Mathematical Logic & Miscellaneous Algebra PYQs Set 5 (PDF)

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Set 5 of 6 · Questions 121-150 of 171

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Mathematical Logic & Miscellaneous Algebra · Set 5 (Q121-150)
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121. $If p, q, r are any real numbers, then$ (1) $max (p, q) < max (p, q, r)$ (2) $min (p, q) = \frac{1}{2} (p + q - \|p - q\|)$ (3) $max (p, q) < min(p, q, r)$ (4) $None of these$ 122. $If 3x = 4x - 1 , then x =$ (1) $\frac{2 l o g _{3} 2}{2 l o g _{3} 2-1}$ (2) $\frac{2}{2- l o g _{2} 3}$ (3) $\frac{1}{1- l o g _{4} 3}$ (4) $\frac{2 l o g _{2} 3}{2 l o g _{2} 3-1}$ 123. $If \| x - 2 \| \leq 1, then$ (1) $x \in (1, 3)$ (2) $x \in (- 1, 3)$ (3) $x \in [1, 3]$ (4) $x \in [- 1, 3)$ 124. $If \| 3 x - 5 \| \leq 2, then$ (1) $- 1 \leq x \leq \frac{7}{3}$ (2) $1 \leq x \leq \frac{9}{3}$ (3) $1 \leq x \leq \frac{7}{3}$ (4) $- 1 \leq x \leq \frac{9}{3}$ 125. $If 3x = 4x - 1 , then x =$ (1) $\frac{2 l o g _{3} 2}{2 l o g _{3} 2-1}$ (2) $\frac{2}{2- l o g _{2} 3}$ (3) $\frac{1}{1- l o g _{4} 3}$ (4) $\frac{2 l o g _{2} 3}{2 l o g _{2} 3-1}$ 126. $Which of the following is a tautology?$ (1) $(\sim p) \land (p \lor q) \to q$ (2) $(\sim q) \lor (p \land q) \to q$ (3) $(p \to q) \land (q \to p)$ (4) $(q \to p) \lor \sim (p \to q)$ 127. $If \| x + 5 \| \geq 10, then$ (1) $x \in (- 15, 5]$ (2) $x \in (- 5, 5]$ (3) $x \in (- \infty, - 15] \cup [5, \infty)$ (4) $x \in (- \infty, - 15] \cup (5, \infty)$ 128. $Negation of p \land (q \land \sim (p \land q)) is$ (1) $(~(p \land q)) \lor p$ (2) $p \lor q$ (3) $\sim (p \lor q)$ (4) $(\sim (p \land q)) \land q$ 129. $The number of real solutions of the equation \|x\|2 - 3\|x\| + 2 = 0 is :$ (1) $4$ (2) $1$ (3) $3$ (4) $2$ 130. $Consider the following three statements: (A) If 3 + 3 = 7 then 4 + 3 = 8. (B) If 5 + 3 = 8 then earth is flat. (C) If both (A) and (B) are true then 5 + 6 = 17.T hen, which of the following statements is correct?$ (1) $(A) is false, but (B) and (C) are true$ (2) $(A) and (C) are true while (B) is false$ (3) $(A) is true while (B) and (C) are false$ (4) $(A) and (B) are false while (C) is true$ 131. $The converse of ((\sim p) \land q) \Rightarrow r is$ (1) $((\sim p) \lor q) \Rightarrow r$ (2) $(\sim r) \Rightarrow p \land q$ (3) $(\sim r) \Rightarrow ((\sim p) \land q)$ (4) $(p \lor (\sim q)) \Rightarrow (\sim r)$ 132. $Negation of (p \to q) \to (q \to p) is$ (1) $(p \sim) \lor p$ (2) $q \land_{(} p_{)}$ (3) $(~q) \land P$ (4) $p \lor_{(} q_{)}$ 133. $Statement (P \Rightarrow Q) \land (R \Rightarrow Q) is logically equivalent to$ (1) $(P \Rightarrow Q) \lor (Q \Rightarrow R)$ (2) $(P \land R) \Rightarrow Q$ (3) $(P \Rightarrow R) \land (Q \Rightarrow R)$ (4) $(P \lor R) \Rightarrow Q$ 134. $The negation of (p \land (- q)) \lor (- p) is equivalent to$ (1) $P \land (-q)$ (2) $p \land q$ (3) $p \lor_{(} q \lor_{(} - p_{))}$ (4) $p \land_{(} q \land_{(} - p_{))}$ 135. $(p \land r) \Leftrightarrow (p \land (\sim q)) is equivalent to (~ p) when r is$ (1) $p$ (2) $~p$ (3) $q$ (4) $~q$ 136. $The statement is equivalent to$ (1) $~(p \land q)$ (2) $(p \land q) \land (~p)$ (3) $(~(p \land q)) \land q$ (4) $~(p \lor q)$ 137. $The negation of the statement is$ (1) $(p \lor r) \land (\sim q)$ (2) $((\sim p) \lor r) \land (\sim q)$ (3) $((\sim p \lor) (\sim q)) \land (\sim r)$ (4) $((\sim p) \lor (\sim q)) \land (\sim r)$ 138. $The statement (p \land q) \Rightarrow (p \land r) is equivalent to :$ (1) $q \Rightarrow (p \land r)$ (2) $p \Rightarrow (p \land r)$ (3) $(p \land r) \Rightarrow (p \land q)$ (4) $(p \land q) \Rightarrow r$ 139. $Let p, q, r be three logical statements. Consider the compound statements S1 :((~p) \lor q) \lor ((~p) \lor r) and S_{2} :p \to (q \lor r) Then, which of the following is NOT true ?$ (1) $If S_{2} is True, then S_{1} is True$ (2) $If S_{2} is False, then S_{1} is False$ (3) $If S_{2} is False, then S_{1} is True$ (4) $If S_{1} is False, then S_{2} is False$ 140. $Which of the following statement pattern is a tautology?$ (1) $p \lor (q \to p)$ (2) $\sim q \to \sim p$ (3) $(q \to p) \lor (\sim p \leftrightarrow q)$ (4) $p \land \sim p$ 141. $Let r \in {p, q, ~p, ~q} be such that the logical statement r \lor_{(} p_{)} \Rightarrow_{(} p \land q_{)} \lor r is a tautology. Then ‘r’ is equal to :$ (1) $p$ (2) $q$ (3) $~p$ (4) $~q$ 142. $The number of choices of ∆ \in_{\land, \lor, \Rightarrow, \Leftrightarrow}, such that (p Δ q) \Rightarrow ((p Δ \sim q) \lor ((\sim p) Δ q)) is a tautology, is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 143. $The statement (~(p \Leftrightarrow ~q)) \land q is :$ (1) $a tautology$ (2) $a contradiction$ (3) $equivalent to (p \Rightarrow q) \land q$ (4) $equivalent to (p \Rightarrow q) \land p$ 144. $The statement [p \land (q \lor r)] \lor [\sim r \land \sim q \land p] is equivalent to$ (1) $\sim r$ (2) $p$ (3) $\sim q$ (4) $q$ 145. $If (p \land \sim r) \to (\sim p \lor q) is false, then the truth values of p, q and r are respectively$ (1) $T, F and F$ (2) $F, F and T$ (3) $F, T and T$ (4) $T, F and T$ 146. $The statement pattern (p \land q) \land [\sim r \lor (p \land q)] \lor (\sim p \land q) is equivalent to$ (1) $r$ (2) $q$ (3) $p \land q$ (4) $p$ 147. $If logp x = α and logq x = β, then the value of logp/q x is$ (1) $\frac{α - β}{α β}$ (2) $\frac{β - α}{α β}$ (3) $\frac{α β}{α - β}$ (4) $\frac{α β}{β - α}$ 148. $Consider the following statements : P : Ramu is intelligent Q : Ramu is rich R : Ramu is not honest The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as :$ (1) $((P \land (~R)) \land Q) \land ((~Q) \land ((~P) \lor R))$ (2) $((P \land R)AQ) \lor ((~Q)A((~P) \lor (~R)))$ (3) $(P \land R) \land Q) \land ((~Q) \land ((~P) \lor (~R)))$ (4) $((P \land (~R)) \land Q) \lor ((~Q) \land ((~P) \lor R))$ 149. $If log0.3 (x - 1) < log0.09 (x - 1), then x lies in the intervel$ (1) $(2, \infty)$ (2) $(1, 2)$ (3) $(-2, -1)$ (4) $none of these$ 150. $Solve the equation lo g_{3/4} lo g_{8} (x^{2} + 7) + lo g_{1/2} lo g_{1/4} (x^{2} + 7)^{- 1} = - 2 .$