Mathematical Logic & Miscellaneous Algebra PYQs Set 4 (PDF)

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Set 4 of 6 · Questions 91-120 of 171

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Mathematical Logic & Miscellaneous Algebra · Set 4 (Q91-120)
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91. $The negation of the Boolean expression ((\sim q) \land p) \Rightarrow ((\sim p) \lor q) is logically equivalent to$ (1) $p \Rightarrow q$ (2) $q \Rightarrow p$ (3) $~(p \Rightarrow q)$ (4) $~(q \Rightarrow p)$ 92. $The shaded region in the figure is the solution set of the inequations$ (1) $5x+4y\leq20,x\leq6,y\leq3,x\geq0,y\geq0$ (2) $5x+4y\geq20,x\leq6,y\geq3,x\geq0,y\geq0$ (3) $5x+4y\geq20,x\leq6,y\leq3,x\geq0,y\geq0$ (4) $5x+4y\geq20,x\geq6,y\leq3,x\geq0,y\geq0$ 93. $The sum of all real values of x satisfying the equation (x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1 is$ (1) $- 4$ (2) $6$ (3) $5$ (4) $3$ 94. $Let ∆ \in_{\land, \lor, \Rightarrow, \Leftrightarrow} be such that (p \land q) Δ ((p \lor q) \Rightarrow q) is a tautology. Then ∆ is equal to$ (1) $\land$ (2) $\lor$ (3) $\Rightarrow$ (4) $\Leftrightarrow$ 95. $In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible 5 opted Chemistry course. Then the number of student who did not opt for any of the three courses is :$ (1) $38$ (2) $42$ (3) $102$ (4) $1$ 96. $The shaded region shown in fig. is given by the inequation$ (1) $14x+5y\geq70,y\leq14 and x-y\geq5$ (2) $14x+5y\geq70,y\geq14 and x-y\geq5$ (3) $14x+5y\geq70,y\leq14 and x-y\leq5$ (4) $14x+5y\leq70,y\leq14 and x-y\geq5$ 97. $The shaded region in the figure is the solution set of the inequations$ (1) $4x+5y\geq20,3x+10y\leq30,x\leq6,x,y\geq0$ (2) $4x+5y\leq20,3x+10y\leq30,x\leq6,xy\geq0$ (3) $4x+5y\geq20,3x+10y\leq30,x\geq6,x,y\geq0$ (4) $4x+5y\leq20,3x+10y\leq30,x\geq6,x,y\geq0$ 98. $Consider the following two propositions: P1 : ~(p \to ~q) P2 : (p \land ~q) \land ((~p) \lor q) If the proposition P \to_{((} p_{)} \lor q_{)} is evaluated as FALSE, then:$ (1) $P1 is TRUE and P2 is FALSE$ (2) $P1 is FALSE and P2 is TRUE$ (3) $Both P1 and P2 are FALSE$ (4) $Both P1 and P2 are TRUE$ 99. $Symbolic form of the given switching circuit is equivalent to$ (1) $p \lor \sim q$ (2) $p \land \sim q$ (3) $p \leftrightarrow q$ (4) $\sim (p \leftrightarrow q)$ 100. $Let ∆, \nabla \in_{\land, \lor} be such that p \nabla q \Rightarrow_{((} p \nabla q_{)} \nabla r_{)} is a tautology. Then (p \nabla q) ∆ r is logically equivalent to :$ (1) $(p Δ r) \lor q$ (2) $(p Δ r) \land q$ (3) $(p \land r) Δ q$ (4) $(p \nabla r) \land q$ 101. $If P and Q are two statements, then which of the following compound statement is a tautology?$ (1) $((P \Rightarrow Q) \land \sim Q) \Rightarrow Q$ (2) $((P \Rightarrow Q) \land \sim Q) \Rightarrow \sim P$ (3) $((P \Rightarrow Q) \land \sim Q) \Rightarrow P$ (4) $((P \Rightarrow Q) \land \sim Q) \Rightarrow (P \land Q)$ 102. $The statement (p \land (~q)) \lor ((~p) \land g) \lor ((~p) \land (~q)) is equivalent to_____$ (1) $(~p) \lor q$ (2) $(~p) \lor (~q)$ (3) $p \lor (-g)$ (4) $p \lor g$ 103. $Let - \frac{π}{6} < θ < - \frac{π}{12} . Suppose α1 and β1 are the roots of the equation x2 - 2x sec θ + 1 = 0 and α2 and β2 are the roots of the equation x2 + 2x tan θ - 1 = 0. If α1 > β1 and α2 > β2 , then α1 + β2 equals$ (1) $2(secθ - tan θ)$ (2) $2sec θ$ (3) $- 2tan θ$ (4) $0$ 104. $Negation of the Boolean statement (p \land q) \Rightarrow ((\sim r) \lor p) is equivalent to:$ (1) $p \land_{(} q_{)} \land r$ (2) $(\sim p) \land (\sim q) \land r$ (3) $(\sim p) \land q \land r$ (4) $p \land q \land_{(} r_{)}$ 105. $The statement (p \Rightarrow q) v (p \Rightarrow r) is NOT equivalent to:$ (1) $(p \land (~r)) \Rightarrow q$ (2) $(~q) \Rightarrow ((~r) \lor p)$ (3) $p \Rightarrow (q \lor r)$ (4) $(p \land (~q)) \Rightarrow r$ 106. $If sum of all the solutions of the equation 8 cosx. (cos (\frac{π}{6} + x) . cos (\frac{π}{6} - x) - \frac{1}{2}) = 1 in [0, π] is kπ, then k is equal to :$ (1) $\frac{8}{9}$ (2) $\frac{20}{9}$ (3) $\frac{2}{3}$ (4) $\frac{13}{9}$ 107. $The Boolean expression (\sim (p \land q)) \lor q is equivalent to :$ (1) $q \to_{(} p \land q_{)}$ (2) $p \to q$ (3) $p \to_{(} p \to q_{)}$ (4) $p \to_{(} p \lor q_{)}$ 108. $\frac{x - 1}{3 x + 4} < \frac{x - 3}{3 x - 2} holds, for all x in the interval$ (1) $(\frac{- 4}{3}, \frac{2}{3})$ (2) $(- \infty, \frac{- 5}{4})$ (3) $(\frac{3}{3}, \infty)$ (4) $(- \infty, \frac{- 5}{4}) \cup (\frac{3}{4}, - \infty)$ 109. $Negation of the Boolean expression p \Leftrightarrow (q \Rightarrow p) is$ (1) $(~p) \land q$ (2) $P \land (~q)$ (3) $(~p) \lor (~q)$ (4) $(~p) \land (~q)$ 110. $Consider the following statements : j A : Rishi is a judge. B : Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is$ (1) $B \to_{(} A \lor C_{)}$ (2) $(\sim B) \land (A \land C)$ (3) $B \to_{((} A_{)} \lor_{(} C_{))}$ (4) $B \to_{(} A \land C_{)}$ 111. $Let cos(α + β) = \frac{4}{5} and let sin(α - β) = \frac{5}{13}, where 0 \leq α, β \leq \frac{π}{4} . Then tan 2α =$ (1) $\frac{56}{33}$ (2) $\frac{19}{12}$ (3) $\frac{20}{7}$ (4) $\frac{25}{16}$ 112. $Let Δ, \nabla \in {\land, \lor} be such that (p \to q) Δ (p \nabla q) is a tautology. Then$ (1) $Δ = \land, \nabla = \lor$ (2) $Δ = \lor, \nabla = \land$ (3) $Δ = \lor, \nabla = \lor$ (4) $Δ = \land, \nabla = \land$ 113. $The statement B \Rightarrow ((A) \lor B) is equivalent to$ (1) $B \Rightarrow (A \Rightarrow B)$ (2) $A \Rightarrow (A \Leftrightarrow B)$ (3) $A \Rightarrow ((A) \Rightarrow B)$ (4) $B \Rightarrow ((A) \Rightarrow B)$ 114. $If the Boolean expression (p \Rightarrow q) \Leftrightarrow (q * (\sim p)) is a tautology, then the Boolean expression p * (\sim q) is equivalent to$ (1) $q \Rightarrow p$ (2) $\sim q \Rightarrow p$ (3) $p \Rightarrow\sim q$ (4) $p \Rightarrow q$ 115. $If p, q and r are three propositions, then which of the following combination of truth values of p, q and r makes the logical expression ((p \lor q) \land ((\sim p) \lor r)) \to ((\sim q) \lor r) false?$ (1) $p=T, q=F, r=T$ (2) $p=T, q=T, r=F$ (3) $p=F, q=T, r=F$ (4) $p=T, q=F,r=F$ 116. $The number of ordered triplets of the truth values of p, q and r such that the truth value of the statement (p \lor q) \land (p \lor r) \Rightarrow (q \lor r) is True, is equal to$ 117. $Suppose a class has 7 students. The average marks of these students in the mathematics examination is 62, and their variance is 20. A student fails in the examination if he/she gets less than 50 marks, then in worst case, the number of students can fail is$ 118. $Find all real numbers x which satisfy the equation 2 log2 log2 x + log1/2 log2 (22 x) = 1.$ 119. $Let the point (p,p + 1) lie inside the region E = {(x, y) : 3 - x \leq y \leq 9 - x^{2}, 0 \leq x \leq 3} . If the set of all values of p is the interval (a, b), then b^{2} + b - a^{2} is equal to_________.$ 120. $The positive integer value of n > 3 satisfying the equation \frac{1}{s i n ( \frac{π}{n} )} = \frac{1}{s i n ( \frac{2 π}{n} )} + \frac{1}{s i n ( \frac{3 π}{n} )} is$