Mathematical Logic & Miscellaneous Algebra PYQs Set 2 (PDF)

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Set 2 of 6 · Questions 31-60 of 171

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Mathematical Logic & Miscellaneous Algebra · Set 2 (Q31-60)
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31. $The statement (p \land (p \to q) \land (q \to r)) \to r is :$ (1) $a tautology$ (2) $equivalent to p \to\sim r$ (3) $a fallacy$ (4) $equivalent to q \to\sim r$ 32. $The compound proposition (p \land q) \to p is$ (1) $A tautology$ (2) $A contradiction$ (3) $Neither (a) nor (b)$ (4) $None of these$ 33. $If (p \land \sim r) \to (\sim p \lor q) has truth value ' F', then truth values of p, q and r respectively$ (1) $T, T, T$ (2) $T, F, F$ (3) $F, F, F$ (4) $F, F, 'T'$ 34. $Let p and q be two statements. Then (p \land (p \Rightarrow q)) is equivalent to$ (1) $p \lor (p \land (p))$ (2) $p \lor ((p) \land q)$ (3) $(p) \lor q$ (4) $p \land (p \land q)$ 35. $If c denotes the contradiction, then dual of the compound statement \sim p \land (q \lor c) is$ (1) $\sim p \lor (q \land t)$ (2) $\sim p \land (q \lor t)$ (3) $p \lor (\sim q \lor t)$ (4) $\sim p \lor (q \land c)$ 36. $The negation of the statement "The number is an odd number if and only if it is divisible by 3 ".$ (1) $The number is an odd number but not divisible by 3 or the number is divisible by 3 but not odd.$ (2) $The number is not an odd number iff it is not divisible by 3 .$ (3) $The number is not an odd number but it is divisible by 3 .$ (4) $The number is not an odd number or is not divisible by 3 but the number is divisible by 3 or odd.$ 37. $The negation of p \land (q \to \sim r) is$ (1) $p \lor (q \land r)$ (2) $\sim p \land (q \land r)$ (3) $\sim p \lor (q \land r)$ (4) $p \lor (q \lor r)$ 38. $The number of the real solutions of the equation: x \| x + 3 \| + \| x - 1 \| - 2 = 0 is$ (1) $2$ (2) $4$ (3) $3$ (4) $5$ 39. $The number of integral values of x satisfying 5 x - 1 < (x + 1)^{2} < 7 x - 3 is$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 40. $Negation of the statement (p \lor r) \Rightarrow (q \lorr) is :$ (1) $p\land\sim q\land\sim r$ (2) $\sim p \land q\land\sim r$ (3) $\sim p \land q \land r$ (4) $p \land q \land r$ 41. $The solution set of the inequation \frac{x ^{2} + 6 x - 7}{\| x + 4 \|} < 0 is$ (1) $(- 7, - 4)$ (2) $(- 7, - 4) \cup (4, 1)$ (3) $(- 7, 1)$ (4) $(- 7, - 4) \cup (- 4, 1)$ 42. $The product of all solutions of the equation e^{5 (log_{e} x)^{2} + 3} = x^{8}, x > 0, is :$ (1) $e^{8/5}$ (2) $e^{6/5}$ (3) $e^{2}$ (4) $e$ 43. $(S1) (p \Rightarrow q) \lor (p \land (q)) is a tautology (S2) ((p) \Rightarrow (q)) \land ((p) \lor q) is a contradiction. Then$ (1) $only (S2) is correct$ (2) $both (S1) and (S2) are correct$ (3) $Both (S1) and (S2) are wrong$ (4) $only (S1) is correct$ 44. $Consider the statement. “For an integer n, if n^{3} - 1 is even, then n is odd.” The contrapositive statement of this statement is:$ (1) $For an integer n, if n is odd, then n^{3} - 1 is even.$ (2) $For an integer n, if n is even, then n^{3} - 1 is even.$ (3) $For an integer n, if n is even, then n^{3} - 1 is odd.$ (4) $For an integer n, if n^{3} - 1 is noteven, then n is even.$ 45. $The negative of the statement (p \land q) \to (\sim p \lor r) is$ (1) $p \lor q \lor \sim r$ (2) $p \land q \land \sim r$ (3) $\sim p \lor q \land r$ (4) $\sim p \lor \sim q \lor \sim r$ 46. $The statement among the following that is a tautology is:$ (1) $A \land (A \lor B)$ (2) $B \to [A \land (A \to B)]$ (3) $A \lor (A \land B)$ (4) $[A \land (A \to B)] \to B$ 47. $The negation of the expression q \lor ((q) \land p) is equivalent to$ (1) $(p) \land (q)$ (2) $p \land (q)$ (3) $(p) \lor) (q)$ (4) $(p) \lor q$ 48. $Which of the following Boolean expressions is not a tautology ?$ (1) $(p \Rightarrow q) \lor (\sim q \Rightarrow p)$ (2) $(q \Rightarrow p) \lor (\sim q \Rightarrow p)$ (3) $(p \Rightarrow\sim q) \lor (\sim q \Rightarrow p)$ (4) $(\sim p \Rightarrow q) \lor (\sim q \Rightarrow p)$ 49. $The negation of the statement " 72 is divisible by 2 and 3 " is$ (1) $72 is not divisible by 2 or 72 is not divisible by 3$ (2) $72 is divisible by 2 or 72 is divisible by 3$ (3) $72 is divisible by 2 and 72 is divisible by 3$ (4) $72 is not divisible by 2 and 3$ 50. $Which of the following is the negation of the statement "for all M > 0, there exists x \in S such that x \geq M”$ (1) $there exists M > 0, such that x < M for all x \in S$ (2) $there exists M > 0, there exists x \in S such that x \geq M$ (3) $there exists M > 0, there exists x \in S such that x < M$ (4) $there exists M > 0, such that x \geq M for all x \in S$ 51. $The negation of the proposition "If 2 is prime, then 3 is odd" is$ (1) $if 2 is not prime then 3 is not odd$ (2) $2 is prime and 3 is not odd$ (3) $2 is not prime and 3 is odd$ (4) $if 2 is not prime then 3 is odd$ 52. $Which of the following statements is a tautology??$ (1) $p \to (p \land (q \to q))$ (2) $(p \land q) \to ((p) \to q))$ (3) $(p \land (p \to q)) \to q$ (4) $p \lor (p \land q)$ 53. $The statement A \to (B \to A) is equivalent to:$ (1) $A \to (A \land B)$ (2) $A \to (A \lor B)$ (3) $A \to (A \to B)$ (4) $A \to (A \leftrightarrow B)$ 54. $Which of the following statements is a tautology ?$ (1) $((~p) v q) \Rightarrow p$ (2) $p \Rightarrow ((~p) v q)$ (3) $((~p) v q) \Rightarrow q$ (4) $q \Rightarrow ((~p) v q)$ 55. $\sim [(\sim p) \land q] is logically equivalent to$ (1) $p \land (\sim q)$ (2) $\sim (p \lor q)$ (3) $p \lor (\sim q)$ (4) $\sim [p \land (\sim q)]$ 56. $Which of the following statement is a tautology?$ (1) $((\sim q) \land p) \land q$ (2) $((\sim q) \land p) \land (p \land (\sim p))$ (3) $((\sim q) \land p) \lor (p \lor (\sim p))$ (4) $(p \land q) \land (\sim (p \land q))$ 57. $The logical statement (p \to q) \land (q \to \sim p) is equivalent to$ (1) $\sim p$ (2) $p$ (3) $p \land q$ (4) $p \lor q$ 58. $Consider the two statements: (S1) : (p \to q) \lor (\sim q \to p) is a tautology. (S2) : (p\land\sim q) \land (\sim p \lor q) is a fallacy. Then :$ (1) $only (S1) is true.$ (2) $both (S1) and (S2) are false.$ (3) $both (S1) and (S2) are true.$ (4) $only (S2) is true.$ 59. $The converse of the contrapositive of the conditional p \to \sim q is$ (1) $\sim q \to p$ (2) $p \to q$ (3) $\sim p \to q$ (4) $\sim p \to\sim q$ 60. $The maximum number of compound propositions, out of p \lor r \lor s, p \lor r \lor ~ s, p \lor ~q \lor s, ~p \lor ~r \lor s, ~p \lor ~r \lor ~s, ~p \lor q \lor ~s, q \lor r \lor ~ s, q \lor ~ r \lor ~ s, ~ p \lor - q \lor ~ s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to$