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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.
| 2551. |
The degree of the differential equation \[\left( \frac{2+\sin x}{1+y} \right)\frac{dy}{dx}=-\cos ,x\ y(0)=1,\] is [Pb. CET 2003] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 6 |
| Answer» C. 3 | |
| 2552. |
Degree of the given differential equation \[{{\left( \frac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}={{\left( 1+\frac{dy}{dx} \right)}^{1/2}}\], is [MP PET 1997] |
| A. | 2 |
| B. | 3 |
| C. | \[\frac{1}{2}\] |
| D. | 6 |
| Answer» E. | |
| 2553. |
The differential equation of all circles of radius a is of order |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | None of these |
| Answer» B. 3 | |
| 2554. |
The differential equation of all circles in the first quadrant which touch the coordinate axes is of order |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» B. 2 | |
| 2555. |
The order of the differential equation whose solution is \[y=a\cos x+b\sin x+c{{e}^{-x}}\]is |
| A. | 3 |
| B. | 2 |
| C. | 1 |
| D. | None of these |
| Answer» B. 2 | |
| 2556. |
The order of the differential equation whose solution is \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\], is [MP PET 1995] |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |
| 2557. |
Three ships A, B and C sail from England to India. If the ratio of their arriving safely are 2 : 5, 3 : 7 and 6 : 11 respectively then the probability of all the ships for arriving safely is [Pb. CET 2000] |
| A. | \[\frac{18}{595}\] |
| B. | \[\frac{6}{17}\] |
| C. | \[\frac{3}{10}\] |
| D. | \[\frac{2}{7}\] |
| Answer» B. \[\frac{6}{17}\] | |
| 2558. |
If odds against solving a question by three students are 2 : 1, \[5:2\] and \[5:3\] respectively, then probability that the question is solved only by one student is [RPET 1999] |
| A. | \[\frac{31}{56}\] |
| B. | \[\frac{24}{56}\] |
| C. | \[\frac{25}{56}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2559. |
The odds against a certain event is 5 : 2 and the odds in favour of another event is 6 : 5. If both the events are independent, then the probability that at least one of the events will happen is [RPET 1997] |
| A. | \[\frac{50}{77}\] |
| B. | \[\frac{52}{77}\] |
| C. | \[\frac{25}{88}\] |
| D. | \[\frac{63}{88}\] |
| Answer» C. \[\frac{25}{88}\] | |
| 2560. |
Let S be a set containing n elements and we select 2 subsets A and B of S at random then the probability that \[A\cup B=S\] and \[A\cap B=\varphi \] is [Orissa JEE 2005] |
| A. | \[{{2}^{n}}\] |
| B. | \[{{n}^{2}}\] |
| C. | 1/n |
| D. | \[1/{{2}^{n}}\] |
| Answer» E. | |
| 2561. |
Let A and B be two events such that \[P\overline{(A\cup B)}=\frac{1}{6},P(A\cap B)=\frac{1}{4}\] and \[P(\bar{A})=\frac{1}{4},\] where \[\bar{A}\] stands for complement of event A. Then events A and B are [AIEEE 2005] |
| A. | Independent but not equally likely |
| B. | Mutually exclusive and independent |
| C. | Equally likely and mutually exclusive |
| D. | Equally likely but not independent |
| Answer» B. Mutually exclusive and independent | |
| 2562. |
A card is drawn from a pack of cards. Find the probability that the card will be a queen or a heart [RPET 2003] |
| A. | \[\frac{4}{3}\] |
| B. | \[\frac{16}{3}\] |
| C. | \[\frac{4}{13}\] |
| D. | \[\frac{5}{3}\] |
| Answer» D. \[\frac{5}{3}\] | |
| 2563. |
In a certain population 10% of the people are rich, 5% are famous and 3% are rich and famous. The probability that a person picked at random from the population is either famous or rich but not both, is equal to [UPSEAT 2004] |
| A. | 0. 07 |
| B. | 0.08 |
| C. | 0. 09 |
| D. | 0. 12 |
| Answer» D. 0. 12 | |
| 2564. |
If \[P(A\cup B)=0.8\] and \[P(A\cap B)=0.3,\] then \[P(\bar{A})+P(\bar{B})=\] [EAMCET 2003] |
| A. | 0.3 |
| B. | 0.5 |
| C. | 0.7 |
| D. | 0.9 |
| Answer» E. | |
| 2565. |
A random variable X has the probability distribution X 1 2 3 4 5 6 7 8 P(X) 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05 For the events \[E=\{X\]is prime number} and \[F=\{X |
| A. | 0.50 |
| B. | 0.77 |
| C. | 0.35 |
| D. | 0.87 |
| Answer» C. 0.35 | |
| 2566. |
If \[P(A)=P(B)=x\] and \[P(A\cap B)=P({A}'\cap {B}')=\frac{1}{3}\], then \[x=\] [UPSEAT 2003] |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{4}\] |
| D. | \[\frac{1}{6}\] |
| Answer» B. \[\frac{1}{3}\] | |
| 2567. |
The probability that at least one of the events A and B occurs is 3/5. If A and B occur simultaneously with probability 1/5, then \[P({A}')+P({B}')\] is [DCE 2002] |
| A. | \[\frac{2}{5}\] |
| B. | \[\frac{4}{5}\] |
| C. | \[\frac{6}{5}\] |
| D. | \[\frac{7}{5}\] |
| Answer» D. \[\frac{7}{5}\] | |
| 2568. |
If A and B are arbitrary events, then [DCE 2002] |
| A. | \[P(A\cap B)\ge P(A)+P(B)\] |
| B. | \[P(A\cup B)\le P(A)+P(B)\] |
| C. | \[P(A\cap B)=P(A)+P(B)\] |
| D. | None of these |
| Answer» C. \[P(A\cap B)=P(A)+P(B)\] | |
| 2569. |
In two events \[P(A\cup B)=5/6\], \[P({{A}^{c}})=5/6\], \[P(B)=2/3,\] then A and B are [UPSEAT 2001] |
| A. | Independent |
| B. | Mutually exclusive |
| C. | Mutually exhaustive |
| D. | Dependent |
| Answer» C. Mutually exhaustive | |
| 2570. |
If \[P(A)=0.25,\,\,P(B)=0.50\] and \[P(A\cap B)=0.14,\] then \[P(A\cap \bar{B})\] is equal to [RPET 2001] |
| A. | 0.61 |
| B. | 0.39 |
| C. | 0.48 |
| D. | None of these |
| Answer» E. | |
| 2571. |
Let A and B are two independent events. The probability that both A and B occur together is 1/6 and the probability that neither of them occurs is 1/3. The probability of occurrence of A is [RPET 2000] |
| A. | 0 or 1 |
| B. | \[\frac{1}{2}\] or \[\frac{1}{3}\] |
| C. | \[\frac{1}{2}\] or \[\frac{1}{4}\] |
| D. | \[\frac{1}{3}\] or \[\frac{1}{4}\] |
| Answer» C. \[\frac{1}{2}\] or \[\frac{1}{4}\] | |
| 2572. |
One card is drawn randomly from a pack of 52 cards, then the probability that it is a king or spade is [RPET 2001] |
| A. | \[\frac{1}{26}\] |
| B. | \[\frac{3}{26}\] |
| C. | \[\frac{4}{13}\] |
| D. | \[\frac{3}{13}\] |
| Answer» D. \[\frac{3}{13}\] | |
| 2573. |
The probability of solving a question by three students are \[\frac{1}{2},\,\,\frac{1}{4},\,\,\frac{1}{6}\] respectively. Probability of question is being solved will be [UPSEAT 1999] |
| A. | \[\frac{33}{48}\] |
| B. | \[\frac{35}{48}\] |
| C. | \[\frac{31}{48}\] |
| D. | \[\frac{37}{48}\] |
| Answer» B. \[\frac{35}{48}\] | |
| 2574. |
Let A and B be events for which \[P(A)=x\], \[P(B)=y,\]\[P(A\cap B)=z,\] then \[P(\bar{A}\cap B)\] equals [AMU 1999] |
| A. | \[(1-x)\,y\] |
| B. | \[1-x+\,y\] |
| C. | y ? z |
| D. | \[1-x+y-z\] |
| Answer» D. \[1-x+y-z\] | |
| 2575. |
The probabilities of occurrence of two events are respectively 0.21 and 0.49. The probability that both occurs simultaneously is 0.16. Then the probability that none of the two occurs is [MP PET 1998] |
| A. | 0.30 |
| B. | 0.46 |
| C. | 0.14 |
| D. | None of these |
| Answer» C. 0.14 | |
| 2576. |
One card is drawn from a pack of 52 cards. The probability that it is a queen or heart is [RPET 1999] |
| A. | \[\frac{1}{26}\] |
| B. | \[\frac{3}{26}\] |
| C. | \[\frac{4}{13}\] |
| D. | \[\frac{3}{13}\] |
| Answer» D. \[\frac{3}{13}\] | |
| 2577. |
Let \[{{E}_{1}},{{E}_{2}},{{E}_{3}}\]be three arbitrary events of a sample space S. Consider the following statements which of the following statements are correct [Pb. CET 2004] |
| A. | P (only one of them occurs) \[=P({{\bar{E}}_{1}}{{E}_{2}}{{E}_{3}}+{{E}_{1}}{{\bar{E}}_{2}}{{E}_{3}}+{{E}_{1}}{{E}_{2}}{{\overline{E}}_{3}})\] |
| B. | P (none of them occurs) \[=P({{\overline{E}}_{1}}+{{\overline{E}}_{2}}+{{\overline{E}}_{3}})\] |
| C. | P (at least one of them occurs) \[=P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\] |
| D. | P (all the three occurs)\[=P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\] where \[P({{E}_{1}})\]denotes the probability of \[{{E}_{1}}\] and \[{{\bar{E}}_{1}}\] denotes complement of \[{{E}_{1}}\]. |
| Answer» D. P (all the three occurs)\[=P({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\] where \[P({{E}_{1}})\]denotes the probability of \[{{E}_{1}}\] and \[{{\bar{E}}_{1}}\] denotes complement of \[{{E}_{1}}\]. | |
| 2578. |
For an event, odds against is 6 : 5. The probability that event does not occur, is |
| A. | \[\frac{5}{6}\] |
| B. | \[\frac{6}{11}\] |
| C. | \[\frac{5}{11}\] |
| D. | \[\frac{1}{6}\] |
| Answer» C. \[\frac{5}{11}\] | |
| 2579. |
A, B, C are any three events. If P (S) denotes the probability of S happening then \[P\,(A\cap (B\cup C))=\] [EAMCET 1994] |
| A. | \[P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)\] |
| B. | \[P(A)+P(B)+P(C)-P(B)\,P(C)\] |
| C. | \[P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)\] |
| D. | None of these |
| Answer» D. None of these | |
| 2580. |
In a class of 125 students 70 passed in Mathematics, 55 in Statistics and 30 in both. The probability that a student selected at random from the class has passed in only one subject is [EAMCET 1993] |
| A. | \[\frac{13}{25}\] |
| B. | \[\frac{3}{25}\] |
| C. | \[\frac{17}{25}\] |
| D. | \[\frac{8}{25}\] |
| Answer» B. \[\frac{3}{25}\] | |
| 2581. |
If \[{{A}_{1}},\,{{A}_{2}},...{{A}_{n}}\] are any n events, then |
| A. | \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})=P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\] |
| B. | \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})>P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\] |
| C. | \[P\,({{A}_{1}}\cup {{A}_{2}}\cup ...\cup {{A}_{n}})\le P\,({{A}_{1}})+P({{A}_{2}})+...+P\,({{A}_{n}})\] |
| D. | None of these |
| Answer» D. None of these | |
| 2582. |
If A and B are any two events, then \[P(A\cup B)=\] [MP PET 1995] |
| A. | \[P(A)+P(B)\] |
| B. | \[P(A)+P(B)+P(A\cap B)\] |
| C. | \[P(A)+P(B)-P(A\cap B)\] |
| D. | \[P(A)\,\,.\,\,P(B)\] |
| Answer» D. \[P(A)\,\,.\,\,P(B)\] | |
| 2583. |
Given two mutually exclusive events A and B such that \[P(A)=0.45\] and \[P(B)=0.35,\] then P (A or B) = [AI CBSE 1979] |
| A. | 0.1 |
| B. | 0.25 |
| C. | 0.15 |
| D. | 0.8 |
| Answer» E. | |
| 2584. |
The probability that a man will be alive in 20 years is \[\frac{3}{5}\] and the probability that his wife will be alive in 20 years is \[\frac{2}{3}\]. Then the probability that at least one will be alive in 20 years, is [Bihar CEE 1994] |
| A. | \[\frac{13}{15}\] |
| B. | \[\frac{7}{15}\] |
| C. | \[\frac{4}{15}\] |
| D. | None of these |
| Answer» B. \[\frac{7}{15}\] | |
| 2585. |
The probability that at least one of A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, then \[P({A}')+P({B}')=\] |
| A. | 0.9 |
| B. | 1.15 |
| C. | 1.1 |
| D. | 1.2 |
| Answer» D. 1.2 | |
| 2586. |
In a city 20% persons read English newspaper, 40% read Hindi newspaper and 5% read both newspapers. The percentage of non-reader either paper is |
| A. | 60% |
| B. | 35% |
| C. | 25% |
| D. | 45% |
| Answer» E. | |
| 2587. |
An event has odds in favour 4 : 5, then the probability that event occurs, is |
| A. | \[\frac{1}{5}\] |
| B. | \[\frac{4}{5}\] |
| C. | \[\frac{4}{9}\] |
| D. | \[\frac{5}{9}\] |
| Answer» D. \[\frac{5}{9}\] | |
| 2588. |
Twelve tickets are numbered 1 to 12. One ticket is drawn at random, then the probability of the number to be divisible by 2 or 3, is |
| A. | \[\frac{2}{3}\] |
| B. | \[\frac{7}{12}\] |
| C. | \[\frac{5}{6}\] |
| D. | \[\frac{3}{4}\] |
| Answer» B. \[\frac{7}{12}\] | |
| 2589. |
The two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur simultaneously is 0.14. Then the probability that neither A nor B occurs is [IIT 1980; MP PET 1994] |
| A. | 0.39 |
| B. | 0.25 |
| C. | 0.904 |
| D. | None of these |
| Answer» B. 0.25 | |
| 2590. |
For two given events A and B, \[P\,(A\cap B)=\] [IIT 1988] |
| A. | Not less than \[P(A)+P\,(B)-1\] |
| B. | Not greater than \[P(A)+P(B)\] |
| C. | Equal to \[P(A)+P(B)-P(A\cup B)\] |
| D. | All of the above |
| Answer» E. | |
| 2591. |
Let A and B be two events such that \[P\,(A)=0.3\] and \[P\,(A\cup B)=0.8\]. If A and B are independent events, then \[P(B)=\] [IIT 1990; UPSEAT 2001, 02] |
| A. | \[\frac{5}{6}\] |
| B. | \[\frac{5}{7}\] |
| C. | \[\frac{3}{5}\] |
| D. | \[\frac{2}{5}\] |
| Answer» C. \[\frac{3}{5}\] | |
| 2592. |
If A and B are two independent events such that \[P\,(A\cap B')=\frac{3}{25}\] and \[P\,(A'\cap B)=\frac{8}{25},\] then \[P(A)=\] [IIT Screening] |
| A. | \[\frac{1}{5}\] |
| B. | \[\frac{3}{8}\] |
| C. | \[\frac{2}{5}\] |
| D. | \[\frac{4}{5}\] |
| Answer» B. \[\frac{3}{8}\] | |
| 2593. |
A and B are two independent events. The probability that both A and B occur is \[\frac{1}{6}\] and the probability that neither of them occurs is \[\frac{1}{3}\]. Then the probability of the two events are respectively [Roorkee 1989] |
| A. | \[\frac{1}{2}\]and \[\frac{1}{3}\] |
| B. | \[\frac{1}{5}\]and \[\frac{1}{6}\] |
| C. | \[\frac{1}{2}\]and \[\frac{1}{6}\] |
| D. | \[\frac{2}{3}\]and \[\frac{1}{4}\] |
| Answer» B. \[\frac{1}{5}\]and \[\frac{1}{6}\] | |
| 2594. |
If A and B are two events such that \[P\,(A\cup B)\,+P\,(A\cap B)=\frac{7}{8}\] and \[P\,(A)=2\,P\,(B),\] then \[P\,(A)=\] |
| A. | \[\frac{7}{12}\] |
| B. | \[\frac{7}{24}\] |
| C. | \[\frac{5}{12}\] |
| D. | \[\frac{17}{24}\] |
| Answer» B. \[\frac{7}{24}\] | |
| 2595. |
If A and B an two events such that \[P\,(A\cup B)=\frac{5}{6}\],\[P\,(A\cap B)=\frac{1}{3}\] and \[P\,(\bar{B})=\frac{1}{3},\] then \[P\,(A)=\] |
| A. | \[\frac{1}{4}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{2}{3}\] |
| Answer» D. \[\frac{2}{3}\] | |
| 2596. |
A card is drawn from a pack of 52 cards. A gambler bets that it is a spade or an ace. What are the odds against his winning this bet |
| A. | 17 : 52 |
| B. | 52 : 17 |
| C. | 9 : 4 |
| D. | 4 : 9 |
| Answer» D. 4 : 9 | |
| 2597. |
If an integer is chosen at random from first 100 positive integers, then the probability that the chosen number is a multiple of 4 or 6, is |
| A. | \[\frac{41}{100}\] |
| B. | \[\frac{33}{100}\] |
| C. | \[\frac{1}{10}\] |
| D. | None of these |
| Answer» C. \[\frac{1}{10}\] | |
| 2598. |
If A and B are two independent events, then \[P\,(A+B)=\] [MP PET 1992] |
| A. | \[P\,(A)+P\,(B)-P\,(A)\,P\,(B)\] |
| B. | \[P\,(A)-P\,(B)\] |
| C. | \[P\,(A)+P\,(B)\] |
| D. | \[P\,(A)+P\,(B)+P\,(A)\,P\,(B)\] |
| Answer» B. \[P\,(A)-P\,(B)\] | |
| 2599. |
If A and B are two independent events such that \[P\,(A)=0.40,\,\,P\,(B)=0.50.\] Find \[P\](neither A nor B) [MP PET 1989; J & K 2005] |
| A. | 0.90 |
| B. | 0.10 |
| C. | 0.2 |
| D. | 0.3 |
| Answer» E. | |
| 2600. |
If \[P\,(A)=\frac{1}{4},\,\,P\,(B)=\frac{5}{8}\] and \[P\,(A\cup B)=\frac{3}{4},\] then \[P\,(A\cap B)=\] |
| A. | \[\frac{1}{8}\] |
| B. | 0 |
| C. | \[\frac{3}{4}\] |
| D. | 1 |
| Answer» B. 0 | |