c) ✔ −∞ < 𝑥 < ∞ , 𝑥 ≠ 𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 2 Domain of 𝒚 = 𝒄𝒔𝒄𝒙 is 2𝑛+1 −∞ < 𝑥 < ∞ (b)✔ −1 ≤ 𝑥 ≤ 1 (c) −∞ < 𝑥 < ∞ , 𝑥 ≠ 2 𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 19 | P a g e 6. (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. (a) 12. (a) 13. (a) 14. (a) 15. (a) 16. (a) 17. (a) 18. (a) 19. (a) 20. (a) 21. (a) Domain of 𝒚 = 𝒄𝒐𝒕𝒙 is 2𝑛+1 −∞ < 𝑥 < ∞ (b)✔ −1 ≤ 𝑥 ≤ 1 (c) −∞ < 𝑥 < ∞ , 𝑥 ≠ 2 𝜋 , 𝑛 ∈ 𝑍 (d) 𝑥 ≥ 1, 𝑥 ≤ −1 Range of 𝒚 = 𝒔𝒊𝒏𝒙 is 𝑅 (b) ✔−1 ≤ 𝑦 ≤ 1 (c) (−∞, 1) ∪ (1, ∞) (d) −1 < 𝑦 < 1 Range of 𝒚 = 𝒄𝒐𝒔𝒙 is 𝑅 (b) ✔ −1 ≤ 𝑦 ≤ 1 (c) (−∞, 1) ∪ (1, ∞) (d) −1 < 𝑦 < 1 Range of 𝒚 = 𝒕𝒂𝒏𝒙 is ✔𝑅 (b) −1 ≤ 𝑦 ≤ 1 (c) 𝑄 (d) 𝑅 − {0} Range of 𝒚 = 𝒄𝒐𝒕𝒙 is ✔𝑅 (b) 𝑅 − [−1,1] (c) 𝑅 − {0} (d) 𝑍 Range of 𝒚 = 𝒔𝒆𝒄𝒙 is 𝑅 (b) ✔ 𝑦 ≥ 1𝑜𝑟 𝑦 ≤ −1 (c) −1 ≤ 𝑦 ≤ 1 (d) 𝑅 − [−1,1] Range of 𝒚 = 𝒄𝒐𝒔𝒆𝒄𝒙 is 𝑅 (b) ✔ 𝑦 ≥ 1𝑜𝑟 𝑦 ≤ −1 (c) −1 ≤ 𝑦 ≤ 1 (d) 𝑅 − [−1,1] Smallest +𝒊𝒗𝒆 number which when added to the original circular measure of the angle gives the same value of the function is called: Domain (b) Range (c) Co domain (d) ✔Period Period of 𝒔𝒊𝒏𝜽 is 𝜋 𝜋 (b) ✔ 2𝜋 (c) −2𝜋 (d) 2 Period of 𝒄𝒐𝒔𝒆𝒄𝜽 is 𝜋 𝜋 (b) ✔2𝜋 (c) −2𝜋 (d) 2 Period of 𝒄𝒐𝒔𝜽 is 𝜋 𝜋 (b) ✔2𝜋 (c) −2𝜋 (d) 2 Period of 𝒕𝒂𝒏𝜽 is 𝜋 ✔𝜋 (b) 2𝜋 (c) −2𝜋 (d) 2 Period of 𝒄𝒐𝒕𝜽 is 𝜋 ✔𝜋 (b) 2𝜋 (c) −2𝜋 (d) 2 Period of 𝒔𝒆𝒄𝜽 is 𝜋 𝜋 (b) ✔2𝜋 (c) −2𝜋 (d) 2 Period of 𝒕𝒂𝒏𝟒𝒙 is 𝜋 𝜋 (b) 2𝜋 (c) −2𝜋 (d) ✔ 4 Period of 𝒄𝒐𝒕𝟑𝒙 is 𝜋 𝜋 𝜋 (b) ✔ (c) −2𝜋 (d) 22. Period of 𝟑𝒄𝒐𝒔 𝒙 𝟓 is 𝜋 3 4 (a) 2𝜋 (b) (c) 𝜋 (d) ✔ 10𝜋 2 23. The graph of trigonometric functions have: (a) Break segments (b) Sharp corners (c) Straight line segments (d) ✔ smooth curves 24. Curves of the trigonometric functions repeat after fixed intervals because trigonometric functions are (a) Simple (b) linear (c) quadratic (d) ✔ periodic 25. The graph of 𝒚 = 𝒄𝒐𝒔𝒙 lies between the horizontal line 𝒚 = −𝟏 and (a) ✔ +1 (b) 0 (c) 2 (d) -2 UNIT # 12 Application of Trigonometry Each question has four possible answer. Tick the correct answer. 1. (a) 2. (a) 3. (a) A “Triangle” has : Two elements (b) 3 elements (c) 4 elements (d) ✔ 6 elmen When we look an object above the horizontal ray, the angle formed is called angle of: ✔Elevation (b) depression (c) incidence (d) reflects When we look an object below the horizontal ray, the angle formed is called angle of: Elevation (b) ✔ depression (c) incidence (d) reflects 20 | P a g e 4. A triangle which is not right is called: (a) ✔Oblique triangle (b) Isosceles triangle triangle 5. To solve an oblique triangle we use: (a) ✔Law of Sine (b) Law of Cosine 6. In any triangle 𝑨𝑩𝑪, (a) 7. (a) 8. (a) 9. (a) ✔𝑐𝑜𝑠𝛼 𝒃𝟐 +𝒄𝟐 −𝒂𝟐 (c) Scalene triangle (c) Law of Tangents (d) All of these = 𝟐𝒃𝒄 (b) 𝑠𝑖𝑛𝛼 (c) 𝑐𝑜𝑠𝛽 (d) 𝑐𝑜𝑠𝛾 Which can be reduced to Pythagoras theorem, Law of sine (b) ✔ law of cosine (c) law of tangents (d) Half angle formulas In any triangle 𝑨𝑩𝑪, if 𝜷 = 𝟗𝟎° , then 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄𝒄𝒐𝒔𝜷 becomes: Law of sine (b) Law of tangents (c) ✔Law of cosine (d) None of these In any triangle 𝑨𝑩𝑪, law of tangent is : 𝑎−𝑏 𝑎+𝑏 tan(𝛼−𝛽) = tan(𝛼+𝛽) 𝑎+𝑏 𝛼 (a) sin 2 𝛼 2 𝛼 𝛽 (b) sin 2 (𝑺−𝒂)(𝒔−𝒄) = 𝒃𝒄 𝛽 (b) sin 2 𝒂𝒄 𝜶 𝟐 𝑠(𝑠−𝑎) 𝑎𝑏 (b) √ 𝑠(𝑠−𝑎) 𝑎𝑏 (b) ✔√ 𝑠(𝑠−𝑎) 𝑎𝑏 (b) √ 𝜷 14. In any triangle 𝑨𝑩𝑪, 𝒄𝒐𝒔 𝟐 = (a) √ 𝜸 15. In any triangle 𝑨𝑩𝑪, 𝒄𝒐𝒔 𝟐 = (a) √ 𝛽 (b) ✔ sin 2 13. In any triangle 𝑨𝑩𝑪, 𝒄𝒐𝒔 = (a) √ = = 12. In any triangle 𝑨𝑩𝑪, √ (a) sin 2 𝒂𝒃 (𝑺−𝒃)(𝒔−𝒄) 11. In any triangle 𝑨𝑩𝑪, √ (a) ✔ sin tan(𝛼+𝛽) (b) 𝑎−𝑏 = tan(𝛼−𝛽) (𝑺−𝒂)(𝒔−𝒃) 10. In any triangle 𝑨𝑩𝑪, √ 𝑠(𝑠−𝑏) 𝑎𝑐 𝑠(𝑠−𝑏) 𝑎𝑐 𝑠(𝑠−𝑏) 𝑎𝑐 𝑎−𝑏 (c) ✔ 𝑎+𝑏 = (c) ✔ sin (a) 𝑎 + 𝑏 + 𝑐 17. In any triangle 𝛾 (a) sin 2 18. In any triangle 𝛾 𝑎+𝑏+𝑐 3 𝒔(𝒔−𝒄) 𝑨𝑩𝑪, √(𝒔−𝒂)(𝒔−𝒃) = 𝛾 (b) cos 2 (𝒔−𝒂)(𝒔−𝒃) 𝑨𝑩𝑪, √ = 𝒔(𝒔−𝒄) 𝛾 (b) cos 2 (b) 𝛼−𝛽 2 𝛼+𝛽 tan 2 tan 𝑎−𝑏 (d) 𝑎+𝑏 = 𝛾 𝛼 𝛾 (d) cos 2 𝛼 𝛾 (d) cos 2 𝛼 (c) sin 2 (c) ✔ √ 𝑠(𝑠−𝑎) 𝑏𝑐 (d) √ 𝑠(𝑠−𝑐) 𝑎𝑏 𝑠(𝑠−𝑐) 𝑎𝑏 (c) √ 𝑠(𝑠−𝑎) 𝑏𝑐 (d) √ (c) √ 𝑠(𝑠−𝑎) 𝑏𝑐 (d) ✔ √ (c) ✔ 𝛼+𝛽 2 𝛼−𝛽 tan 2 tan (d) cos 2 2 (c) sin 2 16. In any triangle 𝑨𝑩𝑪, with usual notations , 𝒔 is equal to (a) 19. (a) 20. (a) 21. (a) 22. (a) 23. (a) (d) Right isosceles 𝑎+𝑏+𝑐 2 𝛾 (c) tan 2 (d) 𝑎𝑏𝑐 2 𝑠(𝑠−𝑐) 𝑎𝑏 (d) ✔ cot 𝛾 𝛾 𝛾 2 sin 2 (c) ✔ tan 2 (d) cot 2 To solve an oblique triangles when measure of three sides are given , we can use: ✔Hero’s formula (b) Law of cosine (c) Law of sine (d) Law of tangents The smallest angle of ∆𝑨𝑩𝑪, when 𝒂 = 𝟑𝟕. 𝟑𝟒 , 𝒃 = 𝟑. 𝟐𝟒 , 𝒄 = 𝟑𝟓. 𝟎𝟔 is 𝛼 (b) ✔𝛽 (c) 𝛾 (d) cannot be determined In any triangle 𝑨𝑩𝑪 Area if triangle is : 1 1 1 𝑏𝑐 sin 𝛼 (b) 2 𝑐𝑎 𝑠𝑖𝑛𝛼 (c) 2 𝑎𝑏 𝑠𝑖𝑛𝛽 (d) ✔ 2 𝑎𝑏𝑠𝑖𝑛𝛾 The circle passing through the thee vertices of a triangle is called: ✔Circum circle (b) in-circle (c) ex-centre (d) escribed circle The point of intersection of the right bisectors of the sides of the triangle is : ✔Circum centre (b) In-centre (c) Escribed center (d) Diameter 21 | P a g e 24. In any triangle 𝑨𝑩𝑪, with usual notations, 𝒂 𝟐𝒔𝒊𝒏𝜶 = (a) 𝑟 (b) 𝑟1 (c) ✔ 𝑅 𝒂 25. In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔𝒊𝒏𝜷 = (a) 26. (a) 27. (a) (d) ∆ 2𝑟 (b)2 𝑟1 (c) ✔2𝑅 In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔𝒊𝒏 𝜸 = 2𝑅 𝑐 (c) 𝑐 𝑅 (b) ✔ 2𝑅 In any triangle 𝑨𝑩𝑪, with usual notations, 𝒂𝒃𝒄 = 𝑅 (b) 𝑅𝑠 (c) ✔4𝑅∆ (d) 2∆ 𝑅 (d) 2 ∆ (d) 𝑠 ∆ 28. In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒂 = (a) 𝑟 (b) 𝑅 (c) ✔ 𝑟1 ∆ 29. In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒃 = (d) 𝑟2 (a) 𝑟 (b) 𝑅 (c) 𝑟1 ∆ 30. In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒄 = (a) 31. (a) 32. (a) 33. (a) (d) ✔ 𝑟2 ✔𝑟3 (b) 𝑅 (c) 𝑟1 In any triangle 𝑨𝑩𝑪, with usual notation , 𝒓: 𝑹: 𝒓𝟏 = 3:2:1 (b) 1:2:2 (c) ✔ 1:2:3 In any triangle 𝑨𝑩𝑪, with usual notation , 𝒓: 𝑹: 𝒓𝟏 : 𝒓𝟐 : 𝒓𝟑 = 3:3:3:2:1 (b) 1:2:2:3:3 (c) ✔ 1:2:3:3:3 In a triangle 𝑨𝑩𝑪, if 𝜷 = 𝟔𝟎° , 𝜸 = 𝟏𝟓° then 𝜶 = 90° (b) 180° (c) 150° (d) 𝑟2 (d) 1:1:1 (d) 1:1:1:1:1 (d) ✔105° UNIT # 13 Inverse trigonometric functions Each question has four possible answer.Tick the correct answer. 1. If 𝒚 = 𝑺𝒊𝒏𝒙, then Domain is : 𝜋 𝜋 (b) 0 ≤ 𝑥 ≤ 𝜋 (a) ✔− ≤ 𝑥 ≤ 2 2 2. If 𝒚 = 𝑪𝒐𝒔𝒙, then Domain is : (a) 3. (a) 4. 𝜋 𝜋 −2 ≤ 𝑥 ≤ 2 (b) ✔ 0 ≤ 𝑥 ≤ 𝜋 If 𝒚 = 𝑺𝒆𝒄𝒙, then Domain is : 𝜋 𝜋 −2 ≤ 𝑥 ≤ 2 (b) 0 ≤ 𝑥 ≤ 𝜋 If 𝒚 = 𝑪𝒐𝒔𝒆𝒄𝒙, then Domain is : 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 (a) − 2 ≤ 𝑥 ≤ 2 (b) 0 ≤ 𝑥 ≤ 𝜋 5. If 𝒚 = 𝑻𝒂𝒏𝒙, then Domain is : (a) − ≤ 𝑥 ≤ (b) ✔0 ≤ 𝑥 ≤ 𝜋 2 2 6. If 𝒚 = 𝑪𝒐𝒕𝒙, then Domain is : (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. (a) 12. (a) 13. (a) −2 ≤ 𝑥 ≤ 2 (b) 0 ≤ 𝑥 ≤ 𝜋 If 𝒚 = 𝑺𝒊𝒏𝒙, then range is : ✔−1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑪𝒐𝒔𝒙, then range is : ✔−1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑻𝒂𝒏𝒙, then range is : −1 ≤ 𝑥 ≤ 1 (b) ✔ (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑪𝒐𝒕𝒙, then range is : −1 ≤ 𝑥 ≤ 1 (b) ✔ (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑪𝒐𝒔𝒆𝒄𝒙, then range is : −1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑺𝒆𝒄𝒙, then range is : −1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑺𝒊𝒏−𝟏 𝒙, then domain is: ✔−1≤𝑥≤1 (b) (−∞, +∞)𝑜𝑟 𝑅 𝜋 𝜋 𝜋 (c) [0, 𝜋], 𝑥 ≠ 2 (d) [− 2 , 2 ] , 𝑥 ≠ 0 𝜋 (c) [0, 𝜋], 𝑥 ≠ 2 𝜋 𝜋 𝜋 (c) ✔ [0, 𝜋], 𝑥 ≠ 2 𝜋 (c) [0, 𝜋], 𝑥 ≠ 2 (c) [0, 𝜋], 𝑥 ≠ 𝜋 2 (d) [− 2 , 2 ] , 𝑥 ≠ 0 𝜋 𝜋 (d) [− 2 , 2 ] , 𝑥 ≠ 0 𝜋 𝜋 (d) ✔ [− 2 , 2 ] , 𝑥 ≠ 0 𝜋 𝜋 𝜋 (d) [− , ] , 𝑥 ≠ 0 2 2 𝜋 𝜋 (c) ✔ [0, 𝜋], 𝑥 ≠ 2 (d) [− 2 , 2 ] , 𝑥 ≠ 0 (c) 𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (c) 𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (c) 𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (c) 𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (c) ✔𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (c) ✔𝑦 ≤ −1 𝑜𝑟 𝑦 ≥ 1 (d) 𝑦 < −1𝑜𝑟 𝑦 > 1 (c) 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 22 | P a g e 14. (a) 15. (a) 16. (a) 17. (a) 18. (a) 19. (a) 20. (a) 21. (a) If 𝒚 = 𝑪𝒐𝒔−𝟏 𝒙, then domain is: ✔−1≤𝑥≤1 (b) (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑻𝒂𝒏−𝟏 𝒙, then domain is: −1 ≤ 𝑥 ≤ 1 (b) ✔ (−∞, +∞)𝑜𝑟 𝑅 If 𝒚 = 𝑪𝒐𝒕−𝟏 𝒙, then domain is: −1 ≤ 𝑥 ≤ 1 (b) ✔ (−∞, +∞)𝑜𝑟 𝑅 −𝟏 If 𝒚 = 𝑺𝒆𝒄 𝒙, then domain is: −1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 −𝟏 If 𝒚 = 𝑪𝒐𝒔𝒆𝒄 𝒙, then domain is: −1 ≤ 𝑥 ≤ 1 (b) (−∞, +∞)𝑜𝑟 𝑅 −𝟏 If 𝒚 = 𝑺𝒊𝒏 𝒙, then range is: 𝜋 𝜋 ✔− ≤𝑥≤ (b) 0 ≤ 𝑥 ≤ 𝜋 2 2 −𝟏 If 𝒚 = 𝑪𝒐𝒔 𝒙, then range is: 𝜋 𝜋 (b) ✔ 0 ≤ 𝑥 ≤ 𝜋 −2 ≤ 𝑥 ≤ 2 −𝟏 If 𝒚 = 𝑻𝒂𝒏 𝒙, then range is: 𝜋 𝜋 − ≤𝑥≤ (b) 0 ≤ 𝑥 ≤ 𝜋 2 2 22. If 𝒚 = 𝑪𝒐𝒕−𝟏 𝒙, then range is: 𝜋 𝜋 (a) − ≤ 𝑥 ≤ (b) 0 ≤ 𝑥 ≤ 𝜋 2 2 −𝟏 23. If 𝒚 = 𝒔𝒆𝒄 𝒙, then range is: 𝜋 𝜋 (a) ✔ − ≤ 𝑥 ≤ (b) 0 ≤ 𝑥 ≤ 𝜋 24. (a) 25. (a) 26. (a) 2 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 (c) 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 (c) 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (c) ✔ 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 𝜋 (c) – 2 < 𝑥 < 𝜋 (c) – 2 < 𝑥 < 𝜋 𝜋 2 𝜋 2 (c) ✔ – 2 < 𝑥 < 𝜋 (c) – 2 < 𝑥 < 𝜋 𝜋 2 (d) 0 < 𝑥 < 𝜋 𝜋 2 𝜋 2 If 𝒚 = 𝑪𝒐𝒔𝒆𝒄 𝒙, then range is: 𝜋 𝜋 𝜋 𝜋 −2 ≤ 𝑥 ≤ 2 (b) ✔ 0 ≤ 𝑥 ≤ 𝜋 (c) – 2 < 𝑥 < 2 Inverse of a function exist only if it is: Trigonometric function (b) ✔(1 − 1) function (c) onto function 𝑺𝒊𝒏−𝟏 𝒙 = 𝜋 𝜋 𝜋 (b) 2 − sin−1 𝑥 (c) 2 + cos−1 𝑥 ✔ − cos−1 𝑥 2 𝜋 (b) ✔ 2 − sin−1 𝑥 28. 𝑪𝒐𝒔−𝟏 𝒙 = 𝜋 (a) ✔ − sec−1 𝑥 (b) 30. 𝑻𝒂𝒏−𝟏 𝒙 = 𝜋 (a) 2 − sec −1 𝑥 (b) 2 −𝟏 𝜋 2 − sin−1 𝑥 𝜋 2 − sin−1 𝑥 29. 𝑺𝒆𝒄 𝒙 = 𝜋 (a) 2 − sec −1 𝑥 (b) 31. 𝑪𝒐𝒕−𝟏 𝒙 = 𝜋 (a) 2 − sec −1 𝑥 (b)✔ 32. 𝑺𝒊𝒏 (𝑪𝒐𝒔−𝟏 𝜋 6 √𝟑 ) 𝟐 = 𝜋 2 (b) ✔ 33. 𝑻𝒂𝒏−𝟏 (√𝟑) = 𝜋 (a) 6 (b) – 6 (a) ✔ (b) 𝟏 34. 𝑺𝒊𝒏 (𝑺𝒊𝒏−𝟏 𝟐) = 1 2 −𝟏 35. 𝑺𝒊𝒏 𝑨 + 𝑺𝒊𝒏−𝟏 𝑩 = 𝜋 𝜋 2 2 3 (a) ✔𝑆𝑖𝑛−1 (𝐴√1 − 𝐵2 + 𝐵√1 − 𝐴2 ) (c) 𝑆𝑖𝑛−1 (𝐵√1 − 𝐴2 + 𝐴√1 − 𝐵2 ) −𝟏 36. 𝑺𝒊𝒏 𝑨 − 𝑺𝒊𝒏−𝟏 𝑩 = (a) 𝑆𝑖𝑛−1 (𝐴√1 − 𝐵2 + 𝐵√1 − 𝐴2 ) (c) 𝑆𝑖𝑛−1 (𝐵√1 − 𝐴2 + 𝐴√1 − 𝐵2 ) 𝜋 (c) 2 + sec −1 𝑥 (c) 2 + sec −1 𝑥 𝜋 − tan−1 𝑥 1 2 𝜋 (c) 2 + cos−1 𝑥 𝜋 − sin−1 𝑥 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 (c) ✔ 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (d) 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 (c) – 2 < 𝑥 < 2 −𝟏 27. 𝑪𝒐𝒔−𝟏 𝒙 = 𝜋 (a) 2 − cos−1 𝑥 (a) (c) 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 (c) ✔ 2 −cot −1 𝑥 𝜋 (c) 2 + sec −1 𝑥 (c) − 1 2 𝜋 (d) 0 < 𝑥 < 𝜋 (d) 0 < 𝑥 < 𝜋 (d) ✔ 0 < 𝑥 < 𝜋 (d) 0 < 𝑥 < 𝜋 (d) 0 < 𝑥 < 𝜋 (d) an into function 𝜋 (d) 2 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 𝜋 (d) 2 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 𝜋 (d) 2 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 𝜋 (d) ✔ 2 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 (d) (d) (d) 𝜋 2 𝜋 2 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 − 𝑐𝑜𝑠𝑒𝑐 −1 𝑥 √3 2 𝜋 (c) – 3 (d) ✔ 3 (c) 2 (d) 1 3 (b) 𝑆𝑖𝑛−1 (𝐴√1 − 𝐴2 − 𝐵√1 − 𝐵2 ) (d) 𝑆𝑖𝑛−1 (𝐴𝐵√(1 − 𝐴2 )(1 − 𝐵2 )) (b) ✔ 𝑆𝑖𝑛−1 (𝐴√1 − 𝐴2 − 𝐵√1 − 𝐵2 ) (d) 𝑆𝑖𝑛−1 (𝐴𝐵√(1 − 𝐴2 )(1 − 𝐵2 )) 23 | P a g e 37. 𝑪𝒐𝒔−𝟏 𝑨 + 𝑪𝒐𝒔−𝟏 𝑩 = (a) 𝐶𝑜𝑠 −1 (𝐴𝐵 − √(1 − 𝐴2 )(1 − 𝐵2 )) (c) 𝐶𝑜𝑠 −1 (𝐴𝐵 − √(1 + 𝐴2 )(1 + 𝐵2 )) 38. 𝑪𝒐𝒔−𝟏 𝑨 + 𝑪𝒐𝒔−𝟏 𝑩 = (b) 𝐶𝑜𝑠 −1 (𝐴𝐵 + √(1 − 𝐴2 )(1 − 𝐵2 )) (d) 𝐶𝑜𝑠 −1 (𝐴𝐵 + √(1 + 𝐴2 )(1 + 𝐵2 )) (a) ✔ 𝐶𝑜𝑠 −1 (𝐴𝐵 − √(1 − 𝐴2 )(1 − 𝐵2 )) (b) ✔ 𝐶𝑜𝑠 −1 (𝐴𝐵 + √(1 − 𝐴2 )(1 − 𝐵2 )) (c) 𝐶𝑜𝑠 −1 (𝐴𝐵 − √(1 + 𝐴2 )(1 + 𝐵2 )) 39. 𝑻𝒂𝒏−𝟏 𝑨 + 𝑻𝒂𝒏−𝟏 𝑩 = 𝐴−𝐵 (a) ✔𝑇𝑎𝑛−1 (1+𝐴𝐵) 𝐴+𝐵 (b) 𝑇𝑎𝑛−1 (1+𝐴𝐵) 40. 𝑻𝒂𝒏−𝟏 𝑨 + 𝑻𝒂𝒏−𝟏 𝑩 = 𝐴−𝐵 𝐴+𝐵 (a) 𝑇𝑎𝑛−1 (1+𝐴𝐵) (b) ✔ 𝑇𝑎𝑛−1 (1+𝐴𝐵) 41. (a) 42. (a) 43. (a) 44. (a) 45. (a) 46. (a) 𝑺𝒊𝒏−𝟏 (−𝒙) = ✔– 𝑆𝑖𝑛−1 𝑥 𝑪𝒐𝒔−𝟏 (−𝒙) = – 𝐶𝑜𝑠 −1 𝑥 𝑻𝒂𝒏−𝟏 (−𝒙) = ✔– 𝑇𝑎𝑛−1𝑥 𝑪𝒐𝒔𝒆𝒄−𝟏 (−𝒙) = ✔– 𝐶𝑜𝑠𝑒𝑐−1 𝑥 𝑺𝒆𝒄−𝟏 (−𝒙) = – 𝑆𝑒𝑐 −1 𝑥 𝑪𝒐𝒕−𝟏 (−𝒙) = −𝐶𝑜𝑡 −1 𝑥 (b) 𝑆𝑖𝑛−1 𝑥 (b) 𝐶𝑜𝑠 −1 𝑥 (d) 𝐶𝑜𝑠 −1 (𝐴𝐵 + √(1 + 𝐴2 )(1 + 𝐵2 )) 𝐴−𝐵 (d) 𝑇𝑎𝑛−1 (1+𝐴𝐵) (c) 𝜋 − 𝑆𝑖𝑛−1 𝑥 (d) 𝜋 − 𝑆𝑖𝑛𝑥 𝐴−𝐵 (c) 𝑇𝑎𝑛−1 (1−𝐴𝐵) (c) ✔ 𝜋 − 𝐶𝑜𝑠 −1 𝑥 (b) 𝑇𝑎𝑛−1 𝑥 (c) 𝜋 − 𝑇𝑎𝑛−1 𝑥 (b) 𝐶𝑜𝑠𝑒𝑐 −1 𝑥 (c) 𝜋 − 𝐶𝑜𝑠𝑒𝑐 −1 𝑥 (b) 𝑆𝑒𝑐 −1 𝑥 (c) ✔ 𝜋 − 𝑆𝑒𝑐 −1 𝑥 (b) 𝐶𝑜𝑡 −1 𝑥 𝐴+𝐵 (c) 𝑇𝑎𝑛−1 (1−𝐴𝐵) (c) ✔ 𝜋 − 𝐶𝑜𝑡 −1 𝑥 𝐴+𝐵 (d) 𝑇𝑎𝑛−1 (1+𝐴𝐵) (d) 𝜋 − 𝐶𝑜𝑠𝑥 (d) 𝜋 − 𝑇𝑎𝑛𝑥 (d) 𝜋 − 𝐶𝑜𝑠𝑒𝑐𝑥 (d) 𝜋 − 𝑆𝑒𝑐𝑥 (d) 𝜋 − 𝐶𝑜𝑡𝑥 UNIT # 14 Solution of Trigonometric Equations Each question has four possible answer. Tick the correct answer. 1. An equation containing at least one trigonometric function is called: (a) Trigonometric function (b) ✔Trigonometric equation (c) Trigonometric value (d) None 𝟏 𝟐 2. If 𝒔𝒊𝒏𝒙 = , then solution in the interval [𝟎, 𝟐𝝅] is: 𝜋 5𝜋 (a) ✔{ , 6 6 } 𝜋 7𝜋 } 6 (b) { 6 , 𝟏 3. If 𝒄𝒐𝒔𝒙 = 𝟐 , then the reference angle is: 𝜋 3 (a) ✔ 𝜋 (b) – 3 𝟏 4. If 𝒔𝒊𝒏𝒙 = 𝟐 , then the reference angle is: (a) 𝜋 3 𝜋 (b) – 3 5. General solution of 𝒕𝒂𝒏𝒙 = 𝟏 is: (a) 6. (a) 7. (a) 8. (a) 9. (a) 𝜋 ✔{4 + 𝑛𝜋, 5𝜋 𝜋 5𝜋 𝜋 4𝜋 } 6 𝜋 2𝜋 } 6 (c) { 3 , (d) { 3 , 𝜋 (c) 6 𝜋 (d) – 6 𝜋 𝜋 (d) – 6 (c) ✔ 6 𝜋 3𝜋 𝜋 3𝜋 + 𝑛𝜋} (b){ 4 + 2𝑛𝜋, 4 + 2𝑛𝜋} (c){ 4 + 𝑛𝜋, 4 + 𝑛𝜋} (d){ 4 + 2𝑛𝜋, 4 + 2𝑛𝜋} If 𝒕𝒂𝒏𝟐𝒙 = −𝟏, then solution in the interval [𝟎, 𝝅]is: 𝜋 𝜋 3𝜋 3𝜋 (b) (c) (d) ✔8 4 8 4 If 𝒔𝒊𝒏𝒙 + 𝒄𝒐𝒔𝒙 = 𝟎 then value of 𝒙 ∈ [𝟎, 𝟐𝝅] 𝜋 3𝜋 𝜋 7𝜋 3𝜋 7𝜋 𝜋 −𝜋 {4 , } (b) { 4 , 4 } (c) ✔ { 4 , 4 } (d) { 4 , 4 } 4 General solution of 𝟒𝒔𝒊𝒏𝒙 − 𝟖 = 𝟎 is: {𝜋 + 2𝑛𝜋} (b) {𝜋 + 𝑛𝜋} (c) {−𝜋 + 𝑛𝜋} (d) ✔not possible General solution of 𝟏 + 𝒄𝒐𝒔𝒙 = 𝟎 is: ✔{𝜋 + 2𝑛𝜋} (b) {𝜋 + 𝑛𝜋} (c) {−𝜋 + 𝑛𝜋} (d) not possible 4 24 | P a g e 10. For the general solution , we first find the solution in the interval whose length is equal to its: (a) Range (b) domain (c) co-domain (d) ✔ period 11. All trigonometric functions are ……………….. functions. (a) ✔Periodic (b) continues (c) injective (d) bijective 12. General solution of every trigonometric equation consists of : (a) One solution only (b) two solutions (c) ✔ infinitely many solutions (d) no real solution 13. Solution of the equation 𝟐𝒔𝒊𝒏𝒙 + √𝟑 = 𝟎 in the 4th quadrant is: −𝜋 −𝜋 11𝜋 𝜋 (b✔) 3 (c) 6 (d) 6 (a) 2 14. If 𝒔𝒊𝒏𝒙 = 𝒄𝒐𝒔𝒙, then general solution is: 𝜋 𝜋 (a) { 4 + 𝑛𝜋, 𝑛 ∈ 𝑍} (b) { 4 + 2𝑛𝜋, 𝑛 ∈ 𝑍} 15. (a) 16. (a) 17. (a) 5𝜋 𝜋 𝜋 18. The solution of the 𝒄𝒐𝒕𝒙 = 𝜋 𝜋 𝟏 √𝟑 20. 𝑺𝒊𝒏𝟐𝒙 = 𝜋 [0, ] 2 √𝟑 𝟐 4 2 in [𝟎, 𝝅] is (a) (b) 4 6 19. One solution of 𝒔𝒆𝒄𝒙 = −𝟐 is : 𝜋 𝜋 (a) 6 (b) (a) 5𝜋 (d){ 4 + 𝑛𝜋, 4 + 𝑛𝜋} (c) ✔ { 4 + 𝑛𝜋, 4 + 𝑛𝜋} In which quadrant is the solution of the equation 𝒔𝒊𝒏𝒙 + 𝟏 = 𝟎 1st and 2nd (b) 2nd and 3rd (c) ✔ 3rd and 4th If 𝒔𝒊𝒏𝒙 = 𝟎 then 𝒙 = 𝑛𝜋 ✔𝑛𝜋 , 𝑛 ∈ 𝑍 (b) 2 , 𝑛 ∈ 𝑍 (c) 0 If 𝒕𝒂𝒏𝒙 = 𝟎 then 𝒙 = 𝑛𝜋 ,𝑛 ∈ 𝑍 (c) 0 ✔𝑛𝜋 , 𝑛 ∈ 𝑍 (b) has two values of 𝒙 in the interval: (b) [0,2𝜋] (c) 2𝜋 3 (c) ✔ 2𝜋 3 𝜋 (c) [– 𝜋, 2 ] (d) Only 1st 𝜋 (d) 2 𝜋 (d) 2 𝜋 (d) ✔ 3 (d) 𝜋 3 𝜋 (d) [− 2 , 0] ----------THE END-------- WITH BEST WISHES BY:MUHAMMAD SALMAN SHERAZI M.Sc(Math) , B.ed 03337727666/03067856232