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values | question stringlengths 17 24.5k | options stringlengths 2 4.26k | correct_option stringclasses 6
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|---|---|---|---|---|---|---|---|---|---|---|---|
xwfoVcp6WOGDPwSwrCee3 | maths | mathematical-reasoning | logical-connectives | If q is false and p $$ \wedge $$ q $$ \leftrightarrow $$ r is true, then which one of the following statements is a tautology ? | [{"identifier": "A", "content": "P $$ \\wedge $$ r"}, {"identifier": "B", "content": "(p $$ \\vee $$ r) $$ \\to $$ (p $$ \\wedge $$ r)"}, {"identifier": "C", "content": "p $$ \\vee $$ r"}, {"identifier": "D", "content": "(p $$ \\wedge $$ r) $$ \\to $$ (p $$ \\vee $$ r)"}] | ["D"] | null | Given q is F and (p $$ \wedge $$ q) $$ \leftrightarrow $$ r is T
<br><br>$$ \Rightarrow $$ p $$ \wedge $$ q is F which implies that r is F
<br><br>$$ \Rightarrow $$ q is F and r is F
<br><br>$$ \Rightarrow $$ (p $$ \wedge $$ r) is always F
<br><br>$$ \Rightarrow $$ (p $$ \wed... | mcq | jee-main-2019-online-11th-january-morning-slot | 6,831 |
SzEiL7YZ0iQAZgnQKyN4R | maths | mathematical-reasoning | logical-connectives | If the Boolean expression
<br/>(p $$ \oplus $$ q) $$\wedge$$ (~ p $$ \odot $$ q) is equivalent
<br/> to p $$\wedge$$ q, where $$ \oplus , \odot \in \left\{ { \wedge , \vee } \right\}$$, then the
<br/>ordered pair $$\left( { \oplus , \odot } \right)$$ is : | [{"identifier": "A", "content": "$$\\left( { \\vee , \\wedge } \\right)$$"}, {"identifier": "B", "content": "$$\\left( { \\vee , \\vee } \\right)$$"}, {"identifier": "C", "content": "$$\\left( { \\wedge , \\vee } \\right)$$"}, {"identifier": "D", "content": "$$\\left( { \\wedge , \\wedge } \\right)$$"}] | ["C"] | null | <img src="https://gateclass.cdn.examgoal.net/qTCfGCsgnLJmurfJO/mTRBMelO4xBQhIbdNCj3SfmvgSIpr/yQ9Y64EkKKeaEzp8Qw7Tue/uploadfile.jpg" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2019 (Online) 9th January Morning Slot Mathematics - Mathematical Reasoning Question 103 ... | mcq | jee-main-2019-online-9th-january-morning-slot | 6,832 |
FrleKOpX9kV4yTWIC0jgy2xukfqcz1nw | maths | mathematical-reasoning | logical-connectives | The statement
<br/>$$\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)$$ is : | [{"identifier": "A", "content": "a tautology"}, {"identifier": "B", "content": "a contradiction"}, {"identifier": "C", "content": "equivalent to (p $$ \\vee $$ q) $$ \\wedge $$ ($$ \\sim $$ p)"}, {"identifier": "D", "content": "equivalent to (p $$ \\wedge $$ q) $$ \\vee $$ ($$ \\sim $$ q)"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267009/exam_images/ug8ifnvkt1ne5r5zuiz4.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 5th September Evening Slot Mathematics - Mathematical Reasoning Question 74 English Explanation">
... | mcq | jee-main-2020-online-5th-september-evening-slot | 6,834 |
hySFpWbwwA3tcTNjAMjgy2xukf8zn82s | maths | mathematical-reasoning | logical-connectives | Given the following two statements:<br/><br/>
$$\left( {{S_1}} \right):\left( {q \vee p} \right) \to \left( {p \leftrightarrow \sim q} \right)$$ is a tautology<br/><br/>
$$\left( {{S_2}} \right): \,\,\sim q \wedge \left( { \sim p \leftrightarrow q} \right)$$ is a fallacy. Then: | [{"identifier": "A", "content": "both (S<sub>1</sub>) and (S<sub>2</sub>) are not correct"}, {"identifier": "B", "content": "only (S<sub>1</sub>) is correct"}, {"identifier": "C", "content": "only (S<sub>2</sub>) is correct"}, {"identifier": "D", "content": "both (S<sub>1</sub>) and (S<sub>2</sub>) are correct"}] | ["A"] | null | <b>Truth table for S<sub>1</sub> :
<br><br></b><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-sty... | mcq | jee-main-2020-online-4th-september-morning-slot | 6,836 |
nOq66cqiaklAbEp9Ztjgy2xukf3yrsm1 | maths | mathematical-reasoning | logical-connectives | Let p, q, r be three statements such that the
truth value of <br/>(p $$ \wedge $$ q) $$ \to $$ ($$ \sim $$q $$ \vee $$ r) is F. Then the
truth values of p, q, r are respectively : | [{"identifier": "A", "content": "T, F, T"}, {"identifier": "B", "content": "F, T, F"}, {"identifier": "C", "content": "T, T, T"}, {"identifier": "D", "content": "T, T, F"}] | ["D"] | null | Given, (p $$ \wedge $$ q) $$ \to $$ ($$ \sim $$q $$ \vee $$ r) is false.<br><br>This statement is false when<br><br>p $$ \wedge $$ q = T<br><br>and ($$ \sim $$q $$ \vee $$ r) = F<br><br>Now, p $$ \wedge $$ q = T when<br><br>both p and q are True.<br><br>As q = T<br><br>$$ \therefore $$ $$ \sim $$q = F<br><br>Now, ($$ \... | mcq | jee-main-2020-online-3rd-september-evening-slot | 6,837 |
nFlqKR4iPQURZOKVmP7k9k2k5khoedh | maths | mathematical-reasoning | logical-connectives | If p $$ \to $$ (p $$ \wedge $$ ~q) is false, then the truth values
of p and q are respectively : | [{"identifier": "A", "content": "T, T"}, {"identifier": "B", "content": "T, F"}, {"identifier": "C", "content": "F, T"}, {"identifier": "D", "content": "F, F"}] | ["A"] | null | p $$ \to $$ (p $$ \wedge $$ ~q) will be false only when p is true and (p $$ \wedge $$ ~q) is false.
<br><br>So, p = T, q = T | mcq | jee-main-2020-online-9th-january-evening-slot | 6,839 |
hAUxfzeSOEKdF4vGZL7k9k2k5hizmsn | maths | mathematical-reasoning | logical-connectives | Which of the following statements is a tautology? | [{"identifier": "A", "content": "~(p $$ \\wedge $$ ~q) $$ \\to $$ p $$ \\vee $$ q"}, {"identifier": "B", "content": "~(p $$ \\vee $$ ~q) $$ \\to $$ p $$ \\vee $$ q"}, {"identifier": "C", "content": "~(p $$ \\vee $$ ~q) $$ \\to $$ p $$ \\wedge $$ q"}, {"identifier": "D", "content": "p $$ \\vee $$ (~q) $$ \\to $$ p $$ \\... | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265134/exam_images/emajs7ltkurmpe8hmtbv.webp"><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263920/exam_images/d5mbdvxcpol7gfsdgwd3.webp" style="max-width: 100%;height: auto;display: block;margi... | mcq | jee-main-2020-online-8th-january-evening-slot | 6,840 |
eNuZyyW4HZ3ebDB5rx7k9k2k5gs3o9i | maths | mathematical-reasoning | logical-connectives | Which one of the following is a tautology? | [{"identifier": "A", "content": "P $$ \\wedge $$ (P $$ \\vee $$ Q)"}, {"identifier": "B", "content": "P $$ \\vee $$ (P $$ \\wedge $$ Q)"}, {"identifier": "C", "content": "Q $$ \\to $$ (P $$ \\wedge $$ (P $$ \\to $$ Q))"}, {"identifier": "D", "content": "(P $$ \\wedge $$ (P $$ \\to $$ Q)) $$ \\to $$ Q"}] | ["D"] | null | <b>Option A :</b>
<br><br><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;}
.tg th{font-family:Arial, sans-serif;font-size:14px;font-wei... | mcq | jee-main-2020-online-8th-january-morning-slot | 6,841 |
L5tV8VTiGQ5Joiaa5M7k9k2k5e4j5it | maths | mathematical-reasoning | logical-connectives | The logical statement (p $$ \Rightarrow $$ q) $$\Lambda $$ ( q $$ \Rightarrow $$ ~p) is equivalent to : | [{"identifier": "A", "content": "q"}, {"identifier": "B", "content": "$$ \\sim $$p"}, {"identifier": "C", "content": "p"}, {"identifier": "D", "content": "$$ \\sim $$q"}] | ["B"] | null | (p $$ \Rightarrow $$ q) $$\Lambda $$ ( q $$ \Rightarrow $$ ~p)
<br><br>$$ \equiv $$ $$\left( { \sim p \vee q} \right) \wedge \left( { \sim q \vee \sim p} \right)$$
<br><br>$$ \equiv $$ $$ \sim p \vee \left( {q \wedge \sim q} \right)$$
<br><br>$$ \equiv $$ $$ \sim $$p
<br><br>As $${q \wedge \sim q}$$ is a fallacy. | mcq | jee-main-2020-online-7th-january-morning-slot | 6,843 |
rkDqKdV7f4jhoxxZQhjgy2xukezbqrn0 | maths | mathematical-reasoning | logical-connectives | Which of the following is a tautology ? | [{"identifier": "A", "content": "$$\\left( { \\sim p} \\right) \\wedge \\left( {p \\vee q} \\right) \\to q$$"}, {"identifier": "B", "content": "$$\\left( {q \\to p} \\right) \\vee \\sim \\left( {p \\to q} \\right)$$"}, {"identifier": "C", "content": "$$\\left( {p \\to q} \\right) \\wedge \\left( {q \\to p} \\right)$$"... | ["A"] | null | ~ p $$ \wedge $$ (p $$ \vee $$ q) $$ \to $$ q<br><br>
$$ \equiv $$ (~ p $$ \wedge $$ p) $$ \vee $$ (~ p $$ \wedge $$ q) $$ \to $$ q<br><br>
$$ \equiv $$ C $$ \vee $$ (~ p $$ \wedge $$ q) $$ \to $$ q [C = contradiction]<br><br>
$$ \equiv $$ (~ p $$ \wedge $$ q) $$ \to $$ q<br><br>
$$ \equiv $$ ~ (~ p $... | mcq | jee-main-2020-online-2nd-september-evening-slot | 6,844 |
e8m7fiFp2ii9uhwxX41klrhnfcs | maths | mathematical-reasoning | logical-connectives | The statement among the following that is a tautology is : | [{"identifier": "A", "content": "$$B \\to \\left[ {A \\wedge \\left( {A \\to B} \\right)} \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {A \\wedge \\left( {A \\to B} \\right)} \\right] \\to B$$"}, {"identifier": "C", "content": "$$\\left[ {A \\wedge \\left( {A \\vee B} \\right)} \\right]$$"}, {"identifier": "... | ["B"] | null | Given, $$\left[ {A \wedge \left( {A \to B} \right)} \right] \to B$$
<br/><br/>= $$A \wedge \left( { \sim A \vee B} \right) \to B$$
<br/><br/>= $$\left[ {\left( {A \wedge \sim A} \right) \vee \left( {A \wedge B} \right)} \right] \to B$$
<br/><br/>= $$\left( {A \wedge B} \right) \to B$$
<br/><br/>= $${ \sim A \vee \sim... | mcq | jee-main-2021-online-24th-february-morning-slot | 6,845 |
ganiqHSEGa3zsqkGqV1klrl43pu | maths | mathematical-reasoning | logical-connectives | The negation of the statement <br/><br/>$$ \sim p \wedge (p \vee q)$$ is : | [{"identifier": "A", "content": "$$p \\vee \\sim q$$"}, {"identifier": "B", "content": "$$ \\sim p \\vee q$$"}, {"identifier": "C", "content": "$$ \\sim p \\wedge q$$"}, {"identifier": "D", "content": "$$p \\wedge \\sim q$$"}] | ["A"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264386/exam_images/pkgg8abdxwnuxm3mksuq.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 24th February Evening Shift Mathematics - Mathematical Reasoning Question 70 English Explanation">... | mcq | jee-main-2021-online-24th-february-evening-slot | 6,846 |
Y4dstDxJfkHb7DfLzh1klrl963n | maths | mathematical-reasoning | logical-connectives | For the statements p and q, consider the following compound statements :<br/><br/>(a) $$( \sim q \wedge (p \to q)) \to \sim p$$<br/><br/>(b) $$((p \vee q) \wedge \sim p) \to q$$<br/><br/>Then which of the following statements is correct? | [{"identifier": "A", "content": "(b) is a tautology but not (a)."}, {"identifier": "B", "content": "(a) and (b) both are not tautologies."}, {"identifier": "C", "content": "(a) and (b) both are tautologies."}, {"identifier": "D", "content": "(a) is a tautology but not (b)."}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263768/exam_images/zyaqwun6l1vaxc3vzpc7.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264303/exam_images/pe5z4lubb8xitalkkhvu.webp"><img src="https://res.c... | mcq | jee-main-2021-online-24th-february-evening-slot | 6,847 |
UiyDtnS5Vrk1mJy1Gf1kls47g1s | maths | mathematical-reasoning | logical-connectives | The statement A $$ \to $$ (B $$ \to $$ A) is equivalent to : | [{"identifier": "A", "content": "A $$ \\to $$ (A $$\\mathrel{\\mathop{\\kern0pt\\longleftrightarrow}\n\\limits_{}} $$ B)"}, {"identifier": "B", "content": "A $$ \\to $$ (A $$ \\vee $$ B)"}, {"identifier": "C", "content": "A $$ \\to $$ (A $$ \\wedge $$ B)"}, {"identifier": "D", "content": "A $$ \\to $$ (A $$ \\to $$ B)"... | ["B"] | null | $$A \to (B \to A)$$<br><br>$$ \Rightarrow A \to ( \sim B \vee A)$$<br><br>$$ \Rightarrow \, \sim A \vee ( \sim B \vee A)$$<br><br>$$ \Rightarrow \, \sim B \vee ( \sim A \vee A)$$<br><br>$$ \Rightarrow \, \sim B \vee t$$<br><br>= t (tantology)<br><br>From options :<br><br>(B) $$A \to (A \vee B)$$<br><br>$$ \Rightarrow \... | mcq | jee-main-2021-online-25th-february-morning-slot | 6,848 |
fPu8c7tDybZ3josxp31kluvu5ch | maths | mathematical-reasoning | logical-connectives | Let F<sub>1</sub>(A, B, C) = (A $$ \wedge $$ $$ \sim $$ B) $$ \vee $$ [$$\sim$$C $$\wedge$$ (A $$\vee$$ B)] $$\vee$$ $$\sim$$ A and <br/>F<sub>2</sub>(A, B) = (A $$\vee$$ B) $$\vee$$ (B $$ \to $$ $$\sim$$A) be two logical expressions. Then : | [{"identifier": "A", "content": "Both F<sub>1</sub> and F<sub>2</sub> are not tautologies"}, {"identifier": "B", "content": "F<sub>1</sub> and F<sub>2</sub> both are tautologies"}, {"identifier": "C", "content": "F<sub>1</sub> is not a tautology but F<sub>2</sub> is a tautology"}, {"identifier": "D", "content": "F<sub>... | ["C"] | null | Truth table for F<sub>1</sub> :
<br> <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265217/exam_images/nh8hwgezkfdnwxzdht25.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265361/exam_images/y6kerhcssgaio... | mcq | jee-main-2021-online-26th-february-evening-slot | 6,849 |
1p2dl8GpcxkStpiRZt1kmhvyzos | maths | mathematical-reasoning | logical-connectives | Which of the following Boolean expression is a tautology? | [{"identifier": "A", "content": "(p $$ \\wedge $$ q) $$ \\vee $$ (p $$ \\to $$ q)"}, {"identifier": "B", "content": "(p $$ \\wedge $$ q) $$ \\vee $$ (p $$\\vee$$ q)"}, {"identifier": "C", "content": "(p $$ \\wedge $$ q) $$ \\to $$ (p $$ \\to $$ q)"}, {"identifier": "D", "content": "(p $$ \\wedge $$ q) $$ \\wedge $$ (p ... | ["C"] | null | $$\matrix{
p & q & {p \wedge q} & {p \vee q} & {p \to q} & {(p \wedge q) \to (p \to q)} \cr
T & T & T & T & T & T \cr
F & T & F & T & T & T \cr
T & F & F & T & F & T \cr
F & F & F & F & T & T \cr ... | mcq | jee-main-2021-online-16th-march-morning-shift | 6,850 |
8hZxyvOr3nEBB8fFxy1kmjapolf | maths | mathematical-reasoning | logical-connectives | If the Boolean expression (p $$ \Rightarrow $$ q) $$ \Leftrightarrow $$ (q * ($$ \sim $$p) is a tautology, then the boolean expression (p * ($$ \sim $$q)) is equivalent to : | [{"identifier": "A", "content": "q $$ \\Rightarrow $$ p"}, {"identifier": "B", "content": "p $$ \\Rightarrow $$ q"}, {"identifier": "C", "content": "p $$ \\Rightarrow $$ $$ \\sim $$ q"}, {"identifier": "D", "content": "$$ \\sim $$q $$ \\Rightarrow $$ p"}] | ["A"] | null | <p>The Boolean expression</p>
<p>$(p \Rightarrow q) \Leftrightarrow\left(q^*(\sim p)\right)$ is a tautology.</p>
<p>Making the truth table for this</p>
<style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;width:100%}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, san... | mcq | jee-main-2021-online-17th-march-morning-shift | 6,851 |
fbDsZRHROBb4k6QKaY1kmkmyoqc | maths | mathematical-reasoning | logical-connectives | If the Boolean expression $$(p \wedge q) \odot (p \otimes q)$$ is a tautology, then $$ \odot $$ and $$ \otimes $$ are respectively given by : | [{"identifier": "A", "content": "$$ \\vee , \\to $$"}, {"identifier": "B", "content": "$$ \\to $$, $$ \\to $$"}, {"identifier": "C", "content": "$$ \\wedge $$, $$ \\vee $$"}, {"identifier": "D", "content": "$$ \\wedge $$, $$ \\to $$"}] | ["B"] | null | $$(p \wedge q)\, \to \,(p \to q)$$<br><br>$$(p \wedge q)\, \to \,( \sim p \vee q)$$<br><br>$$( \sim p \vee \sim q)\, \vee ( \sim p \vee q)$$<br><br>$$ \sim p \vee ( \sim q \vee q) \Rightarrow $$ Tautology<br><br>$$ \Rightarrow \odot \Rightarrow \to $$<br><br>$$ \otimes \Rightarrow \to $$ | mcq | jee-main-2021-online-17th-march-evening-shift | 6,852 |
nzAOiYmH4hkTFVNC2R1kmm3sgyz | maths | mathematical-reasoning | logical-connectives | If P and Q are two statements, then which of the following compound statement is a tautology? | [{"identifier": "A", "content": "((P $$ \\Rightarrow $$ Q) $$ \\wedge $$ $$ \\sim $$ Q) $$ \\Rightarrow $$ (P $$ \\wedge $$ Q)"}, {"identifier": "B", "content": "((P $$ \\Rightarrow $$ Q) $$ \\wedge $$ $$ \\sim $$ Q) $$ \\Rightarrow $$ Q"}, {"identifier": "C", "content": "((P $$ \\Rightarrow $$ Q) $$ \\wedge $$ $$ \\si... | ["D"] | null | <p>LHS of all the options are same i.e.</p>
<p>$$((P \to Q) \wedge \sim Q)$$</p>
<p>$$ \equiv ( \sim P \vee Q) \wedge \sim Q$$</p>
<p>$$ \equiv ( \sim P \wedge \sim Q) \vee (Q \wedge \sim Q)$$</p>
<p>$$ \equiv \sim P \wedge \sim Q$$</p>
<p>(A) $$( \sim P \wedge \sim Q) \to Q$$</p>
<p>$$ \equiv \sim ( \sim P \we... | mcq | jee-main-2021-online-18th-march-evening-shift | 6,853 |
1krpripml | maths | mathematical-reasoning | logical-connectives | The Boolean expression $$(p \wedge \sim q) \Rightarrow (q \vee \sim p)$$ is equivalent to : | [{"identifier": "A", "content": "$$q \\Rightarrow p$$"}, {"identifier": "B", "content": "$$p \\Rightarrow q$$"}, {"identifier": "C", "content": "$$ \\sim q \\Rightarrow p$$"}, {"identifier": "D", "content": "$$p \\Rightarrow \\, \\sim q$$"}] | ["B"] | null | <table class="tg">
<thead>
<tr>
<th class="tg-baqh">p</th>
<th class="tg-baqh">q</th>
<th class="tg-baqh">$$ \sim p$$</th>
<th class="tg-baqh">$$ \sim q$$</th>
<th class="tg-baqh">$$p \wedge \sim q$$</th>
<th class="tg-baqh">$$q \vee \sim p$$</th>
<th class="tg-baqh">$$(p \wedge \sim q)... | mcq | jee-main-2021-online-20th-july-morning-shift | 6,854 |
1kru3ommn | maths | mathematical-reasoning | logical-connectives | Which of the following Boolean expressions is not a tautology? | [{"identifier": "A", "content": "(p $$\\Rightarrow$$ q) $$ \\vee $$ ($$ \\sim $$ q $$\\Rightarrow$$ p)"}, {"identifier": "B", "content": "(q $$\\Rightarrow$$ p) $$ \\vee $$ ($$ \\sim $$ q $$\\Rightarrow$$ p)"}, {"identifier": "C", "content": "(p $$\\Rightarrow$$ $$ \\sim $$ q) $$ \\vee $$ ($$ \\sim $$ q $$\\Rightarrow$... | ["D"] | null | (1) (p $$\to$$ q) $$\vee$$ ($$\sim$$ q $$\to$$ p)<br><br>= ($$\sim$$ p $$\vee$$ q) $$\vee$$ (q $$\vee$$ p)<br><br>= ($$\sim$$ p $$\vee$$ p) $$\vee$$ q<br><br>= t $$\vee$$ q = t<br><br>(2) (q $$\to$$ p) $$\vee$$ ($$\sim$$ q $$\to$$ p)<br><br>= ($$\sim$$ q $$\vee$$ p) $$\vee$$ (q $$\vee$$ p)<br><br>= ($$\sim$$ q $$\vee$$... | mcq | jee-main-2021-online-22th-july-evening-shift | 6,855 |
1ks07sngu | maths | mathematical-reasoning | logical-connectives | The compound statement $$(P \vee Q) \wedge ( \sim P) \Rightarrow Q$$ is equivalent to : | [{"identifier": "A", "content": "$$P \\vee Q$$"}, {"identifier": "B", "content": "$$P \\wedge \\sim Q$$"}, {"identifier": "C", "content": "$$ \\sim (P \\Rightarrow Q)$$"}, {"identifier": "D", "content": "$$ \\sim (P \\Rightarrow Q) \\Leftrightarrow P \\wedge \\sim Q$$"}] | ["D"] | null | Using Truth Table :<br><br> <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264245/exam_images/uemdlbsr1t66wwcno3ny.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266789/exam_images/jobctzefoxr8ftpbzqtr.w... | mcq | jee-main-2021-online-27th-july-morning-shift | 6,857 |
1ktbgkdhv | maths | mathematical-reasoning | logical-connectives | If the truth value of the Boolean expression $$\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)$$ is false, then the truth values of the statements p, q, r respectively can be : | [{"identifier": "A", "content": "T F T"}, {"identifier": "B", "content": "F F T"}, {"identifier": "C", "content": "T F F "}, {"identifier": "D", "content": "F T F"}] | ["C"] | null | <table class="tg">
<thead>
<tr>
<th class="tg-baqh">p</th>
<th class="tg-baqh">q</th>
<th class="tg-baqh">r</th>
<th class="tg-baqh">$$\underbrace {p \vee q}_a$$</th>
<th class="tg-baqh">$$\underbrace {q \to r}_b$$</th>
<th class="tg-baqh">$${a \wedge b}$$</th>
<th class="tg-baqh">$${ \sim... | mcq | jee-main-2021-online-26th-august-morning-shift | 6,858 |
1ktcznobe | maths | mathematical-reasoning | logical-connectives | Consider the two statements :<br/><br/>(S1) : (p $$\to$$ q) $$ \vee $$ ($$ \sim $$ q $$\to$$ p) is a tautology .<br/><br/>(S2) : (p $$ \wedge $$ $$ \sim $$ q) $$ \wedge $$ ($$\sim$$ p $$\wedge$$ q) is a fallacy.<br/><br/>Then : | [{"identifier": "A", "content": "only (S1) is true."}, {"identifier": "B", "content": "both (S1) and (S2) are false."}, {"identifier": "C", "content": "both (S1) and (S2) are true."}, {"identifier": "D", "content": "only (S2) is true."}] | ["C"] | null | S<sub>1</sub> : ($$\sim$$ p $$ \vee $$ q) $$ \vee $$ (q $$ \vee $$ p) = (q $$ \vee $$ $$\sim$$ p) $$ \vee $$ (q $$ \vee $$ p)<br><br>S<sub>1</sub> = q $$ \vee $$ ($$\sim$$ p $$ \vee $$ p) = qvt = t = tautology<br><br>S<sub>2</sub> : (p $$ \wedge $$ $$\sim$$ q) $$ \wedge $$ ($$\sim$$ p $$\vee$$ q) = (p $$ \wedge $$ $$\s... | mcq | jee-main-2021-online-26th-august-evening-shift | 6,859 |
1ktekw6qp | maths | mathematical-reasoning | logical-connectives | The statement (p $$ \wedge $$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r is : | [{"identifier": "A", "content": "a tautology"}, {"identifier": "B", "content": "equivalent to p $$\\to$$ $$\\sim$$ r"}, {"identifier": "C", "content": "a fallacy"}, {"identifier": "D", "content": "equivalent to q $$\\to$$ $$\\sim$$ r"}] | ["A"] | null | (p $$ \wedge $$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r<br><br>$$\equiv$$ (p $$\wedge$$ ($$\sim$$ p $$\vee$$ q) $$\vee$$ ($$\sim$$ q $$\vee$$ r)) $$\to$$ r<br><br>$$\equiv$$ ((p $$\wedge$$ q) $$\wedge$$ ($$\sim$$ p $$\vee$$ r)) $$\to$$ r<br><br>$$\equiv$$ (p $$\wedge$$ q $$\wedge$$ r) $$\to$$ r<br><br>$$\equi... | mcq | jee-main-2021-online-27th-august-morning-shift | 6,860 |
1ktg2lye2 | maths | mathematical-reasoning | logical-connectives | The Boolean expression (p $$\wedge$$ q) $$\Rightarrow$$ ((r $$\wedge$$ q) $$\wedge$$ p) is equivalent to : | [{"identifier": "A", "content": "(p $$\\wedge$$ q) $$\\Rightarrow$$ (r $$\\wedge$$ q)"}, {"identifier": "B", "content": "(q $$\\wedge$$ r) $$\\Rightarrow$$ (p $$\\wedge$$ q)"}, {"identifier": "C", "content": "(p $$\\wedge$$ q) $$\\Rightarrow$$ (r $$\\vee$$ q)"}, {"identifier": "D", "content": "(p $$\\wedge$$ r) $$\\Ri... | ["A"] | null | given statement says<br><br>"if p and q both happen then p and q and r will happen"<br><br>it simply implies "If p and q both happen then 'r' too will happen"<br><br>i.e.<br><br>"if p and q both happen then r and p too will happen<br><br>i.e.<br><br>(p $$\wedge$$ q) $$\Rightarrow$$ (r $$\wedge$$ p) | mcq | jee-main-2021-online-27th-august-evening-shift | 6,861 |
1ktk5iipi | maths | mathematical-reasoning | logical-connectives | Negation of the statement (p $$\vee$$ r) $$\Rightarrow$$ (q $$\vee$$ r) is : | [{"identifier": "A", "content": "p $$\\wedge$$ $$\\sim$$ q $$\\wedge$$ $$\\sim$$ r"}, {"identifier": "B", "content": "$$\\sim$$ p $$\\wedge$$ q $$\\wedge$$ $$\\sim$$ 4"}, {"identifier": "C", "content": "$$\\sim$$ p $$\\wedge$$ q $$\\wedge$$ r"}, {"identifier": "D", "content": "p $$\\wedge$$ q $$\\wedge$$ r"}] | ["A"] | null | <p>Negative of (p $$\vee$$ r) $$\Rightarrow$$ (q $$\vee$$ r)</p>
<p>$$ \equiv $$ $$\sim$$ ((p $$\vee$$ r) $$\Rightarrow$$ (q $$\vee$$ r)) $$\equiv$$ (p $$\vee$$ r) $$ \wedge $$ ($$\sim$$ (q $$\vee$$ r))</p>
<p>$$\equiv$$ (p $$\vee$$ r) $$\wedge$$ ($$\sim$$ q $$\wedge$$ $$\sim$$ r) $$\equiv$$ (p $$\vee$$ r) $$\wedge$$ $... | mcq | jee-main-2021-online-31st-august-evening-shift | 6,863 |
1kto27r9o | maths | mathematical-reasoning | logical-connectives | Which of the following is equivalent to the Boolean expression p $$\wedge$$ $$\sim$$ q ? | [{"identifier": "A", "content": "$$\\sim$$ (q $$\\to$$ p)"}, {"identifier": "B", "content": "$$\\sim$$ p $$\\to$$ $$\\sim$$ q"}, {"identifier": "C", "content": "$$\\sim$$ (p $$\\to$$ $$\\sim$$ q)"}, {"identifier": "D", "content": "$$\\sim$$ (p $$\\to$$ q)"}] | ["D"] | null | <table class="tg">
<thead>
<tr>
<th class="tg-baqh">p</th>
<th class="tg-baqh">q</th>
<th class="tg-baqh">$$ \sim $$ p</th>
<th class="tg-baqh">$$ \sim $$ q</th>
<th class="tg-baqh">p $$ \to $$ q</th>
<th class="tg-baqh">$$ \sim $$ (p $$ \to $$ q)</th>
<th class="tg-baqh">q $$ \to $$ p</th... | mcq | jee-main-2021-online-1st-september-evening-shift | 6,864 |
1l5450e9z | maths | mathematical-reasoning | logical-connectives | <p>Let $$\Delta$$ $$\in$$ {$$\wedge$$, $$\vee$$, $$\Rightarrow$$, $$\Leftrightarrow$$} be such that (p $$\wedge$$ q) $$\Delta$$ ((p $$\vee$$ q) $$\Rightarrow$$ q) is a tautology. Then $$\Delta$$ is equal to :</p> | [{"identifier": "A", "content": "$$\\wedge$$"}, {"identifier": "B", "content": "$$\\vee$$"}, {"identifier": "C", "content": "$$\\Rightarrow$$"}, {"identifier": "D", "content": "$$\\Leftrightarrow$$"}] | ["C"] | null | $(p \vee q) \Rightarrow q$
<br/><br/>
$$
\begin{aligned}
& \sim(p \vee q) \vee q \\\\
& =(\sim p \wedge \sim q) \vee q \\\\
& =(\sim p \vee q) \wedge(\sim q \vee q) \\\\
& =(\sim p \vee q) \wedge T \\\\
& =\sim p \vee q
\end{aligned}
$$
<br/><br/>
Now $(p \wedge q) \Delta(\sim p \vee q)$
<br/><br/>
$$
\begin{array}{ccc... | mcq | jee-main-2022-online-29th-june-morning-shift | 6,865 |
1l54tfxa2 | maths | mathematical-reasoning | logical-connectives | <p>Negation of the Boolean statement (p $$\vee$$ q) $$\Rightarrow$$ (($$\sim$$ r) $$\vee$$ p) is equivalent to :</p> | [{"identifier": "A", "content": "p $$\\wedge$$ ($$\\sim$$ q) $$\\wedge$$ r"}, {"identifier": "B", "content": "($$\\sim$$ p) $$\\wedge$$ ($$\\sim$$ q) $$\\wedge$$ r"}, {"identifier": "C", "content": "($$\\sim$$ p) $$\\wedge$$ q $$\\wedge$$ r"}, {"identifier": "D", "content": "p $$\\wedge$$ q $$\\wedge$$ ($$\\sim$$ r)"}] | ["C"] | null | <p>Given,</p>
<p>(p $$\vee$$ q) $$\Rightarrow$$ (($$\sim$$ r) $$\vee$$ p)</p>
<p>Negation is</p>
<p>$$\sim$$ ((p $$\vee$$ q) $$\Rightarrow$$ ($$\sim$$ r) $$\vee$$ p))</p>
<p>= (p $$\vee$$ q) $$\wedge$$ $$\sim$$ (($$\sim$$ r) $$\vee$$ p)</p>
<p>= (p $$\vee$$ q) $$\wedge$$ (r $$\wedge$$ $$\sim$$ p)</p>
<p>[(p $$\wedge$$ ... | mcq | jee-main-2022-online-29th-june-evening-shift | 6,866 |
1l55jdggk | maths | mathematical-reasoning | logical-connectives | <p>The maximum number of compound propositions, out of p$$\vee$$r$$\vee$$s, p$$\vee$$r$$\vee$$$$\sim$$s, p$$\vee$$$$\sim$$q$$\vee$$s, $$\sim$$p$$\vee$$$$\sim$$r$$\vee$$s, $$\sim$$p$$\vee$$$$\sim$$r$$\vee$$$$\sim$$s, $$\sim$$p$$\vee$$q$$\vee$$$$\sim$$s, q$$\vee$$r$$\vee$$$$\sim$$s, q$$\vee$$$$\sim$$r$$\vee$$$$\sim$$s, $... | [] | null | 9 | There are total 9 compound propositions, out of which 6 contain $\sim s$. So if we assign $s$ as false, these 6 propositions will be true.
<br/><br/>
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions wi... | integer | jee-main-2022-online-28th-june-evening-shift | 6,867 |
1l567e91x | maths | mathematical-reasoning | logical-connectives | <p>Let p, q, r be three logical statements. Consider the compound statements</p>
<p>$${S_1}:(( \sim p) \vee q) \vee (( \sim p) \vee r)$$ and</p>
<p>$${S_2}:p \to (q \vee r)$$</p>
<p>Then, which of the following is NOT true?</p> | [{"identifier": "A", "content": "If S<sub>2</sub> is True, then S<sub>1</sub> is True"}, {"identifier": "B", "content": "If S<sub>2</sub> is False, then S<sub>1</sub> is False"}, {"identifier": "C", "content": "If S<sub>2</sub> is False, then S<sub>1</sub> is True"}, {"identifier": "D", "content": "If S<sub>1</sub> is ... | ["C"] | null | <p>$${S_1}:( \sim p \vee q) \vee ( \sim p \vee r)$$</p>
<p>$$ \cong ( \sim p \vee q \vee r)$$</p>
<p>$${S_2}: \sim p \vee (q \vee r)$$</p>
<p>Both are same</p>
<p>So, option (C) is incorrect.</p> | mcq | jee-main-2022-online-28th-june-morning-shift | 6,868 |
1l56rnxj7 | maths | mathematical-reasoning | logical-connectives | <p>Which of the following statement is a tautology?</p> | [{"identifier": "A", "content": "$$(( \\sim q) \\wedge p) \\wedge q$$"}, {"identifier": "B", "content": "$$(( \\sim q) \\wedge p) \\wedge (p \\wedge ( \\sim p))$$"}, {"identifier": "C", "content": "$$(( \\sim q) \\wedge p) \\vee (p \\vee ( \\sim p))$$"}, {"identifier": "D", "content": "$$(p \\wedge q) \\wedge ( \\sim p... | ["C"] | null | <p>$$\because$$ (($$\sim$$ q) $$\wedge$$ p) $$\vee$$ (p $$\vee$$ ($$\sim$$ p))</p>
<p>= ($$\sim$$ q $$\wedge$$ p) $$\vee$$ t (t is tautology)</p>
<p>$$\equiv$$ t</p>
<p>$$\therefore$$ option (C) is correct.</p> | mcq | jee-main-2022-online-27th-june-evening-shift | 6,869 |
1l57ouws7 | maths | mathematical-reasoning | logical-connectives | <p>The boolean expression $$( \sim (p \wedge q)) \vee q$$ is equivalent to :</p> | [{"identifier": "A", "content": "$$q \\to (p \\wedge q)$$"}, {"identifier": "B", "content": "$$p \\to q$$"}, {"identifier": "C", "content": "$$p \\to (p \\to q)$$"}, {"identifier": "D", "content": "$$p \\to (p \\vee q)$$"}] | ["D"] | null | <p>Making truth table</p>
<p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:solid;border-wi... | mcq | jee-main-2022-online-27th-june-morning-shift | 6,870 |
1l58abzlk | maths | mathematical-reasoning | logical-connectives | <p>Let $$\Delta$$, $$\nabla $$ $$\in$$ {$$\wedge$$, $$\vee$$} be such that p $$\nabla$$ q $$\Rightarrow$$ ((p $$\Delta$$ q) $$\nabla$$ r) is a tautology. Then (p $$\nabla$$ q) $$\Delta$$ r is logically equivalent to :</p> | [{"identifier": "A", "content": "(p $$\\Delta$$ r) $$\\vee$$ q"}, {"identifier": "B", "content": "(p $$\\Delta$$ r) $$\\wedge$$ q"}, {"identifier": "C", "content": "(p $$\\wedge$$ r) $$\\Delta$$ q"}, {"identifier": "D", "content": "(p $$\\nabla$$ r) $$\\wedge$$ q"}] | ["A"] | null | <b>Case-I</b><br/><br/>
If $\nabla$ is same as $\wedge$<br/><br/>
Then $(p \wedge q) \Rightarrow((p \Delta q) \wedge r)$<br/><br/> is equivalent to $\sim(p \wedge q) \vee$ $((p \Delta q) \wedge r)$<br/><br/> is equivalent to $(\sim(p \wedge q) \vee(p \Delta q)) \wedge(\sim(p \wedge$ $q) \vee r)$<br/><br/>
Which cannot ... | mcq | jee-main-2022-online-26th-june-morning-shift | 6,871 |
1l58grngd | maths | mathematical-reasoning | logical-connectives | <p>Let r $$\in$$ {p, q, $$\sim$$p, $$\sim$$q} be such that the logical statement</p>
<p>r $$\vee$$ ($$\sim$$p) $$\Rightarrow$$ (p $$\wedge$$ q) $$\vee$$ r</p>
<p>is a tautology. Then r is equal to :</p> | [{"identifier": "A", "content": "p"}, {"identifier": "B", "content": "q"}, {"identifier": "C", "content": "$$\\sim$$p"}, {"identifier": "D", "content": "$$\\sim$$q"}] | ["C"] | null | <p>Clearly r must be equal to $$\sim$$ p</p>
<p>$$\because$$ $$\sim$$ p $$\vee$$ $$\sim$$ p = $$\sim$$ p</p>
<p>and (p $$\wedge$$ q) $$\vee$$ $$\sim$$ p = p</p>
<p>$$\therefore$$ $$\sim$$ p $$\Rightarrow$$ p = tautology.</p> | mcq | jee-main-2022-online-26th-june-evening-shift | 6,872 |
1l59kxe1t | maths | mathematical-reasoning | logical-connectives | <p>The negation of the Boolean expression (($$\sim$$ q) $$\wedge$$ p) $$\Rightarrow$$ (($$\sim$$ p) $$\vee$$ q) is logically equivalent to :</p> | [{"identifier": "A", "content": "$$p \\Rightarrow q$$"}, {"identifier": "B", "content": "$$q \\Rightarrow p$$"}, {"identifier": "C", "content": "$$ \\sim (p \\Rightarrow q)$$"}, {"identifier": "D", "content": "$$ \\sim (q \\Rightarrow p)$$"}] | ["C"] | null | <p>Let $$S:(( \sim q) \wedge p) \Rightarrow (( \sim p) \vee q)$$</p>
<p>$$ \Rightarrow S:\, \sim (( \sim q) \wedge p) \vee (( \sim p) \vee q)$$</p>
<p>$$ \Rightarrow S:(q \vee ( \sim p)) \vee (( \sim p) \vee q)$$</p>
<p>$$ \Rightarrow S:( \sim p) \vee q$$</p>
<p>$$ \Rightarrow S:p \Rightarrow q$$</p>
<p>So, negation of... | mcq | jee-main-2022-online-25th-june-evening-shift | 6,873 |
1l5ai3ak1 | maths | mathematical-reasoning | logical-connectives | <p>Consider the following two propositions:</p>
<p>$$P1: \sim (p \to \sim q)$$</p>
<p>$$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$$</p>
<p>If the proposition $$p \to (( \sim p) \vee q)$$ is evaluated as FALSE, then :</p> | [{"identifier": "A", "content": "P1 is TRUE and P2 is FALSE"}, {"identifier": "B", "content": "P1 is FALSE and P2 is TRUE"}, {"identifier": "C", "content": "Both P1 and P2 are FALSE"}, {"identifier": "D", "content": "Both P1 and P2 are TRUE"}] | ["C"] | null | Given $p \rightarrow(\sim p \vee q)$ is false <br/><br/>$\Rightarrow \sim p \vee q$ is false and $p$ is true<br/><br/>
Now $p=$ True.<br/><br/>
$\sim T \vee q=F$<br/><br/>
$F \vee q=F \Rightarrow q$ is false<br/><br/>
$P 1: \sim(T \rightarrow \sim F) \equiv \sim(T \rightarrow T) \equiv$ False.<br/><br/>
$$
\begin{align... | mcq | jee-main-2022-online-25th-june-morning-shift | 6,874 |
1l5c1xxom | maths | mathematical-reasoning | logical-connectives | <p>The number of choices for $$\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \} $$, such that <br/><br/>$$(p\Delta q) \Rightarrow ((p\Delta \sim q) \vee (( \sim p)\Delta q))$$ is a tautology, is :</p> | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "4"}] | ["B"] | null | Let $x:(p \Delta q) \Rightarrow(p \Delta \sim q) \vee(\sim p \Delta q)$
<br/><br/>
<b>Case-I</b>
<br/><br/>
When $\Delta$ is same as $v$
<br/><br/>
Then $(p \Delta \sim q) \vee(\sim p \Delta q)$ becomes
<br/><br/>
$(p \vee \sim q) \vee(\sim p \vee q)$ which is always true, so $x$ becomes a tautology.
<br/><br/>
<b>Case... | mcq | jee-main-2022-online-24th-june-morning-shift | 6,875 |
1l5w0obkl | maths | mathematical-reasoning | logical-connectives | <p>The conditional statement</p>
<p>$$((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q))$$ is :</p> | [{"identifier": "A", "content": "a tautology"}, {"identifier": "B", "content": "a contadiction"}, {"identifier": "C", "content": "equivalent to $$p \\wedge q$$"}, {"identifier": "D", "content": "equivalent to $$( \\sim p) \\vee r$$"}] | ["A"] | null | <p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial,... | mcq | jee-main-2022-online-30th-june-morning-shift | 6,876 |
1l6dwvb9t | maths | mathematical-reasoning | logical-connectives | <p>Which of the following statements is a tautology ?</p> | [{"identifier": "A", "content": "$$((\\sim \\mathrm{p}) \\vee \\mathrm{q}) \\Rightarrow \\mathrm{p}$$"}, {"identifier": "B", "content": "$$p \\Rightarrow((\\sim p) \\vee q)$$"}, {"identifier": "C", "content": "$$((\\sim p) \\vee q) \\Rightarrow q$$"}, {"identifier": "D", "content": "$$q \\Rightarrow((\\sim p) \\vee q)$... | ["D"] | null | Truth Table<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l97slo37/df9ec0ac-a06c-4e60-b3b5-9f73d35cb598/e7215830-4b5c-11ed-bfde-e1cb3fafe700/file-1l97slo38.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l97slo37/df9ec0ac-a06c-4e60-b3b5-9f73d35cb598/e7215830-4b5c-11ed-... | mcq | jee-main-2022-online-25th-july-morning-shift | 6,877 |
1l6gj6vbd | maths | mathematical-reasoning | logical-connectives | <p>The statement $$(\sim(\mathrm{p} \Leftrightarrow \,\sim \mathrm{q})) \wedge \mathrm{q}$$ is :</p> | [{"identifier": "A", "content": "a tautology"}, {"identifier": "B", "content": "a contradiction"}, {"identifier": "C", "content": "equivalent to $$(p \\Rightarrow q) \\wedge q$$"}, {"identifier": "D", "content": "equivalent to $$(p \\Rightarrow q) \\wedge p$$"}] | ["D"] | null | <p>$$\sim$$ (p $$ \Leftrightarrow $$ $$\sim$$ q) $$\wedge$$ q</p>
<p>= (p $$ \Leftrightarrow $$ q) $$\wedge$$ q</p>
<p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;p... | mcq | jee-main-2022-online-26th-july-morning-shift | 6,878 |
1l6hz70ir | maths | mathematical-reasoning | logical-connectives | <p>Negation of the Boolean expression $$p \Leftrightarrow(q \Rightarrow p)$$ is</p> | [{"identifier": "A", "content": "$$(\\sim p) \\wedge q$$"}, {"identifier": "B", "content": "$$p \\wedge(\\sim q)$$"}, {"identifier": "C", "content": "$$(\\sim p) \\vee(\\sim q)$$"}, {"identifier": "D", "content": "$$(\\sim p) \\wedge(\\sim q)$$"}] | ["D"] | null | <p>$$p \Leftrightarrow (q \Rightarrow p)$$</p>
<p>$$ \sim (p \Leftrightarrow (q \Leftrightarrow p))$$</p>
<p>$$ \equiv p \Leftrightarrow \, \sim (q \Rightarrow p)$$</p>
<p>$$ \equiv p \Leftrightarrow (q \wedge \sim p)$$</p>
<p>$$ \equiv (p \Rightarrow (q \wedge \sim p)) \wedge ((q \wedge \sim p) \Rightarrow p))$$</p... | mcq | jee-main-2022-online-26th-july-evening-shift | 6,879 |
1l6kl6wn5 | maths | mathematical-reasoning | logical-connectives | <p>If the truth value of the statement $$(P \wedge(\sim R)) \rightarrow((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is $$\mathrm{F}$$ ?</p> | [{"identifier": "A", "content": "$$\\mathrm{P} \\vee \\mathrm{Q} \\rightarrow \\,\\sim \\mathrm{R}$$"}, {"identifier": "B", "content": "$$\\mathrm{R} \\vee \\mathrm{Q} \\rightarrow \\,\\sim \\mathrm{P}$$"}, {"identifier": "C", "content": "$$\\sim(\\mathrm{P} \\vee \\mathrm{Q}) \\rightarrow \\sim \\mathrm{R}$$"}, {"iden... | ["D"] | null | $\mathrm{X} \Rightarrow \mathrm{Y}$ is a false<br/><br/>
when $X$ is true and $Y$ is false<br/><br/>
So, $\mathrm{P} \rightarrow \mathrm{T}, \mathrm{Q} \rightarrow \mathrm{F}, \mathrm{R} \rightarrow \mathrm{F}$<br/><br/>
(A) $\mathrm{P} \vee \mathrm{Q} \rightarrow \sim \mathrm{R}$ is $\mathrm{T}$<br/><br/>
(B) $\mathrm... | mcq | jee-main-2022-online-27th-july-evening-shift | 6,881 |
1l6m5e0bg | maths | mathematical-reasoning | logical-connectives | <p>Let the operations $$*, \odot \in\{\wedge, \vee\}$$. If $$(\mathrm{p} * \mathrm{q}) \odot(\mathrm{p}\, \odot \sim \mathrm{q})$$ is a tautology, then the ordered pair $$(*, \odot)$$ is :</p> | [{"identifier": "A", "content": "$$(\\vee, \\wedge)$$"}, {"identifier": "B", "content": "$$(\\vee, \\vee)$$"}, {"identifier": "C", "content": "$$(\\wedge, \\wedge)$$"}, {"identifier": "D", "content": "$$(\\wedge, \\vee)$$"}] | ["B"] | null | <p>$$ * ,\, \odot \in \{ \wedge ,\, \vee \} $$</p>
<p>Now for $$(p * q) \odot (p \odot \sim q)$$ is tautology</p>
<p>(A) $$( \vee , \wedge ):(p \vee q) \wedge (p \wedge \sim q)$$ not a tautology</p>
<p>(B) $$( \vee , \vee ):(p \vee q) \vee (p \vee \sim q)$$</p>
<p>$$ = P \vee T$$ is tautology</p>
<p>(C) $$( \wedge... | mcq | jee-main-2022-online-28th-july-morning-shift | 6,882 |
1l6p294z9 | maths | mathematical-reasoning | logical-connectives | <p>The statement $$(p \wedge q) \Rightarrow(p \wedge r)$$ is equivalent to :</p> | [{"identifier": "A", "content": "$$q \\Rightarrow(p \\wedge r)$$"}, {"identifier": "B", "content": "$$p\\Rightarrow(\\mathrm{p} \\wedge \\mathrm{r})$$"}, {"identifier": "C", "content": "$$(\\mathrm{p} \\wedge \\mathrm{r}) \\Rightarrow(\\mathrm{p} \\wedge \\mathrm{q})$$"}, {"identifier": "D", "content": "$$(p \\wedge q)... | ["D"] | null | <p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial,... | mcq | jee-main-2022-online-29th-july-morning-shift | 6,883 |
1ldo5et0w | maths | mathematical-reasoning | logical-connectives | <p>Which of the following statements is a tautology?</p> | [{"identifier": "A", "content": "$$\\mathrm{p\\vee(p\\wedge q)}$$"}, {"identifier": "B", "content": "$$(\\mathrm{p\\wedge(p\\to q))\\to\\,\\sim q}$$"}, {"identifier": "C", "content": "$$\\mathrm{p\\to (p\\wedge (p\\to q))}$$"}, {"identifier": "D", "content": "$$(\\mathrm{p\\wedge q)\\to(\\sim (p)\\to q)}$$"}] | ["D"] | null | $\begin{aligned} & \sim p \rightarrow q \equiv \sim(\sim p) \vee q \equiv p \vee q \\\\ & p \wedge q \rightarrow(\sim p \rightarrow q) \\\\ & \equiv p \wedge q \rightarrow(p \vee q) \\\\ & \equiv \sim(p \wedge q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee \sim q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee(p \vee q)) \vee(\... | mcq | jee-main-2023-online-1st-february-evening-shift | 6,885 |
ldo8e0p7 | maths | mathematical-reasoning | logical-connectives | The number of values of $\mathrm{r} \in\{\mathrm{p}, \mathrm{q}, \sim \mathrm{p}, \sim \mathrm{q}\}$ for which $((\mathrm{p} \wedge \mathrm{q}) \Rightarrow(\mathrm{r} \vee \mathrm{q})) \wedge((\mathrm{p} \wedge \mathrm{r}) \Rightarrow \mathrm{q})$ is a tautology, is : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "3"}] | ["A"] | null | $((p \wedge q) \Rightarrow(r \vee q)) \wedge((p \wedge r) \Rightarrow q) \equiv T$ (given)
<br/><br/>$\equiv((\sim p \vee \sim q) \vee(r \vee q)) \wedge(\sim p \vee \sim r \vee q)$
<br/><br/>$\equiv((\sim p \vee r) \vee(\sim q \vee q)) \wedge(\sim p \vee \sim r \vee q)$
<br/><br/>$\equiv \sim p \vee \sim r \vee q$
... | mcq | jee-main-2023-online-31st-january-evening-shift | 6,886 |
1ldpt87pz | maths | mathematical-reasoning | logical-connectives | <p>$$(\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))$$ is a tautology</p>
<p>$$(\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$$ is a contradiction.</p>
<p>Then</p> | [{"identifier": "A", "content": "only (S2) is correct"}, {"identifier": "B", "content": "both (S1) and (S2) are correct"}, {"identifier": "C", "content": "only (S1) is correct"}, {"identifier": "D", "content": "both (S1) and (S2) are wrong"}] | ["C"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lek3mdx9/896912c9-3981-4e34-b46a-5223559fea7c/7c31fad0-b51e-11ed-accc-792fddd82133/file-1lek3mdxa.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lek3mdx9/896912c9-3981-4e34-b46a-5223559fea7c/7c31fad0-b51e-11ed-accc-792fddd82133/fi... | mcq | jee-main-2023-online-31st-january-morning-shift | 6,888 |
1ldr5iqx7 | maths | mathematical-reasoning | logical-connectives | <p>Among the statements :</p>
<p>$$(\mathrm{S} 1)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow(\mathrm{p} \Rightarrow \mathrm{r})$$</p>
<p>$$(\mathrm{S} 2)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow((\mathrm{p} \Rightarrow \mathrm{r}) \vee(\mathrm{q} \Rightarrow \math... | [{"identifier": "A", "content": "only (S1) is a tautology"}, {"identifier": "B", "content": "neither (S1) nor (S2) is a tautology"}, {"identifier": "C", "content": "both (S1) and (S2) are tautologies"}, {"identifier": "D", "content": "only (S2) is a tautology"}] | ["B"] | null | <p>$${S1}:\left( {(p \vee q) \Rightarrow r} \right) \Leftrightarrow (p \Rightarrow r)$$</p>
<p>$${S2}:\left( {(p \vee q) \Rightarrow r} \right) \Leftrightarrow \left( {(p \Rightarrow r) \vee (q \Rightarrow r)} \right)$$</p>
<p>In $$S1:$$ If $$p=F, q=T, r=F$$ then $$S_1$$ is false</p>
<p>In $$S2:$$ If $$p=T, q=F, r=F$$ ... | mcq | jee-main-2023-online-30th-january-morning-shift | 6,889 |
1ldsegiln | maths | mathematical-reasoning | logical-connectives | <p>The statement $$B \Rightarrow \left( {\left( { \sim A} \right) \vee B} \right)$$ is equivalent to :</p> | [{"identifier": "A", "content": "$$B \\Rightarrow \\left( {\\left( { \\sim A} \\right) \\Rightarrow B} \\right)$$"}, {"identifier": "B", "content": "$$A \\Rightarrow \\left( {A \\Leftrightarrow B} \\right)$$"}, {"identifier": "C", "content": "$$A \\Rightarrow \\left( {\\left( { \\sim A} \\right) \\Rightarrow B} \\right... | null | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lfq8buii/842105db-2a88-4b01-bc04-d2093d4bdf40/d00809a0-cc49-11ed-b18d-8994bf82aa9d/file-1lfq8buij.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lfq8buii/842105db-2a88-4b01-bc04-d2093d4bdf40/d00809a0-cc49-11ed-b18d-8994bf82aa9d/fi... | mcqm | jee-main-2023-online-29th-january-evening-shift | 6,890 |
1ldwx7ql1 | maths | mathematical-reasoning | logical-connectives | <p>Let p and q be two statements. Then $$ \sim \left( {p \wedge (p \Rightarrow \, \sim q)} \right)$$ is equivalent to</p> | [{"identifier": "A", "content": "$$\\left( { \\sim p} \\right) \\vee q$$"}, {"identifier": "B", "content": "$$p \\vee \\left( {p \\wedge ( \\sim q)} \\right)$$"}, {"identifier": "C", "content": "$$p \\vee \\left( {p \\wedge q} \\right)$$"}, {"identifier": "D", "content": "$$p \\vee \\left( {\\left( { \\sim p} \\right) ... | ["A"] | null | <p>Making truth table $(E \equiv \sim(p \wedge(p \Rightarrow \sim q))$</p>
<p><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;width:100%}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:norm... | mcq | jee-main-2023-online-24th-january-evening-shift | 6,894 |
lgnwpeal | maths | mathematical-reasoning | logical-connectives | Negation of $p \wedge(q \wedge \sim(p \wedge q))$ is : | [{"identifier": "A", "content": "$(\\sim(p \\wedge q)) \\wedge q$"}, {"identifier": "B", "content": "$(\\sim(p \\wedge q)) \\vee p$"}, {"identifier": "C", "content": "$p \\vee q$"}, {"identifier": "D", "content": "$\\sim(p \\vee q)$"}] | ["B"] | null | <p>The given statement is:</p>
<p>$p \wedge (q \wedge \sim(p \wedge q))$</p>
<p>We want to find the negation of this statement. To do that, we negate the whole expression:</p>
<p>$\sim (p \wedge (q \wedge \sim(p \wedge q)))$</p>
<p>Now, we will use De Morgan's laws to simplify the expression. De Morgan's laws s... | mcq | jee-main-2023-online-15th-april-morning-shift | 6,896 |
1lgoxllvb | maths | mathematical-reasoning | logical-connectives | <p>The statement $$(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))$$ is equivalent to _________.</p> | [{"identifier": "A", "content": "$$(\\sim p) \\vee(\\sim q)$$"}, {"identifier": "B", "content": "$$p \\vee(\\sim q)$$"}, {"identifier": "C", "content": "$$\\mathrm{p} \\vee \\mathrm{q}$$"}, {"identifier": "D", "content": "$$(\\sim p) \\vee q$$"}] | ["A"] | null | $$
\begin{array}{|c|c|c|c|c|c|c|}
\hline \mathbf{p} & \mathbf{q} & \sim \mathbf{q} & \sim \mathbf{p} & \mathbf{p} \wedge \sim \mathbf{q} & \mathbf{\sim p} \wedge \mathbf{q} & \mathbf{\sim p} \wedge \mathbf{\sim q} \\
\hline T & T & F & F & F & F & F \\
T & F & T & F & T & F & F \\
F & T & F & T & F & T & F \\
F & F & T... | mcq | jee-main-2023-online-13th-april-evening-shift | 6,897 |
1lgpxp5mt | maths | mathematical-reasoning | logical-connectives | <p>The negation of the statement $$((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$$ is</p> | [{"identifier": "A", "content": "equivalent to $$B ~\\vee \\sim C$$"}, {"identifier": "B", "content": "equivalent to $$\\sim A$$"}, {"identifier": "C", "content": "equivalent to $$\\sim C$$"}, {"identifier": "D", "content": "a fallacy"}] | ["B"] | null | $$((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$$
<br/><br/>$$ \equiv $$ $$
[\sim(\mathrm{A} \wedge(\mathrm{B} \vee \mathrm{C})) \vee(\mathrm{A} \vee \mathrm{B})] \Rightarrow \mathrm{A}
$$
<br/><br/>$$ \equiv $$ $$
\sim(\sim(A \wedge(B \vee C)) \vee(A \vee B)) \vee A
$$
<br/><br/>$$ \equiv $$ $$
(A \wedge(... | mcq | jee-main-2023-online-13th-april-morning-shift | 6,898 |
1lgrejueh | maths | mathematical-reasoning | logical-connectives | <p>Among the two statements</p>
<p>$$(\mathrm{S} 1):(p \Rightarrow q) \wedge(p \wedge(\sim q))$$ is a contradiction and</p>
<p>$$(\mathrm{S} 2):(p \wedge q) \vee((\sim p) \wedge q) \vee(p \wedge(\sim q)) \vee((\sim p) \wedge(\sim q))$$ is a tautology</p> | [{"identifier": "A", "content": "both are false."}, {"identifier": "B", "content": "only (S1) is true."}, {"identifier": "C", "content": "both are true."}, {"identifier": "D", "content": "only (S2) is true."}] | ["C"] | null | $$
\begin{aligned}
S_1: & (p \Rightarrow q) \wedge(p \wedge \sim q) \\\\
& \equiv(\sim p \vee q) \wedge(p \wedge \sim q) \\\\
& \equiv(\sim p \wedge p \wedge \sim q) \vee(q \wedge p \wedge \sim q) \\\\
& \equiv(f \wedge \sim q) \vee(f \wedge p) \\\\
& \equiv f \vee f \equiv f \\\\
S_2: & (p \wedge q) \vee(\sim p \wedge... | mcq | jee-main-2023-online-12th-april-morning-shift | 6,899 |
1lgsuhzhe | maths | mathematical-reasoning | logical-connectives | <p>The converse of $$((\sim p) \wedge q) \Rightarrow r$$ is</p> | [{"identifier": "A", "content": "$$((\\sim p) \\vee q) \\Rightarrow r$$"}, {"identifier": "B", "content": "$$(\\sim \\mathrm{r}) \\Rightarrow \\mathrm{p} \\wedge \\mathrm{q}$$"}, {"identifier": "C", "content": "$$(\\mathrm{p} \\vee(\\sim \\mathrm{q})) \\Rightarrow(\\sim \\mathrm{r})$$"}, {"identifier": "D", "content": ... | ["C"] | null | Converse of $((\sim p) \wedge q) \Rightarrow r$
<br/><br/>$$
\begin{aligned}
& \equiv \mathrm{r} \Rightarrow(\sim \mathrm{p} \wedge \mathrm{q}) \\\\
& \equiv \sim \mathrm{r} \vee(\sim \mathrm{p} \wedge \mathrm{q}) \\\\
& \equiv \sim \mathrm{r} \vee(\mathrm{p} \vee \sim \mathrm{q}) \equiv(\mathrm{p} \vee \sim \mathrm{q}... | mcq | jee-main-2023-online-11th-april-evening-shift | 6,900 |
1lgvpc3a0 | maths | mathematical-reasoning | logical-connectives | <p>The statement $$\sim[p \vee(\sim(p \wedge q))]$$ is equivalent to :</p> | [{"identifier": "A", "content": "$$(\\sim(p \\wedge q)) \\wedge q$$"}, {"identifier": "B", "content": "$$\\sim(p \\vee q)$$"}, {"identifier": "C", "content": "$$(p \\wedge q) \\wedge(\\sim p)$$"}, {"identifier": "D", "content": "$$\\sim(p \\wedge q)$$"}] | ["C"] | null | $$
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline p & q & p \wedge q & \sim(p \wedge q) & \begin{array}{c}
p \vee \sim \\
(p \wedge q)
\end{array} & \begin{array}{l}
\sim[p \vee \sim \\
(p \wedge q)]
\end{array} & p \vee q & \begin{array}{l}
\sim(p \\
\vee q)
\end{array} & \begin{array}{l}
(p \wedge q) \wedge \\
\quad(\sim ... | mcq | jee-main-2023-online-10th-april-evening-shift | 6,901 |
1lgxhachl | maths | mathematical-reasoning | logical-connectives | <p>The negation of the statement $$(p \vee q) \wedge (q \vee ( \sim r))$$ is :</p> | [{"identifier": "A", "content": "$$(( \\sim p) \\vee r)) \\wedge ( \\sim q)$$"}, {"identifier": "B", "content": "$$(p \\vee r) \\wedge ( \\sim q)$$"}, {"identifier": "C", "content": "$$(( \\sim p) \\vee ( \\sim q)) \\vee ( \\sim r)$$"}, {"identifier": "D", "content": "$$(( \\sim p) \\vee ( \\sim q)) \\wedge ( \\sim r)$... | ["A"] | null | The negation of the statement $(p \vee q) \wedge(q \vee(\sim r))$ is
<br/><br/>$$
\begin{aligned}
& =\sim[(p \vee q) \wedge(q \vee(\sim r))] \\\\
& =\sim[(p \vee q) \vee \sim(q \vee(\sim r)] \\\\
& =((\sim p) \wedge \sim q)) \vee((\sim q) \wedge r))
\end{aligned}
$$
<br/><br/>Apply distribution law,
<br/><br/>$$
\begin... | mcq | jee-main-2023-online-10th-april-morning-shift | 6,902 |
1lgylheex | maths | mathematical-reasoning | logical-connectives | <p>The negation of $$(p \wedge(\sim q)) \vee(\sim p)$$ is equivalent to :</p> | [{"identifier": "A", "content": "$$p \\wedge q$$"}, {"identifier": "B", "content": "$$p \\wedge(\\sim q)$$"}, {"identifier": "C", "content": "$$p \\wedge(q \\wedge(\\sim p))$$"}, {"identifier": "D", "content": "$$p \\vee(q \\vee(\\sim p))$$"}] | ["A"] | null | $$
\begin{aligned}
& (p \wedge(\sim q)) \vee(\sim p) \\\\
& \equiv(p \vee \sim p) \wedge(\sim q \vee \sim p) \\\\
& \equiv \mathrm{T} \wedge(\sim q \vee \sim p) \\\\
& \equiv \sim q \vee \sim p \text { negation } p \wedge q
\end{aligned}
$$ | mcq | jee-main-2023-online-8th-april-evening-shift | 6,903 |
1lgzzynuh | maths | mathematical-reasoning | logical-connectives | <p>Negation of $$(p \Rightarrow q) \Rightarrow(q \Rightarrow p)$$ is :</p> | [{"identifier": "A", "content": "$$(\\sim q) \\wedge p$$"}, {"identifier": "B", "content": "$$q \\wedge(\\sim p)$$"}, {"identifier": "C", "content": "$$p \\vee(\\sim q)$$"}, {"identifier": "D", "content": "$$(\\sim p) \\vee q$$"}] | ["B"] | null | Given: $(p \rightarrow q) \rightarrow(q \rightarrow p)$
<br/><br/>Negation of above statement is :
<br/><br/>$$
\begin{aligned}
& \sim[(p \rightarrow q) \rightarrow(q \rightarrow p)] \\\\
& \equiv \sim[\sim p \rightarrow q \wedge q \rightarrow p] \\\\
& \equiv p \rightarrow q \wedge \sim q \rightarrow p \\\\
& \equiv ... | mcq | jee-main-2023-online-8th-april-morning-shift | 6,904 |
1lh2yajqy | maths | mathematical-reasoning | logical-connectives | <p>Among the statements</p>
<p>(S1) : $$(p \Rightarrow q) \vee((\sim p) \wedge q)$$ is a tautology</p>
<p>(S2) : $$(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$$ is a contradiction</p> | [{"identifier": "A", "content": "neither (S1) and (S2) is True"}, {"identifier": "B", "content": "only (S2) is True"}, {"identifier": "C", "content": "both $$(\\mathrm{S} 1)$$ and $$(\\mathrm{S} 2)$$ are True"}, {"identifier": "D", "content": "only (S1) is True"}] | ["A"] | null | (S1) : $$(p \Rightarrow q) \vee((\sim p) \wedge q)$$
<br/><br/>$$
\begin{array}{|c|c|c|c|c|c|}
\hline \mathrm{P} & \mathrm{Q} & \sim p & \sim p \wedge q & p \Rightarrow q & \begin{array}{c}
(p \Rightarrow q) \vee \\
(\sim p \wedge q)
\end{array} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} &... | mcq | jee-main-2023-online-6th-april-evening-shift | 6,906 |
IKF8zEfPChsloJ2J | maths | mathematical-reasoning | logical-statement | Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”,
and r be the statement “x is a rational number iff y is a transcendental number”.
<br/><br/> <b>Statement –1:</b> r is equivalent to either q or p.
<br/><br/><b>Statement –2:</b> r is equivalent to $$ \sim \left( {p ... | [{"identifier": "A", "content": "Statement \u2212 1 is false, Statement \u2212 2 is false"}, {"identifier": "B", "content": "Statement \u22121 is false, Statement \u22122 is true"}, {"identifier": "C", "content": "Statement \u22121 is true, Statement \u22122 is true, Statement \u22122 is a correct explanation for State... | ["A"] | null | <p>p : x is an irrational number</p>
<p>q : y is a transcendental number</p>
<p>r : x is a rational number, if y is a transcendental number</p>
<p>$$\Rightarrow$$ r : $$\sim$$ p $$\leftrightarrow$$ q</p>
<p>S<sub>1</sub> : r $$\equiv$$ q $$\vee$$ p</p>
<p>and S<sub>2</sub> : r $$\equiv$$ $$\sim$$ (p $$\leftrightarrow$$... | mcq | aieee-2008 | 6,907 |
O0QHhaazXNzgQQym | maths | mathematical-reasoning | logical-statement | Let S be a non-empty subset of R. Consider the following statement:
<br/>P : There is a rational number x ∈ S such that x > 0.
<br/>Which of the following statements is the negation of the statement P? | [{"identifier": "A", "content": "There is no rational number x \u2208 S such that x \u2264 0"}, {"identifier": "B", "content": "Every rational number x \u2208 S satisfies x \u2264 0"}, {"identifier": "C", "content": "x \u2208 S and x \u2264 0 $$ \\Rightarrow $$ x is not rational"}, {"identifier": "D", "content": "There... | ["B"] | null | <p>Given that S is a non-empty subset of R.</p>
<p>$$\bullet$$ P : There is a rational number x $$\in$$ S such that x > 0.</p>
<p>Now, we need to find the negation of P. Clearly, P is equivalent to saying that "There is a positive rational number in S.</p>
<p>So, its negation ($$\sim$$ P) is "There is no positive ratio... | mcq | aieee-2010 | 6,908 |
aZuPum2ZGIl6pDOy | maths | mathematical-reasoning | logical-statement | Consider the following statements
<br/>P : Suman is brilliant
<br/>Q : Suman is rich
<br/>R : Suman is honest
<br/>The negation of the statement,
<br/><br/>“Suman is brilliant and dishonest if and only if Suman is rich” can be
expressed as : | [{"identifier": "A", "content": "$$ \\sim \\left[ {Q \\leftrightarrow \\left( {P \\wedge \\sim R} \\right)} \\right]$$"}, {"identifier": "B", "content": "$$ \\sim Q \\leftrightarrow P \\wedge R$$"}, {"identifier": "C", "content": "$$ \\sim \\left( {P \\wedge \\sim R} \\right) \\leftrightarrow Q$$"}, {"identifier": "D... | ["A"] | null | "Suman is brilliant and dishonest" an be expressed as : $${P \wedge \sim R}$$
<br><br>So “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as :
<br><br>$$\left( {P \wedge \sim R} \right) \leftrightarrow Q$$
<br><br>Now negation of this = $$ \sim \left[ {\left( {P \wedge \sim R} \right)... | mcq | aieee-2011 | 6,909 |
LQt8RJq61sF54YAA | maths | mathematical-reasoning | logical-statement | The negation of the statement “If I become a teacher, then I will open a school” is : | [{"identifier": "A", "content": "I will become a teacher and I will not open a school"}, {"identifier": "B", "content": "Either I will not become a teacher or I will not open a school"}, {"identifier": "C", "content": "Neither I will become a teacher nor I will open a school"}, {"identifier": "D", "content": "I will no... | ["A"] | null | <p>Let, p : I become a teacher</p>
<p>q : I will open a school</p>
<p>$$\therefore$$ "If I become a teacher, then I will open a school", is p $$\to$$ q</p>
<p>So, negation of $$(p \to q) = \sim (p \to q) = p \wedge \sim q$$ = I will become a teacher and I will not open a school</p> | mcq | aieee-2012 | 6,910 |
pnAtpDdWdcKtXojQpGxye | maths | mathematical-reasoning | logical-statement | Consider the following two statements :
<br/><br/><b>P :</b> If 7 is an odd number, then 7 is
divisible by 2.
<br/><b>Q :</b> If 7 is a prime number, then 7 is an
odd number
<br/><br/>If V<sub>1</sub> is the truth value of the contrapositive of P and V<sub>2</sub> is the truth value of contrapositive of Q, the... | [{"identifier": "A", "content": "(T, T)"}, {"identifier": "B", "content": "(T, F)"}, {"identifier": "C", "content": "(F, T)"}, {"identifier": "D", "content": "(F, F)"}] | ["C"] | null | Contrapositive of P : If 7 is not divisible by 2,
then 7 is not an odd number.
<br><br>This statement is false.
<br><br>$$ \therefore $$ V<sub>1</sub> = False (F)
<br><br>Contrapositive of Q : If 7 is not an odd number
,then 7 is not a prime number.
<br><br>This statement is true.
<br><br>$$ \theref... | mcq | jee-main-2016-online-9th-april-morning-slot | 6,911 |
yhonGffUZ1g1RreXDYZq2 | maths | mathematical-reasoning | logical-statement | The contrapositive of the following statement,
<br/><br/>“If the side of a square doubles, then its area increases four times”, is : | [{"identifier": "A", "content": "If the side of a square is not doubled, then its area does not increase four times."}, {"identifier": "B", "content": "If the area of a square increases four times, then its side is doubled."}, {"identifier": "C", "content": "If the area of a square increases four times, then its side i... | ["D"] | null | Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>Here,
<br><br>Let
<br><br> p = Side of a square is doubles.
<br><br> q = Area of square increases four times.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim $$p = If the area o... | mcq | jee-main-2016-online-10th-april-morning-slot | 6,912 |
5pVUlnZkfL3JgxJQQJldg | maths | mathematical-reasoning | logical-statement | Contrapositive of the statement
<br/><br/>‘If two numbers are not equal, then their squares are not equal’, is : | [{"identifier": "A", "content": "If the squares of two numbers are equal, then the numbers are equal."}, {"identifier": "B", "content": "If the squares of two numbers are equal, then the numbers are not equal."}, {"identifier": "C", "content": "If the squares of two numbers are not equal, then the numbers are not equal... | ["A"] | null | Let,
<br><br>p : two numbers are not equal
<br><br>q : squares of two numbers are not equal
<br><br>Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim $$p means "If the squares of two numbers are equal, then the numbers are equal". | mcq | jee-main-2017-online-9th-april-morning-slot | 6,913 |
v9SVtI3nIde0ejoVgeEB0 | maths | mathematical-reasoning | logical-statement | Consider the following two statements :
<br/><br/><b>Statement p :</b>
<br/>The value of sin 120<sup>o</sup> can be derived by taking $$\theta = {240^o}$$ in the equation
<br/>2sin$${\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $$
<br/><br/><b>Statement q :</b>
<br/>The angles A, B, C and... | [{"identifier": "A", "content": "F, T"}, {"identifier": "B", "content": "T, F"}, {"identifier": "C", "content": "T, T"}, {"identifier": "D", "content": "F, F"}] | ["A"] | null | <b>Statement p :</b>
<br>sin 120<sup>o</sup> = cos 30<sup>o</sup> = $${{\sqrt 3 } \over 2}$$ $$ \Rightarrow $$ 2 sin 120<sup>o</sup> = $$\sqrt 3 $$
<br><br>So, $$\sqrt {1 + \sin {{240}^o}} - \sqrt {1 - \sin {{240}^o}} $$
<br><br> $$ = \sqrt {{{1 - \sqrt 3 } \over 2}} - \sqrt {{{1 + \sqrt 3 } \over 2}} \ne \sqrt 3 $$... | mcq | jee-main-2018-online-15th-april-evening-slot | 6,914 |
WGPsr0o7H2TKzUFjIMsK9 | maths | mathematical-reasoning | logical-statement | Consider the statement : "P(n) : n<sup>2</sup> – n + 41 is prime". Then which one of the following is true ? | [{"identifier": "A", "content": "P(5) is false but P(3) is true"}, {"identifier": "B", "content": "Both P(3) and P(5) are true"}, {"identifier": "C", "content": "P(3) is false but P(5) is true"}, {"identifier": "D", "content": "Both P(3) and P(5) are false"}] | ["B"] | null | P(n) : n<sup>2</sup> $$-$$ n + 41 is prime
<br><br>P(5) = 61 which is prime
<br><br>P(3) = 47 which is also prime | mcq | jee-main-2019-online-10th-january-morning-slot | 6,915 |
XGWuG9V4dktxwFEkSEcmf | maths | mathematical-reasoning | logical-statement | Consider the following three statements :
<br/><br/>P : 5 is a prime number
<br/><br/>Q : 7 is a factor of 192
<br/><br/>R : L.C.M. of 5 and 7 is 35
<br/><br/>Then the truth value of which one of the following statements is true ? | [{"identifier": "A", "content": "(P $$ \\wedge $$ Q) $$ \\vee $$ ($$ \\sim $$ R)"}, {"identifier": "B", "content": "P $$ \\vee $$ ($$ \\sim $$ Q $$ \\wedge $$ R)"}, {"identifier": "C", "content": "(~ P) $$ \\wedge $$ ($$ \\sim $$ Q $$ \\wedge $$ R)"}, {"identifier": "D", "content": "($$ \\sim $$ P) $$ \\vee $$ (Q $$ \\... | ["B"] | null | It is obvious | mcq | jee-main-2019-online-10th-january-evening-slot | 6,916 |
oBwMmAnQrawvLlbPhGr5h | maths | mathematical-reasoning | logical-statement | Contrapositive of the statement " If two numbers are not equal, then their squares are not equal." is : | [{"identifier": "A", "content": "If the squares of two numbers are equal, then the numbers are not equal"}, {"identifier": "B", "content": "If the squares of two numbers are equal, then the numbers are equal "}, {"identifier": "C", "content": "If the squares of two numbers are not equal, then the numbers are equal"}, {... | ["B"] | null | Let,
<br><br>p : two numbers are not equal
<br><br>q : squares of two numbers are not equal
<br><br>Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim $$p means "If the squares of two numbers are equal, then the numbers are equal". | mcq | jee-main-2019-online-11th-january-evening-slot | 6,917 |
lCnssXZcN2ubzefec6xFx | maths | mathematical-reasoning | logical-statement | The contrapositive of the statement "If you are
born in India, then you are a citizen of India", is :
| [{"identifier": "A", "content": "If you are not a citizen of India, then you are\nnot born in India."}, {"identifier": "B", "content": "If you are born in India, then you are not a\ncitizen of India."}, {"identifier": "C", "content": "If you are a citizen of India, then you are born\nin India."}, {"identifier": "D", "c... | ["A"] | null | Let p = you are born in India.
<br><br>q = you are a citizen of India.
<br><br>$$ \therefore $$ $$ \sim $$ p = you are not born in India.
<br><br>$$ \sim $$ q = you are not a citizen of India.
<br><br>We know Contrapositive of p $$ \to $$ q is ~q $$ \to $$ ~p
<br><br>So contrapositive of statement will be :
<br><br>“If... | mcq | jee-main-2019-online-8th-april-morning-slot | 6,918 |
dA7B1752ytqOXgSd827k9k2k5irgtwi | maths | mathematical-reasoning | logical-statement | Negation of the statement :
<br/><br/>$$\sqrt 5 $$ is an integer or 5 is an irrational is : | [{"identifier": "A", "content": "$$\\sqrt 5 $$ is not an integer and 5 is not irrational."}, {"identifier": "B", "content": "$$\\sqrt 5 $$ is irrational or 5 is an integer."}, {"identifier": "C", "content": "$$\\sqrt 5 $$ is an integer and 5 is irrational."}, {"identifier": "D", "content": "$$\\sqrt 5 $$ is not an inte... | ["A"] | null | p = $$\sqrt 5 $$ is an integer.
<br><br>q : 5 is irrational
<br><br>$$ \sim $$$$\left( {p \vee q} \right)$$ $$ \equiv $$ $$ \sim $$p $$ \wedge $$ $$ \sim $$q
<br><br>= $$\sqrt 5 $$ is not an integer and 5 is not irrational. | mcq | jee-main-2020-online-9th-january-morning-slot | 6,919 |
hLzGTd37OiUHlAyvkcjgy2xukg38l1kq | maths | mathematical-reasoning | logical-statement | Consider the statement : <br/>‘‘For an integer n, if n<sup>3</sup> – 1 is even, then n is odd.’’<br/> The contrapositive statement of this statement is :
| [{"identifier": "A", "content": "For an integer n, if n is even, then n<sup>3</sup> \u2013 1 is even."}, {"identifier": "B", "content": "For an integer n, if n<sup>3</sup> \u2013 1 is not even, then n is not odd."}, {"identifier": "C", "content": "For an integer n, if n is odd, then n<sup>3</sup> \u2013 1 is even."}, {... | ["D"] | null | Let,
p : n<sup>3</sup>–1 is even,
<br>q : n is odd
<br><br>Contrapositive of p $$ \to $$ q = $$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \Rightarrow $$ If n is not odd then n<sup>3</sup> – 1 is not even.
<br><br>$$ \Rightarrow $$ For an integer n, if n is even, then n<sup>3</sup> – 1 is odd. | mcq | jee-main-2020-online-6th-september-evening-slot | 6,920 |
mQhzgCsAFvfKM3rNxDjgy2xukfah5akq | maths | mathematical-reasoning | logical-statement | Contrapositive of the statement :<br/>
‘If a function f is differentiable at a, then it is also continuous at a’, is: | [{"identifier": "A", "content": "If a function f is continuous at a, then it is not differentiable at a."}, {"identifier": "B", "content": "If a function f is not continuous at a, then it is differentiable at a."}, {"identifier": "C", "content": "If a function f is not continuous at a, then it is not differentiable at ... | ["C"] | null | p = function is differentiable at a
<br><br>q = function is continuous at a
<br><br>Contrapositive of statements p $$ \to $$ q is
<br><br>$$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \therefore $$ Contrapositive statement is :
<br><br> If a function f is not continuous at a, then it is not differentiable at a.
| mcq | jee-main-2020-online-4th-september-evening-slot | 6,921 |
DnY3UxFkGprb04C8H1jgy2xukewqg1g8 | maths | mathematical-reasoning | logical-statement | The contrapositive of the statement <br/>"If I reach the
station in time, then I will catch the train" is : | [{"identifier": "A", "content": "If I will catch the train, then I reach the station\nin time."}, {"identifier": "B", "content": "If I do not reach the station in time, then I will\nnot catch the train."}, {"identifier": "C", "content": "If I will not catch the train, then I do not reach\nthe station in time."}, {"iden... | ["C"] | null | Let p denotes statement
<br><br>p : I reach the station in time.
<br><br>q : I will catch the train.
<br><br>Contrapositive of p $$ \to $$ q
is $$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \sim $$q $$ \to $$ $$ \sim $$p : If I will not catch the train, then I do not reach the station in time. | mcq | jee-main-2020-online-2nd-september-morning-slot | 6,922 |
LWR517jHskS2az6BJG1klt77hyt | maths | mathematical-reasoning | logical-statement | The contrapositive of the statement "If you will work, you will earn money" is : | [{"identifier": "A", "content": "If you will not earn money, you will not work"}, {"identifier": "B", "content": "If you will earn money, you will work"}, {"identifier": "C", "content": "You will earn money, if you will not work"}, {"identifier": "D", "content": "To earn money, you need to work"}] | ["A"] | null | Contrapositive of p $$ \to $$ q is ~q $$ \to $$ ~p
<br><br>p : you will work
<br><br>q : you will earn money
<br><br>~q : you will not earn money
<br><br>~p : you will not work
<br><br>$$ \therefore $$ ~q $$ \to $$ ~p : If you will not earn money, you will not work | mcq | jee-main-2021-online-25th-february-evening-slot | 6,923 |
1krrp2k1g | maths | mathematical-reasoning | logical-statement | Consider the following three statements :<br/><br/>(A) If 3 + 3 = 7 then 4 + 3 = 8<br/><br/>(B) If 5 + 3 = 8 then earth is flat.<br/><br/>(C) If both (A) and (B) are true then 5 + 6 = 17.<br/><br/>Then, which of the following statements is correct? | [{"identifier": "A", "content": "(A) is false, but (B) and (C) re true"}, {"identifier": "B", "content": "(A) and (C) are true while (B) is false"}, {"identifier": "C", "content": "(A) is true while (B) and (C) are false"}, {"identifier": "D", "content": "(A) and (B) are false while (C) is true"}] | ["B"] | null | Truth Table <br><br><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:solid;border-width:1px;f... | mcq | jee-main-2021-online-20th-july-evening-shift | 6,924 |
1krzlvswd | maths | mathematical-reasoning | logical-statement | Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following : | [{"identifier": "A", "content": "The match will not be played and weather is not good and ground is wet."}, {"identifier": "B", "content": "If the match will not be played, then either weather is not good or ground is wet."}, {"identifier": "C", "content": "The match will be played and weather is not good or ground is ... | ["C"] | null | p : weather is good<br><br>q : ground is not wet<br><br>$$\sim$$ (p $$ \wedge $$ q) $$ \equiv $$ $$\sim$$ p $$ \vee $$ $$\sim$$ q<br><br>$$\equiv$$ weather is not good or ground is wet | mcq | jee-main-2021-online-25th-july-evening-shift | 6,926 |
1l5baqdkw | maths | mathematical-reasoning | logical-statement | <p>Consider the following statements:</p>
<p>A : Rishi is a judge.</p>
<p>B : Rishi is honest.</p>
<p>C : Rishi is not arrogant.</p>
<p>The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is</p> | [{"identifier": "A", "content": "B $$\\to$$ (A $$\\vee$$ C)"}, {"identifier": "B", "content": "($$\\sim$$B) $$\\wedge$$ (A $$\\wedge$$ C)"}, {"identifier": "C", "content": "B $$\\to$$ (($$\\sim$$A) $$\\vee$$ ($$\\sim$$C))"}, {"identifier": "D", "content": "B $$\\to$$ (A $$\\wedge$$ C)"}] | ["B"] | null | <p>$$\because$$ Given statement is</p>
<p>(A $$\wedge$$ C) $$\to$$ B</p>
<p>Then its negation is</p>
<p>$$\sim$$ {(A $$\wedge$$ C) $$\to$$ B}</p>
<p>or $$\sim$$ {$$\sim$$ (A $$\wedge$$ C) $$\vee$$ B}</p>
<p>$$\therefore$$ (A $$\wedge$$ C) $$\wedge$$ ($$\sim$$ B)</p>
<p>or ($$\sim$$ B) $$\wedge$$ (A $$\wedge$$ C)</p> | mcq | jee-main-2022-online-24th-june-evening-shift | 6,927 |
1l6f32pyv | maths | mathematical-reasoning | logical-statement | <p>Consider the following statements:</p>
<p>P : Ramu is intelligent.</p>
<p>Q : Ramu is rich.</p>
<p>R : Ramu is not honest.</p>
<p>The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:</p> | [{"identifier": "A", "content": "$$((P \\wedge(\\sim R)) \\wedge Q) \\wedge((\\sim Q) \\wedge((\\sim P) \\vee R))$$"}, {"identifier": "B", "content": "$$((P \\wedge R) \\wedge Q) \\vee((\\sim Q) \\wedge((\\sim P) \\vee(\\sim R)))$$"}, {"identifier": "C", "content": "$$((P \\wedge R) \\wedge Q) \\wedge((\\sim Q) \\wedge... | ["D"] | null | <p>P : Ramu is intelligent</p>
<p>Q : Ramu is rich</p>
<p>R : Ramu is not honest</p>
<p>Given statement, "Ramu is intelligent and honest if and only if Ramu is not rich"</p>
<p>$$ = (P \wedge \sim R) \Leftrightarrow \, \sim Q$$</p>
<p>So, negation of the statement is</p>
<p>$$ \sim [(P \wedge \sim R) \Leftrightarrow ... | mcq | jee-main-2022-online-25th-july-evening-shift | 6,928 |
1l6notty1 | maths | mathematical-reasoning | logical-statement | <p>Let</p>
<p>$$\mathrm{p}$$ : Ramesh listens to music.</p>
<p>$$\mathrm{q}$$ : Ramesh is out of his village.</p>
<p>$$\mathrm{r}$$ : It is Sunday.</p>
<p>$$\mathrm{s}$$ : It is Saturday.</p>
<p>Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as</p... | [{"identifier": "A", "content": "$$((\\sim q) \\wedge(r \\vee s)) \\Rightarrow p$$"}, {"identifier": "B", "content": "$$(\\mathrm{q} \\wedge(\\mathrm{r} \\vee \\mathrm{s})) \\Rightarrow \\mathrm{p}$$"}, {"identifier": "C", "content": "$$p \\Rightarrow(q \\wedge(r \\vee s))$$"}, {"identifier": "D", "content": "$$\\mathr... | ["D"] | null | <p>p : Ramesh listens to music</p>
<p>q : Ramesh is out of his village</p>
<p>r : It is Sunday</p>
<p>s : It is Saturday</p>
<p>p $$\to$$ q conveys the same p only if q</p>
<p>Statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday"</p>
<p>$$p \Rightarrow (( \sim \,q)\, \wedge \,(r\... | mcq | jee-main-2022-online-28th-july-evening-shift | 6,929 |
ldqvxf4d | maths | mathematical-reasoning | logical-statement | <p>Consider the following statements:</p>
<p>P : I have fever</p>
<p>Q: I will not take medicine</p>
<p>$\mathrm{R}$ : I will take rest.</p>
<p>The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to :</p> | [{"identifier": "A", "content": "$((\\sim P) \\vee \\sim Q) \\wedge((\\sim P) \\vee \\sim R)$"}, {"identifier": "B", "content": "$(P \\vee \\sim Q) \\wedge(P \\vee \\sim R)$"}, {"identifier": "C", "content": "$((\\sim P) \\vee \\sim Q) \\wedge((\\sim P) \\vee R)$"}, {"identifier": "D", "content": "$(P \\vee Q) \\wedge(... | ["C"] | null | <p>The given expression is</p>
<p>$$P \to \sim Q \wedge R$$</p>
<p>$$ \equiv ( \sim P) \vee ( \sim Q \wedge R)$$</p>
<p>$$ \equiv ( \sim P \vee \sim Q) \wedge ( \sim P \vee R)$$</p> | mcq | jee-main-2023-online-30th-january-evening-shift | 6,930 |
1lguwsge5 | maths | mathematical-reasoning | logical-statement | <p>The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$$ is True, is equal to ___________.</p> | [] | null | 7 | $$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{r} & \boldsymbol{p} \vee \boldsymbol{q} & \boldsymbol{p} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge \\
(\boldsymbol{p} \vee \boldsymbol{r})
\end{array} & \boldsymbol{q} \vee \boldsymbol{r} & ... | integer | jee-main-2023-online-11th-april-morning-shift | 6,931 |
QFyqBI1sBBGMrWsu | maths | matrices-and-determinants | adjoint-of-a-matrix | If $$A = \left[ {\matrix{
{5a} & { - b} \cr
3 & 2 \cr
} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to : | [{"identifier": "A", "content": "$$4$$ "}, {"identifier": "B", "content": "$$13$$"}, {"identifier": "C", "content": "$$-1$$ "}, {"identifier": "D", "content": "$$5$$"}] | ["D"] | null | $$A\left( {Adj\,\,A} \right) = A\,{A^T}$$
<br><br>$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$
<br><br>$$Adj\,\,A = {A^T}$$
<br><br>$$ \Rightarrow \left[ {\matrix{
2 & b \cr
{ - 3} & {5a} \cr
} } \right] = \left[ {\matrix{
{5a} & 3 \cr
{ - b} & 2 \cr
... | mcq | jee-main-2016-offline | 6,932 |
2bwNHWvvDdm8rsiP | maths | matrices-and-determinants | adjoint-of-a-matrix | If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,
<br/><br/>then adj(3A<sup>2</sup> + 12A) is equal to | [{"identifier": "A", "content": "$$\\left[ {\\matrix{\n {51} & {63} \\cr \n {84} & {72} \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\left[ {\\matrix{\n {51} & {84} \\cr \n {63} & {72} \\cr \n\n } } \\right]$$"}, {"identifier": "C", "content": "$$\\left[ {\\matrix{\n {72}... | ["A"] | null | We have, $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$
<br><br>$$ \therefore $$ A<sup>2</sup> = A.A = $$\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$
<br><br>= $$\lef... | mcq | jee-main-2017-offline | 6,933 |
8yFgTFWWqZrfjWfpecKpP | maths | matrices-and-determinants | adjoint-of-a-matrix | Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is <b>not</b> always true ? | [{"identifier": "A", "content": "adj (A) = $$\\left| \\right.$$A$$\\left| \\right.$$.A<sup>$$-$$1</sup>"}, {"identifier": "B", "content": "adj (adj(A)) = $$\\left| \\right.$$A$$\\left| \\right.$$.A"}, {"identifier": "C", "content": "adj (adj(A)) = $$\\left| \\right.$$A$$\\left| \\right.$$<sup>2</sup>.(adj(A))<sup>$$... | ["D"] | null | We know, the formula
<br><br>A<sup>-1</sup> = $${{adj\left( A \right)} \over {\left| A \right|}}$$
<br><br>$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A<sup>$$-$$1</sup>
<br><br><b>So, Option (A) is true.</b>
<br><br>We know, the formula
<br><br>adj (adj (A)) = $${\left| A \right|^{n - 2}}.A$$
<br... | mcq | jee-main-2017-online-8th-april-morning-slot | 6,934 |
1lgpy4kyx | maths | matrices-and-determinants | adjoint-of-a-matrix | <p>Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A|=2$$. Then
$$\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alph... | [{"identifier": "A", "content": "32"}, {"identifier": "B", "content": "$$-$$16"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "16"}] | ["B"] | null | $$
B=\left[\begin{array}{lll}
1 & 3 & \alpha \\
1 & 2 & 3 \\
\alpha & \alpha & 4
\end{array}\right], \alpha>2
$$
<br/><br/>And $\operatorname{adj}(A)=B,|A|=2$
<br/><br/>$$
\begin{aligned}
& \Rightarrow|\operatorname{adj}(A)|=|B| \\\\
& \Rightarrow 2^2=(8-3 \alpha)-3(4-3 \alpha)+\alpha(-\alpha) \\\\
& \Rightarrow \alpha... | mcq | jee-main-2023-online-13th-april-morning-shift | 6,935 |
1lgzxiiqh | maths | matrices-and-determinants | adjoint-of-a-matrix | <p>Let $$A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$$. If $$|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}$$, then $$n$$ is equal to :</p> | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "12"}] | ["C"] | null | We have,
<br/><br/>$$
\begin{aligned}
& |\mathrm{A}|=\left|\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{array}\right|=2(4-1)-1(2-0)+0 \\\\
& =6-2=4 \\\\
& \text { So, }|2 \mathrm{~A}|=2^3|\mathrm{~A}|=8 \times 4=32 \\\\
& \text { Now, }|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 \mathrm... | mcq | jee-main-2023-online-8th-april-morning-shift | 6,936 |
lv0vxbzn | maths | matrices-and-determinants | adjoint-of-a-matrix | <p>Let $$\alpha \in(0, \infty)$$ and $$A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$$. If $$\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$$, then $$(\operatorname{det}(A))^2$$ is equa... | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "36"}, {"identifier": "C", "content": "49"}, {"identifier": "D", "content": "1"}] | ["A"] | null | <p>$$\begin{aligned}
& \left|\operatorname{adj}\left(A-2 A^T\right) \cdot \operatorname{adj}\left(2 A-A^T\right)\right|=2^8 \\
& P=A-2 A^{\top} \\
& Q=2 A^T-A \Rightarrow Q^T=2 A^T-A=-P \\
& |\operatorname{adj}(P) \operatorname{adj}(Q)| \Rightarrow|P Q|=-2^4 \\
& \Rightarrow|P|(-|P|)=-2^4 \Rightarrow|P|=4 \text { and }... | mcq | jee-main-2024-online-4th-april-morning-shift | 6,939 |
lv2eryn4 | maths | matrices-and-determinants | adjoint-of-a-matrix | <p>Let $$A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$$ and $$B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$$.
Then, the sum of all the elements of the matrix $$B$$ is:</p> | [{"identifier": "A", "content": "$$-$$110"}, {"identifier": "B", "content": "22"}, {"identifier": "C", "content": "$$-$$124"}, {"identifier": "D", "content": "$$-$$88"}] | ["D"] | null | <p>$$\begin{aligned}
& \operatorname{adj}(A)=\left[\begin{array}{ll}
1 & -2 \\
0 & 1
\end{array}\right] \\
& (\operatorname{adj} A)^2=\left[\begin{array}{ll}
1 & -4 \\
0 & 1
\end{array}\right] \\
& (\operatorname{adj} A)^3=\left[\begin{array}{cc}
1 & -6 \\
0 & 1
\end{array}\right] \\
& (\operatorname{adj} A)^4=\left[\b... | mcq | jee-main-2024-online-4th-april-evening-shift | 6,940 |
lvb294f8 | maths | matrices-and-determinants | adjoint-of-a-matrix | <p>If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}... | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "6"}] | ["B"] | null | <p>$$\begin{aligned}
& |A|=3 \\
& \left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}(2 A)^{-1}\right)\right)\right)\right| \\
& =\left|-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A)^{-1}\right)\right)\right)\right|^2
\end{align... | mcq | jee-main-2024-online-6th-april-evening-shift | 6,942 |
vLYN6IboT26J7IDy | maths | matrices-and-determinants | basic-of-matrix | The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is : | [{"identifier": "A", "content": "$$5$$ "}, {"identifier": "B", "content": "$$6$$ "}, {"identifier": "C", "content": "at least $$7$$ "}, {"identifier": "D", "content": "less than $$4$$ "}] | ["C"] | null | $$\left[ {\matrix{
1 & {...} & {...} \cr
{...} & 1 & {...} \cr
{...} & {...} & 1 \cr
} } \right]\,\,$$ are $$6$$ non-singular matrices because $$6$$
<br><br>blanks will be filled by $$5$$ zeros and $$1$$ one.
<br><br>Similarly, $$\left[ {\matrix{
{...} & {...} & 1 \c... | mcq | aieee-2010 | 6,943 |
1l6f3ahmk | maths | matrices-and-determinants | basic-of-matrix | <p>Let $$A=\left[\begin{array}{lll}
1 & a & a \\
0 & 1 & b \\
0 & 0 & 1
\end{array}\right], a, b \in \mathbb{R}$$. If for some <br/><br/>$$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc}
1 & 48 & 2160 \\
0 & 1 & 96 \\
0 & 0 & 1
\end{array}\right]
$$ then $$n+a+b$$ is equ... | [] | null | 24 | <p>$$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr
} } \right] + \left[ {\matrix{
0 & a & a \cr
0 & 0 & b \cr
0 & 0 & 0 \cr
} } \right] = I + B$$</p>
<p>$${B^2} = \left[ {\matrix{
0 & a & a \cr
0 & 0 & b \cr
0 & 0 & 0 \cr
} } \right] + \left[ {\matrix{... | integer | jee-main-2022-online-25th-july-evening-shift | 6,944 |
tULOO1HXFB32hWVY | maths | matrices-and-determinants | expansion-of-determinant | If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then
<p>$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr
} } \right|$$ is equal to</p> | [{"identifier": "A", "content": "$${\\omega ^2}$$ "}, {"identifier": "B", "content": "$$0$$"}, {"identifier": "C", "content": "$$1$$ "}, {"identifier": "D", "content": "$$\\omega $$ "}] | ["B"] | null | $$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr
} } \right|$$
<br><br>$$ = 1\left( {{\omega ^{3n}} - 1} \right) - {\omega ^n}\left( {{\omega ^{2n}} - {\omega ^{2n}}} \right) +... | mcq | aieee-2003 | 6,946 |
cR2lVel1EGtJOygR | maths | matrices-and-determinants | expansion-of-determinant | If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant
<p>$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} &... | [{"identifier": "A", "content": "$$-2$$ "}, {"identifier": "B", "content": "$$1$$"}, {"identifier": "C", "content": "$$2$$ "}, {"identifier": "D", "content": "$$0$$"}] | ["D"] | null | $$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr
} } \right|$$
<br><br>$$ = \left| {\matrix{
{\log {a_1}r{}^{n - 1}} ... | mcq | aieee-2004 | 6,947 |
GpuKipzHONqFXg6t | maths | matrices-and-determinants | expansion-of-determinant | If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant
$$$\Delta = \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {... | [{"identifier": "A", "content": "$$1$$ "}, {"identifier": "B", "content": "$$0$$"}, {"identifier": "C", "content": "$$4$$ "}, {"identifier": "D", "content": "$$2$$ "}] | ["B"] | null | As $$\,\,\,\,{a_1},{a_2},{a_3},.........$$ are in $$G.P.$$
<br><br>$$\therefore$$ Using $${a_n} = a{r^{n - 1}},\,\,\,$$ we get the given determinant,
<br><br>as $$\,\,\,\,\,\,\,\left| {\matrix{
{\log a{r^{n - 1}}} & {\log a{r^n}} & {\log a{r^{n + 1}}} \cr
{\log a{r^{n + 2}}} & {\log a{r^{n + 3}}} ... | mcq | aieee-2005 | 6,948 |
xFgehdfYl6MahZIw | maths | matrices-and-determinants | expansion-of-determinant | If $${a^2} + {b^2} + {c^2} = - 2$$ and
<p>f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {\left( {1... | [{"identifier": "A", "content": "$$1$$ "}, {"identifier": "B", "content": "$$0$$ "}, {"identifier": "C", "content": "$$3$$ "}, {"identifier": "D", "content": "$$2$$"}] | ["D"] | null | Applying, $${C_1} \to {C_1} + {C_2} + {C_3}\,\,\,$$ we get
<br><br>$$f\left( x \right) = \left| {\matrix{
{1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {1 + {b^2}x} & {\le... | mcq | aieee-2005 | 6,949 |
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