Mathematical Logic & Miscellaneous Algebra PYQs Set 3 (PDF)

Back to Mathematical Logic & Miscellaneous Algebra

Set 3 of 6 · Questions 61-90 of 171

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Mathematical Logic & Miscellaneous Algebra · Set 3 (Q61-90)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
61. $The set of all values of x which satisfy both the inequations x^{2} - 1 \leq 0 and x^{2} - x - 2 \geq 0 simultaneously is$ (1) $(- 1, 2)$ (2) $(- 1, 1)$ (3) $(- 2, - 1)$ (4) ${- 1}$ 62. $The contrapositive of the statement "If two lines do not intersect in the same plane then they are parallel." is$ (1) $If two lines are not parallel then they do not intersect in the same plane.$ (2) $If two lines are not parallel then they intersect in the same plane.$ (3) $If two lines are parallel then they do not intersect in the same plane.$ (4) $If two lines are parallel then they intersect in the same plane.$ 63. $The statement (p \to (q \to p)) \to (p \to (p \lor q)) is$ (1) $a tautology$ (2) $equivalent to (p \lor q) \land (\sim p)$ (3) $a contradiction$ (4) $equivalent to (p \land q) \lor (\sim q)$ 64. $The negation of the statement: "Getting above 95 % marks is necessary condition for Hema to get the admission in good college".$ (1) $Hema gets above 95 % marks but she does not get the admission in good college$ (2) $Hema does not get above 95 % marks and she gets admission in good college$ (3) $If Hema does not get above 95 % marks then she will not get the admission in good college$ (4) $Hema does not get above 95 % marks or she gets the admission in good college$ 65. $Given that a, b and x are real numbers and a < b, x < 0 then$ (1) $\frac{a}{x} < \frac{b}{x}$ (2) $\frac{a}{x} > \frac{b}{x}$ (3) $\frac{a}{x} \leq \frac{b}{x}$ (4) $\frac{a}{x} \geq \frac{b}{x}$ 66. $Which one of the following is a tautology?$ (1) $(p \land (p \to q) \to q$ (2) $q \to p \land (p \to q)$ (3) $p \lor (p \land q)$ (4) $p \land (p \lor q)$ 67. $If p and q are true and r and s are false statements, then which of the following is true?$ (1) $(q \land r) \lor (\sim p \land s)$ (2) $(\sim p \to q) \leftrightarrow (r \land s)$ (3) $(p \to q) \lor (r \leftrightarrow s)$ (4) $(p \land \sim r) \land (\sim q \lor s)$ 68. $If p \to (p \land \sim q) false, then the truth values of p an q are respectively.$ (1) $F, F$ (2) $F, T$ (3) $T, F$ (4) $T, T$ 69. $Among the statements. (S1) ((p \lor q) \Rightarrow r) \Leftrightarrow (p \Rightarrow r) (S2) ((p \lor q) \Rightarrow r) \Leftrightarrow ((p \Rightarrow r) \lor (q \Rightarrow r))$ (1) $Only (S1) is a tautology$ (2) $Neither (S1) nor (S2) is a tautology$ (3) $Only (S2) is a tautology$ (4) $Both (S1) and (S2) are tautologies$ 70. $The negation of the proposition (\sim p \land \sim q) \lor (p \land \sim q) is$ (1) $(p \land q) \lor (\sim p \land q)$ (2) $(p \lor q) \land (\sim p \lor q)$ (3) $(p \lor q) \lor \sim (p \land q)$ (4) $\sim (p \lor q) \land (\sim p \lor q)$ 71. $The contrapositive of the converse of the statement "If x is a prime number then x is odd" is$ (1) $If x is not a prime number then x is odd.$ (2) $If x is not an odd number then x is not a prime number.$ (3) $If x is a prime number then it is not odd.$ (4) $If x is not a prime number then x is not odd.$ 72. $The statement pattern (\sim p \land q) is logically equivalent to$ (1) $(p \lor q) \lor \sim p$ (2) $(p \lor q) \land \sim p$ (3) $(p \land q) \to p$ (4) $(p \lor q) \to p$ 73. $In a ΔPQR, if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to :$ (1) $\frac{5 π}{6}$ (2) $\frac{π}{6}$ (3) $\frac{π}{4}$ (4) $\frac{3 π}{4}$ 74. $The contrapositive of the statement “If I reach the station in time, then I will catch the train” is:$ (1) $If I will catch the train then I reach the station in time$ (2) $If I do not reach the station in time then I will catch the train.$ (3) $If I do not reach the station in time then I will not catch the train.$ (4) $If I will not catch the train, then I do not reach the station in time.$ 75. $Consider the following statements P : Suman is brilliant Q : Suman is rich R : Suman is honest The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as$ (1) $\sim Q \leftrightarrow \sim P \land R$ (2) $\sim (P \land \sim R) \leftrightarrow Q$ (3) $\sim P \land (Q \leftrightarrow \sim R)$ (4) $\sim (Q \leftrightarrow (P \land \sim R))$ 76. $The statement (p \land (q)) \Rightarrow (p \Rightarrow (q)) is$ (1) $is equivalent to (p) \lor (q)$ (2) $a tautology$ (3) $equivalent to p \lor q$ (4) $a contradiction$ 77. $The negation of the statement \sim p \land (p \lor q) is:$ (1) $\sim p \land q$ (2) $p \land \sim q$ (3) $\sim p \lor q$ (4) $p \lor \sim q$ 78. $The negation of 5 is an integer or 5 is an irrational number is$ (1) $5 is an integer and 5 is not an irrational number.$ (2) $5 is not an integer and 5 is an irrational number.$ (3) $5 is an integer and 5 is an irrational number.$ (4) $5 is not an integer and 5 is not an irrational number.$ 79. $p \to \sim q can also be written as$ (1) $\sim q \to \sim p$ (2) $\sim p \lor \sim q$ (3) $p \to q `$ (4) $q \to p$ 80. $Let A, B, C and D be four non-empty sets. The contra-positive statement of “If A \subseteq B and B \subseteq D, then A \subseteq C ” is:$ (1) $If A⊈C then A \subseteq B and B\subseteqD$ (2) $If A ⊈ C then A ⊈ B and B \subseteq D$ (3) $If A⊈C then B \subset A and D \subset B$ (4) $If A⊈C then A⊈B and B ⊈ D$ 81. $The negation of the Boolean expression p \lor (\sim p \land q) is equivalent to:$ (1) $\sim p \lor q$ (2) $p \land \sim q$ (3) $\sim p \lor \sim q$ (4) $\sim p \land \sim q$ 82. $Which of the following Boolean expression is a tautology?$ (1) $(p \land q) \lor (p \lor q)$ (2) $(p \land q) \lor (p \to q)$ (3) $(p \land q) \land (p \to q)$ (4) $(p \land q) \to (p \to q)$ 83. $The solution set of \| x \|^{2} - 5 \| x \| + 4 < 0 is$ (1) $(- 4, - 1)$ (2) $(1, 4)$ (3) $(- 4, - 1) \cup (1, 4) .$ (4) $(- 4, 4)$ 84. $The Boolean expression (p\land\sim q) \Rightarrow (q\lor\sim p) is equivalent to:$ (1) $q \Rightarrow p$ (2) $p \Rightarrow q$ (3) $\sim q \Rightarrow p$ (4) $p \Rightarrow\sim q$ 85. $The number of values of r ϵ (p, q, p, q) for Which ((p \land q) \Rightarrow (r \lor q)) \land ((p \land r) \Rightarrow q) is a tautology, is$ (1) $3$ (2) $2$ (3) $1$ (4) $4$ 86. $Let F_{1} (A, B, C) = (A \land \sim B) \lor [\sim C \land (A \lor B)] \lor \sim A and F_{2} (A, B) = (A \lor B) \lor (B \to \sim A) be to logical expressions. Then :$ (1) $F1is not a tautology but F_{2} is a tautology$ (2) $F1is a tautology but F_{2} is not a tautology$ (3) $F1and F_{2} both area tautologies$ (4) $Both F_{1} and F_{2} are not tautologies$ 87. $Let (x0, y0) be the solution of the following equations (2x)λn2 = (3y)λn3 3λnx = 2λny . Then x0 is$ (1) $\frac{1}{6}$ (2) $\frac{1}{3}$ (3) $\frac{1}{2}$ (4) $6$ 88. $The number of distinct solutions of the equation \frac{5}{4} cos2 2x + cos4 x + sin4 x + cos6 x + sin6x = 2 in the interval [0, 2π] is$ 89. $The maximum value of the expression \frac{1}{s i n ^{2} θ + 3 s in θ cos θ + 5 co s ^{2} θ} is$ 90. $The number of values of θ in the interval (- \frac{π}{2}, \frac{π}{2}) such that θ \neq \frac{nπ}{5} for n = 0, \pm1, \pm 2 and tanθ = cot 5θ as well as sin 2θ = cos 4θ is$