MCQOPTIONS
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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 3351. |
The logically equivalent proposition of \[p\Leftrightarrow q\] is [Karnataka CET 2000] |
| A. | \[(p\wedge q)\vee (p\wedge q)\] |
| B. | \[(p\Rightarrow q)\wedge (q\Rightarrow p)\] |
| C. | \[(p\wedge q)\vee (q\Rightarrow p)\] |
| D. | \[(p\wedge q)\Rightarrow (q\vee p)\] |
| Answer» C. \[(p\wedge q)\vee (q\Rightarrow p)\] | |
| 3352. |
If \[p\Rightarrow (q\vee r)\] is false, then the truth values of p, q, r are respectively [Karnataka CET 2000] |
| A. | T, F, F |
| B. | F, F, F |
| C. | F, T, T |
| D. | T, T, F |
| Answer» B. F, F, F | |
| 3353. |
The contrapositive of \[(p\vee q)\Rightarrow r\] is [Karnataka CET 1999] |
| A. | \[r\Rightarrow (p\vee q)\] |
| B. | \[\tilde{\ }r\Rightarrow (p\vee q)\] |
| C. | \[\tilde{\ }r\Rightarrow \ \tilde{\ }p\ \wedge \tilde{\ }q\] |
| D. | \[p\Rightarrow (q\vee r)\] |
| Answer» D. \[p\Rightarrow (q\vee r)\] | |
| 3354. |
Which of the following is always true [Karnataka CET 1998] |
| A. | \[(p\Rightarrow q)\equiv \ \tilde{\ }q\Rightarrow \ \tilde{\ }p\] |
| B. | \[\tilde{\ }(p\vee q)\equiv \vee \ p\ \vee \tilde{\ }q\] |
| C. | \[\tilde{\ }(p\Rightarrow q)\equiv p\ \wedge \tilde{\ }q\] |
| D. | \[\tilde{\ }(p\vee q)\equiv \ \tilde{\ }p\ \ \wedge \tilde{\ }q\] |
| Answer» D. \[\tilde{\ }(p\vee q)\equiv \ \tilde{\ }p\ \ \wedge \tilde{\ }q\] | |
| 3355. |
The propositions \[(p\Rightarrow \ \tilde{\ }p)\wedge (\tilde{\ }p\Rightarrow p)\] is a [Karnataka CET 1997] |
| A. | Tautology and contradiction |
| B. | Neither tautology nor contradiction |
| C. | Contradiction |
| D. | Tautology |
| Answer» D. Tautology | |
| 3356. |
The negative of \[q\ \vee \tilde{\ }(p\wedge r)\] is [Karnataka CET 1997] |
| A. | \[\tilde{\ }q\ \wedge \tilde{\ }(p\wedge r)\] |
| B. | \[\tilde{\ }q\wedge (p\wedge r)\] |
| C. | \[\tilde{\ }q\vee (p\wedge r)\] |
| D. | None of these |
| Answer» C. \[\tilde{\ }q\vee (p\wedge r)\] | |
| 3357. |
When does the current flow through the following circuit |
| A. | p, q, r should be closed |
| B. | p, q, r should be open |
| C. | Always |
| D. | None of these |
| Answer» B. p, q, r should be open | |
| 3358. |
\[\tilde{\ }(p\vee q)\vee (\tilde{\ }p\wedge q)\] is logically equivalent to |
| A. | ~p |
| B. | p |
| C. | q |
| D. | ~q |
| Answer» B. p | |
| 3359. |
The inverse of the proposition \[(p\ \wedge \tilde{\ }q)\Rightarrow r\] is |
| A. | \[\tilde{\ }r\Rightarrow \ \tilde{\ }p\vee q\] |
| B. | \[\tilde{\ }p\vee q\Rightarrow \ \tilde{\ }r\] |
| C. | \[r\Rightarrow p\ \wedge \tilde{\ }q\] |
| D. | None of these |
| Answer» C. \[r\Rightarrow p\ \wedge \tilde{\ }q\] | |
| 3360. |
Which of the following is true |
| A. | \[p\Rightarrow q\equiv \ \tilde{\ }p\Rightarrow \ \tilde{\ }q\] |
| B. | \[\tilde{\ }(p\Rightarrow \ \tilde{\ }q)\equiv \ \tilde{\ }p\wedge q\] |
| C. | \[\tilde{\ }(\tilde{\ }p\Rightarrow \,\tilde{\ }q)\equiv \tilde{\ }p\wedge q\] |
| D. | \[\tilde{\ }(p\Leftrightarrow q)\equiv [\tilde{\ }(p\Rightarrow q)\wedge \tilde{\ }(q\Rightarrow p)]\] |
| Answer» D. \[\tilde{\ }(p\Leftrightarrow q)\equiv [\tilde{\ }(p\Rightarrow q)\wedge \tilde{\ }(q\Rightarrow p)]\] | |
| 3361. |
Which of the following is not a statement |
| A. | Roses are red |
| B. | New Delhi is in India |
| C. | Every square is a rectangle |
| D. | Alas ! I have failed |
| Answer» E. | |
| 3362. |
The negation of the compound proposition \[p\vee (\tilde{\ }p\vee q)\] is |
| A. | \[(p\ \wedge \tilde{\ }q)\ \wedge \tilde{\ }p\] |
| B. | \[(p\ \wedge \tilde{\ }q)\ \vee \tilde{\ }p\] |
| C. | \[(p\ \vee \tilde{\ }q)\ \vee \tilde{\ }p\] |
| D. | None of these |
| Answer» B. \[(p\ \wedge \tilde{\ }q)\ \vee \tilde{\ }p\] | |
| 3363. |
Which of the following is not logically equivalent to the proposition : ?A real number is either rational or irrational?. |
| A. | If a number is neither rational nor irrational then it is not real |
| B. | If a number is not a rational or not an irrational, then it is not real |
| C. | If a number is not real, then it is neither rational nor irrational |
| D. | If a number is real, then it is rational or irrational |
| Answer» C. If a number is not real, then it is neither rational nor irrational | |
| 3364. |
If p : It rains today, q : I go to school, r : I shall meet any friends and s : I shall go for a movie, then which of the following is the proposition : If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie. |
| A. | \[\tilde{\ }(p\wedge q)\Rightarrow (r\wedge s)\] |
| B. | \[\tilde{\ }(p\ \wedge \tilde{\ }q)\Rightarrow (r\wedge s)\] |
| C. | \[\tilde{\ }(p\ \wedge q)\ \Rightarrow (r\vee s)\] |
| D. | None of these |
| Answer» B. \[\tilde{\ }(p\ \wedge \tilde{\ }q)\Rightarrow (r\wedge s)\] | |
| 3365. |
\[\tilde{\ }(p\Rightarrow q)\Leftrightarrow \tilde{\ }p\ \vee \tilde{\ }q\] is |
| A. | A tautology |
| B. | A contradiction |
| C. | Neither a tautology nor a contradiction |
| D. | Cannot come to any conclusion |
| Answer» D. Cannot come to any conclusion | |
| 3366. |
\[(p\ \wedge \tilde{\ }q)\wedge (\tilde{\ }p\vee q)\] is |
| A. | A contradiction |
| B. | A tautology |
| C. | Either A or B |
| D. | Neither A nor B |
| Answer» B. A tautology | |
| 3367. |
If p, q, r are simple propositions, then \[(p\wedge q)\wedge (q\wedge r)\] is true then |
| A. | p, q, r are all false |
| B. | p, q, r are all true |
| C. | p, q are true and r is false |
| D. | p is true and q and r are false |
| Answer» C. p, q are true and r is false | |
| 3368. |
If p, q, r are simple propositions with truth values T, F, T, then the truth value of \[(\tilde{\ }p\vee q)\ \wedge \tilde{\ }r\Rightarrow p\] is |
| A. | True |
| B. | False |
| C. | True if r is false |
| D. | True if q is true |
| Answer» B. False | |
| 3369. |
If \[(p\ \wedge \tilde{\ }r)\Rightarrow (q\vee r)\] is false and q and r are both false, then p is |
| A. | True |
| B. | False |
| C. | May be true or false |
| D. | Data insufficient |
| Answer» B. False | |
| 3370. |
\[p\Rightarrow q\] can also be written as |
| A. | \[p\Rightarrow \ \tilde{\ }q\] |
| B. | \[\tilde{\ }p\vee q\] |
| C. | \[\tilde{\ }q\Rightarrow \tilde{\ }p\] |
| D. | None of these |
| Answer» C. \[\tilde{\ }q\Rightarrow \tilde{\ }p\] | |
| 3371. |
\[\tilde{\ }(p\Leftrightarrow q)\] is |
| A. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| B. | \[\tilde{\ }p\ \vee \tilde{\ }q\] |
| C. | \[(p\ \wedge \tilde{\ }q)\vee (\tilde{\ }p\ \wedge q)\] |
| D. | None of these |
| Answer» D. None of these | |
| 3372. |
\[\tilde{\ }((\tilde{\ }p)\ \wedge q)\] is equal to |
| A. | \[p\vee (\tilde{\ }q)\] |
| B. | \[p\vee q\] |
| C. | \[p\wedge (\tilde{\ }q)\] |
| D. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| Answer» B. \[p\vee q\] | |
| 3373. |
\[\tilde{\ }(p\vee (\tilde{\ }q))\] is equal to |
| A. | \[\tilde{\ }p\vee q\] |
| B. | \[(\tilde{\ }p)\wedge q\] |
| C. | \[\tilde{\ }p\ \vee \tilde{\ }p\] |
| D. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| Answer» C. \[\tilde{\ }p\ \vee \tilde{\ }p\] | |
| 3374. |
\[(\tilde{\ }(\tilde{\ }p))\wedge q\] is equal to |
| A. | \[\tilde{\ }p\wedge q\] |
| B. | \[p\wedge q\] |
| C. | \[p\ \wedge \tilde{\ }q\] |
| D. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| Answer» C. \[p\ \wedge \tilde{\ }q\] | |
| 3375. |
\[\tilde{\ }(p\wedge q)\] is equal to |
| A. | \[\tilde{\ }p\ \vee \tilde{\ }q\] |
| B. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| C. | \[\tilde{\ }p\wedge q\] |
| D. | \[p\ \wedge \tilde{\ }q\] |
| Answer» B. \[\tilde{\ }p\ \wedge \tilde{\ }q\] | |
| 3376. |
\[\tilde{\ }(p\vee q)\] is equal to |
| A. | \[\tilde{\ }p\ \vee \tilde{\ }q\] |
| B. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| C. | \[\tilde{\ }p\vee q\] |
| D. | \[p\ \vee \tilde{\ }q\] |
| Answer» C. \[\tilde{\ }p\vee q\] | |
| 3377. |
Which of the following is a contradiction |
| A. | \[(p\wedge q)\wedge \tilde{\ }(p\vee q)\] |
| B. | \[p\vee (\tilde{\ }p\wedge q)\] |
| C. | \[(p\Rightarrow q)\Rightarrow p\] |
| D. | None of these |
| Answer» B. \[p\vee (\tilde{\ }p\wedge q)\] | |
| 3378. |
Which of the following is logically equivalent to \[\tilde{\ }(\tilde{\ }p\Rightarrow q)\] |
| A. | \[p\wedge q\] |
| B. | \[p\wedge \tilde{\ }q\] |
| C. | \[\tilde{\ }p\wedge q\] |
| D. | \[\tilde{\ }p\ \wedge \tilde{\ }q\] |
| Answer» E. | |
| 3379. |
The conditional \[(p\wedge q)\] Þ p is |
| A. | A tautology |
| B. | A fallacy i.e., contradiction |
| C. | Neither tautology nor fallacy |
| D. | None of these |
| Answer» B. A fallacy i.e., contradiction | |
| 3380. |
Negation of ?Ram is in Class X or Rashmi is in Class XII? is |
| A. | Ram is not in class X but Ram is in class XII |
| B. | Ram is not in class X but Rashmi is not in class XII |
| C. | Either Ram is not in class X or Ram is not in class XII |
| D. | None of these |
| Answer» E. | |
| 3381. |
Negation is ?2 + 3 = 5 and 8 < 10? is |
| A. | 2 + 3 ¹ 5 and < 10 |
| B. | 2 + 3 = 5 and 8 ≮ 10 |
| C. | 2 + 3 ¹ 5 or 8 ≮ 10 |
| D. | None of these |
| Answer» D. None of these | |
| 3382. |
Which of the following is a statement |
| A. | Open the door |
| B. | Do your homework |
| C. | Switch on the fan |
| D. | Two plus two is four |
| Answer» E. | |
| 3383. |
If n is a natural number then \[{{\left( \frac{n+1}{2} \right)}^{n}}\ge n\,!\] is true when |
| A. | n > 1 |
| B. | n ³ 1 |
| C. | n > 2 |
| D. | n ³ 2 |
| Answer» C. n > 2 | |
| 3384. |
For each \[n\in N\], the correct statement is |
| A. | \[{{2}^{n}}<n\] |
| B. | \[{{n}^{2}}>2n\] |
| C. | \[{{n}^{4}}<{{10}^{n}}\] |
| D. | \[{{2}^{3n}}>7n+1\] |
| Answer» D. \[{{2}^{3n}}>7n+1\] | |
| 3385. |
For natural number n, \[{{2}^{n}}\,(n-1)\,! |
| A. | n < 2 |
| B. | n > 2 |
| C. | n ³ 2 |
| D. | Never |
| Answer» C. n ³ 2 | |
| 3386. |
For every natural number n |
| A. | \[n>{{2}^{n}}\] |
| B. | \[n<{{2}^{n}}\] |
| C. | \[n\ge {{2}^{n}}\] |
| D. | \[n\le {{2}^{n}}\] |
| Answer» C. \[n\ge {{2}^{n}}\] | |
| 3387. |
If \[n\in N\], then \[{{11}^{n+2}}+{{12}^{2n+1}}\] is divisible by [Roorkee 1982] |
| A. | 113 |
| B. | 123 |
| C. | 133 |
| D. | None of these |
| Answer» D. None of these | |
| 3388. |
If\[n\in N\], then\[{{7}^{2n}}+{{2}^{3n-3}}\]. \[{{3}^{n-1}}\] is always divisible by [IIT 1982] |
| A. | 25 |
| B. | 35 |
| C. | 45 |
| D. | None of these |
| Answer» B. 35 | |
| 3389. |
\[{{10}^{n}}+3\,({{4}^{n+2}})+5\] is divisible by \[(n\in N)\][Kerala (Engg.) 2005] |
| A. | 7 |
| B. | 5 |
| C. | 9 |
| D. | 17 |
| E. | 13 |
| Answer» D. 17 | |
| 3390. |
For a positive integer n, Let\[a\,(n)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+....+\frac{1}{({{2}^{n}})-1}\]. Then [IIT 1999] |
| A. | \[a\,(100)\le 100\] |
| B. | \[a\,(100)>100\] |
| C. | \[a\,(200)\le 100\] |
| D. | \[a\,(200)>100\] |
| Answer» B. \[a\,(100)>100\] | |
| 3391. |
When \[{{2}^{301}}\] is divided by 5, the least positive remainder is [Karnataka CET 2005] |
| A. | 4 |
| B. | 8 |
| C. | 2 |
| D. | 6 |
| Answer» D. 6 | |
| 3392. |
The remainder when \[{{5}^{99}}\] is divided by 13 is |
| A. | 6 |
| B. | 8 |
| C. | 9 |
| D. | 10 |
| Answer» C. 9 | |
| 3393. |
If \[n\in N\], then \[{{x}^{2n-1}}+{{y}^{2n-1}}\] is divisible by |
| A. | \[x+y\] |
| B. | \[x-y\] |
| C. | \[{{x}^{2}}+{{y}^{2}}\] |
| D. | \[{{x}^{2}}+xy\] |
| Answer» B. \[x-y\] | |
| 3394. |
The statement P(n) ?\[1\times 1\,!\,+\,2\times 2\,!\,+\,3\times 3\,!\,+.....+n\times n\,!=(n+1)\,!\,-1\]? is |
| A. | True for all n > 1 |
| B. | Not true for any n |
| C. | True for all n Î N |
| D. | None of these |
| Answer» D. None of these | |
| 3395. |
For every natural number n, n(n + 1) is always |
| A. | Even |
| B. | Odd |
| C. | Multiple of 3 |
| D. | Multiple of 4 |
| Answer» B. Odd | |
| 3396. |
If P(n) = 2 + 4 + 6 +?.+ 2n, n Î N, then P(k) = k(k + 1) + 2 Þ P(k + 1) = (k + 1)(k + 2) + 2 for all k Î N. So we can conclude that P(n) = n(n + 1) + 2 for |
| A. | All n Î N |
| B. | n > 1 |
| C. | n > 2 |
| D. | Nothing can be said |
| Answer» E. | |
| 3397. |
\[x({{x}^{n-1}}-n{{a}^{n-1}})+{{a}^{n}}(n-1)\] is divisible by \[{{(x-a)}^{2}}\] for |
| A. | n > 1 |
| B. | n > 2 |
| C. | All n Î N |
| D. | None of these |
| Answer» D. None of these | |
| 3398. |
If p is a prime number, then \[{{n}^{p}}-n\] is divisible by p when n is a |
| A. | Natural number greater than 1 |
| B. | Irrational number |
| C. | Complex number |
| D. | Odd number |
| Answer» B. Irrational number | |
| 3399. |
Let P (n) denote the statement that \[{{n}^{2}}+n\] is odd. It is seen that \[P(n)\Rightarrow P(n+1)\], \[{{P}_{n}}\] is true for all [IIT JEE 1996] |
| A. | n > 1 |
| B. | n |
| C. | n > 2 |
| D. | None of these |
| Answer» E. | |
| 3400. |
For natural number n, \[{{(n\,!)}^{2}}>{{n}^{n}}\], if |
| A. | n > 3 |
| B. | n > 4 |
| C. | n ³ 4 |
| D. | n ³ 3 |
| Answer» E. | |