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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.
| 2901. |
Let \[f:\left[ -\frac{\pi }{3},\frac{2\pi }{3} \right]\to [0,4]\] be a function defined as \[f(x)=\sqrt{3}\sin x-\cos x+2\].Then \[{{f}^{-1}}(x)\]is given by |
| A. | \[{{\sin }^{-1}}\left( \frac{x-2}{2} \right)-\frac{\pi }{6}\] |
| B. | \[{{\sin }^{-1}}\left( \frac{x-2}{2} \right)+\frac{\pi }{6}\] |
| C. | \[\frac{2\pi }{3}+{{\cos }^{-1}}\left( \frac{x-2}{2} \right)\] |
| D. | none of these |
| Answer» C. \[\frac{2\pi }{3}+{{\cos }^{-1}}\left( \frac{x-2}{2} \right)\] | |
| 2902. |
A function f form the set of natural numbers to integers defined by \[f(n)=\left\{ \begin{matrix} \frac{n-1}{2},\,\,\,\text{when}\,\,n\,\,\text{odd} \\ -\frac{n}{2}\,,\,\,\,\text{when}\,\,n\,\,\text{is}\,\,\text{odd} \\ \end{matrix}\,\,\,\text{is} \right.\] |
| A. | one-one but not onto. |
| B. | onto but not one-one |
| C. | one-one and onto both. |
| D. | neither one-one nor onto. |
| Answer» D. neither one-one nor onto. | |
| 2903. |
The range of following function is \[f(x)=\sqrt{(1-\cos x)\sqrt{(1-\cos x)\sqrt{(1-\cos x)\sqrt{...\infty }}}}\] |
| A. | \[\left[ 0,\text{ }1 \right]\] |
| B. | \[\left[ 0,\text{ }1/2 \right]\] |
| C. | \[\left[ 0,\text{ }2 \right]\] |
| D. | none of these |
| Answer» D. none of these | |
| 2904. |
The range of \[f(x)=co{{s}^{-1}}\left( \frac{1+{{x}^{2}}}{2x} \right)+\sqrt{2-{{x}^{2}}}\]is |
| A. | \[\left\{ 0,\,\,1+\frac{\pi }{2} \right\}\] |
| B. | \[\left\{ 0,\,\,1+\pi \right\}\] |
| C. | \[\left\{ 1,\,\,1+\frac{\pi }{2} \right\}\] |
| D. | \[\left\{ 1,\,\,1+\pi \right\}\] |
| Answer» D. \[\left\{ 1,\,\,1+\pi \right\}\] | |
| 2905. |
The range of the function f defined by \[f(x)=\left[ \frac{1}{\sin \left\{ x \right\}} \right]\] (where [.] and \[\left\{ . \right\}\], respectively, denote the greatest integer and the fractional part functions) is |
| A. | I, the set of integers |
| B. | N, the set of natural numbers |
| C. | W, the set of whole numbers |
| D. | \[\left\{ 1,2,3,4.... \right\}\] |
| Answer» E. | |
| 2906. |
Let \[g(x)=f(x)-1.\] If \[f(x)+f(1-x)=2\forall x\in R\], then \[g(x)\] is symmetrical about |
| A. | the origin |
| B. | the line\[x=\frac{1}{2}\] |
| C. | the point (1, 0) |
| D. | the point \[\left( \frac{1}{2},0 \right)\] |
| Answer» E. | |
| 2907. |
The domain of the functions \[f(x)=\frac{1}{\sqrt{\left\{ \operatorname{sinx} \right\}+\left\{ \sin (\pi +x) \right\}}}\] where \[\left\{ \cdot \right\}\] denotes the fractional part, is |
| A. | \[[0,\pi ]\] |
| B. | \[(2n+1)\pi /2,n\in Z\] |
| C. | \[(0,\pi )\] |
| D. | none of these |
| Answer» E. | |
| 2908. |
Let\[f :X\to y\,f(x)=sinx+cosx+2\sqrt{2}\]be invertible. Then which \[X\to Y\]is not possible? |
| A. | \[\left[ \frac{\pi }{4},\frac{5\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] |
| B. | \[\left[ -\frac{3\pi }{4},\frac{\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] |
| C. | \[\left[ -\frac{3\pi }{4},\frac{3\pi }{4} \right]\to [\sqrt{2},3\sqrt{2}]\] |
| D. | none of these |
| Answer» D. none of these | |
| 2909. |
There are 20 cards. Ten of these cards have the letter "I" printed on them and the other 10 have the letter "T" printed on them. If three cards are picked up at random and kept in the same order, the probability of making word IIT is |
| A. | 4/27 |
| B. | 14001 |
| C. | 1/8 |
| D. | 29465 |
| Answer» C. 1/8 | |
| 2910. |
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 |
| A. | 18/595 |
| B. | 42887 |
| C. | 3/10 |
| D. | 44379 |
| Answer» B. 42887 | |
| 2911. |
A and B toss a fair coin each simultaneously 50 times. The probability that both of them will not get tail at the same toss is |
| A. | \[{{(3/4)}^{50}}\] |
| B. | \[{{(2/7)}^{50}}\] |
| C. | \[{{(1/8)}^{50}}\] |
| D. | \[{{(7/8)}^{50}}\] |
| Answer» B. \[{{(2/7)}^{50}}\] | |
| 2912. |
The probabilities of winning a race by three persons A, B, and C are 1/2, 1/4, and 1/4, respectively. They run two races. The probability of A winning the second race when B, wins the first race is |
| A. | 1/3 |
| B. | ½ |
| C. | 1/4 |
| D. | 44257 |
| Answer» C. 1/4 | |
| 2913. |
A sample space consists of 3 sample points with associated probabilities given as \[2p,\,\,{{p}^{2}},\,\,4p-1\]. Then the value of p is |
| A. | \[p=\sqrt{11}-3\] |
| B. | \[\sqrt{10}-3\] |
| C. | \[\frac{1}{4}<p<\frac{1}{2}\] |
| D. | none |
| Answer» B. \[\sqrt{10}-3\] | |
| 2914. |
A natural number is chosen at random from the first 100 natural numbers. The probability that \[x+\frac{100}{x}>50\] is |
| A. | 1/10 |
| B. | 18568 |
| C. | 11/20 |
| D. | none of these |
| Answer» E. | |
| 2915. |
There are 10 prizes, five A's, three B's, and two C's, placed in identical sealed envelopes for the top 10 contestants in a mathematics contest. The prizes are awarded by allowing winners to select an envelope at random from those remaining. When the 8th contestant goes to select the prize, the probability that the remaining three prizes are one A, one B and one C is |
| A. | 1/4 |
| B. | 44256 |
| C. | 1/12 |
| D. | 44470 |
| Answer» B. 44256 | |
| 2916. |
A man has 3 pairs of black socks and 2 pairs of brown socks kept together in a box. If he dressed hurriedly in the dark, the probability that after he has put on a black sock, he will then put on another black sock is |
| A. | 1/3 |
| B. | 44257 |
| C. | 3/5 |
| D. | 42036 |
| Answer» B. 44257 | |
| 2917. |
Forty teams play a tournament. Each team plays every other team just once. Each game results in a win for one team. If each team has a 50% chance of winning each game, the probability that at the end of the tournament, every team has won a different number of games is |
| A. | \[1/780\] |
| B. | \[40!/{{2}^{780}}\] |
| C. | \[40!/{{3}^{780}}\] |
| D. | none of these |
| Answer» C. \[40!/{{3}^{780}}\] | |
| 2918. |
A fair die is tossed repeatedly. A wins if it is 1 or 2 on two consecutive tosses and B wins if it is 3, 4, 5 or 6 on two consecutive tosses. The probability that A wins if the die is tossed indefinitely is |
| A. | 1/3 |
| B. | 44317 |
| C. | 1/4 |
| D. | 44318 |
| Answer» C. 1/4 | |
| 2919. |
Let A and B are events of an experiment and \[P(A)=1/4,\,\,P(A\cup B)=1/2\] then value of \[P(B/{{A}^{c}})\] is |
| A. | 2/3 |
| B. | 44256 |
| C. | 5/6 |
| D. | 44228 |
| Answer» C. 5/6 | |
| 2920. |
A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighborhood are sick with the flu, denoted by F, while 10% are sick with the measles, denoted by M. A well-known symptom of measles is a rash, denoted by R. The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flu also develop a rash, with conditional probability 0.08. Upon examination the child, the doctor finds a rash. Then what is the probability that the child has the measles? |
| A. | 91/165 |
| B. | 90/163 |
| C. | 82/161 |
| D. | 95/167 |
| Answer» E. | |
| 2921. |
If any four numbers are selected and they are multiplied, then the probability that the last digit will be 1, 3, 5 or 7 is |
| A. | 4/625 |
| B. | 18/625 |
| C. | 16/625 |
| D. | none of these |
| Answer» D. none of these | |
| 2922. |
In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. If a person selected at random from the town has brown hair, the probability that he also has brown eyes is |
| A. | 1/5 |
| B. | 44411 |
| C. | 1/3 |
| D. | 44257 |
| Answer» C. 1/3 | |
| 2923. |
If a is an integer lying in [\[-\]5, 30], then the probability that the graph \[y={{x}^{2}}+2\,(a+4)x-5a+64\] is strictly above the x-axis is |
| A. | 1/6 |
| B. | 13332 |
| C. | 2/9 |
| D. | 44319 |
| Answer» D. 44319 | |
| 2924. |
The probability that a bulb produced by a factory will fuse after 150 days if used is 0.50. What is the probability that out of 5 such bulbs none will fuse after 150 days of use? |
| A. | \[1-{{(19/20)}^{5}}\] |
| B. | \[{{(19/20)}^{5}}\] |
| C. | \[{{(3/4)}^{5}}\] |
| D. | \[90\,{{(1/4)}^{5}}\] |
| Answer» C. \[{{(3/4)}^{5}}\] | |
| 2925. |
A box contains tickets numbered from 1 to 20. Three tickets are drawn from the box with replacement. The probability that the largest number on the tickets is 7 is |
| A. | 2/19 |
| B. | 44013 |
| C. | \[1-{{(7/20)}^{3}}\] |
| D. | none of these |
| Answer» E. | |
| 2926. |
\[\frac{(2n)!}{{{2}^{2n}}{{(n!)}^{2}}}\,\,is\,\,\le \] |
| A. | \[\frac{1}{3n+1}\] |
| B. | \[\frac{1}{{{(3n+1)}^{1/2}}}\] |
| C. | \[\frac{1}{{{(3n+1)}^{2}}}\] |
| D. | \[\frac{1}{{{(3n+1)}^{1/2}}}\] |
| Answer» C. \[\frac{1}{{{(3n+1)}^{2}}}\] | |
| 2927. |
If \[n\in N\], \[p(n)={{2}^{n-1}}\] then p(n)\[\le \] |
| A. | n |
| B. | \[{{n}^{2}}\] |
| C. | (n-1)! |
| D. | n ! |
| Answer» E. | |
| 2928. |
The sum of the series 1\[\times \]3+3\[\times \]5+5\[\times \]7+..to n terms is |
| A. | \[\frac{n(4{{n}^{2}}+6n)-1}{3}\] |
| B. | \[\frac{n(n+1)(n+2)}{2}\] |
| C. | \[\frac{(2n+1)(n+1)}{2}\] |
| D. | \[\frac{{{n}^{2}}+1}{4}\] |
| Answer» B. \[\frac{n(n+1)(n+2)}{2}\] | |
| 2929. |
The sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+.....\]to n terms is |
| A. | \[n-\frac{1}{2}(1-{{3}^{-n}})\] |
| B. | \[n-\frac{1}{2}({{3}^{-n}}-1)\] |
| C. | \[n-\frac{1}{2}({{3}^{n}}-1)\] |
| D. | \[n+\frac{1}{2}({{3}^{n}}-1)\] |
| Answer» B. \[n-\frac{1}{2}({{3}^{-n}}-1)\] | |
| 2930. |
\[p(n)={{n}^{3}}+{{(n+1)}^{3}}{{(n+2)}^{3}}\]where \[n\in N\]is divisible by |
| A. | 9 |
| B. | 7 |
| C. | 13 |
| D. | 15 |
| Answer» B. 7 | |
| 2931. |
Sum of \[\frac{{{1}^{3}}}{1}+\frac{{{1}^{3}}+{{2}^{3}}}{1+2}+...\]to n terms is |
| A. | \[\frac{(n+1)(n+2)}{3}\] |
| B. | \[n(n+1)(n+2)\] |
| C. | \[\frac{n(n+1)(n+2)}{6}\] |
| D. | \[\frac{n(n+1)(n+2)}{2}\] |
| Answer» D. \[\frac{n(n+1)(n+2)}{2}\] | |
| 2932. |
If 'n' is a positive integer, then \[{{n}^{3}}+2n\]is divisible by |
| A. | 3 |
| B. | 2 |
| C. | 6 |
| D. | 15 |
| Answer» B. 2 | |
| 2933. |
If \[p(n)={{n}^{2}}+n,\]\[\forall \,n\in N\], then p(n) is |
| A. | even positive integer |
| B. | prime number |
| C. | odd integer |
| D. | multiple of 4 |
| Answer» B. prime number | |
| 2934. |
The sum to n terms of 1+3+5..is |
| A. | \[n\] |
| B. | \[2n-1\] |
| C. | \[{{n}^{2}}\] |
| D. | \[{{n}^{3}}\] |
| Answer» D. \[{{n}^{3}}\] | |
| 2935. |
If \[p(n)={{n}^{3}}+{{(n+1)}^{3}}+{{(n+2)}^{3}}\]where \[n\in N\]then p(n) is divisible by |
| A. | 9 |
| B. | 7 |
| C. | 13 |
| D. | 15 |
| Answer» B. 7 | |
| 2936. |
The sum of n terms of \[1+\frac{1+2}{2}+\frac{1+2+3}{3}+...\]is |
| A. | \[\frac{m(n+3)}{4}\] |
| B. | \[\frac{(n+3)}{3}\] |
| C. | \[\frac{n(n-3)}{3}\] |
| D. | \[\frac{n(n-3)}{4}\] |
| Answer» B. \[\frac{(n+3)}{3}\] | |
| 2937. |
The number \[({{49}^{2}}-4)({{49}^{3}}-49)\]is divisible by__ |
| A. | 6! |
| B. | 5! |
| C. | 7! |
| D. | 9! |
| Answer» C. 7! | |
| 2938. |
The sum \[\underbrace{{{(666..6)}^{2}}}_{n\,digits}+\underbrace{(888..8)}_{n\,digits}\]is equal to |
| A. | \[\frac{4}{9}{{({{10}^{n}}-1)}^{2}}\] |
| B. | \[\frac{4}{9}({{10}^{n}}-1)\] |
| C. | \[\frac{4}{9}({{10}^{2n}}-1)\] |
| D. | \[4({{10}^{n}}+1)\] |
| Answer» D. \[4({{10}^{n}}+1)\] | |
| 2939. |
Sum of 1+2+\[{{2}^{2}}\]+\[{{2}^{n-1}}\]is |
| A. | \[{{2}^{n}}\] |
| B. | \[{{2}^{n}}+1\] |
| C. | \[{{2}^{n}}-1\] |
| D. | \[{{2}^{n-1}}\] |
| Answer» D. \[{{2}^{n-1}}\] | |
| 2940. |
If\[n\in N\],then \[\sin \theta +\sin 2\theta +\sin 3\theta +....+\sin n\theta =\] |
| A. | \[\sin \left[ \frac{n(n+1)\theta }{2} \right]\] |
| B. | \[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]\sin \theta }{\sin \left( \frac{\theta }{2} \right)}\] |
| C. | \[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]\sin \left( \frac{n\theta }{2} \right)}{\sin \left( \frac{\theta }{2} \right)}\] |
| D. | \[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]{{\sin }^{n}}\left( \frac{\theta }{2} \right)}{\sin \left( \frac{\theta }{2} \right)}\] |
| Answer» D. \[\frac{\sin \left[ \frac{(n+1)}{2}\theta \right]{{\sin }^{n}}\left( \frac{\theta }{2} \right)}{\sin \left( \frac{\theta }{2} \right)}\] | |
| 2941. |
The sum of 1+3+5+7..upto n terms is |
| A. | 2n+1 |
| B. | \[{{n}^{2}}\] |
| C. | \[2{{n}^{2}}-1\] |
| D. | \[2{{n}^{2}}\] |
| Answer» C. \[2{{n}^{2}}-1\] | |
| 2942. |
Number of ways in which a lawn-tennis mixed double be made from seven married couples if no husband and wife play in the same set is |
| A. | 240 |
| B. | 420 |
| C. | 720 |
| D. | none of these |
| Answer» C. 720 | |
| 2943. |
A class contains three girls and four boys. Every Saturday, five go on a picnic (a different group of student is sent every week). During the picnic, eac girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is |
| A. | 21 |
| B. | 45 |
| C. | 27 |
| D. | 24 |
| Answer» C. 27 | |
| 2944. |
How many numbers can be made with the digits 3, 4, 5, 6, 7, 8 lying between 3000 and 4000, which are divisible by 5 while repetition of any digit not allowed in any number? |
| A. | 60 |
| B. | 12 |
| C. | 120 |
| D. | 24 |
| Answer» C. 120 | |
| 2945. |
The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the numbers is 10 and the sums of the numbers placed diagonally are equal is |
| A. | 4 |
| B. | 8 |
| C. | 24 |
| D. | 6 |
| Answer» C. 24 | |
| 2946. |
Total number less than 3 x \[{{10}^{8}}\] and can be formed using the digits 1,2,3 is equal to |
| A. | \[\frac{1}{2}({{3}^{9}}+4\times {{3}^{8}})\] |
| B. | \[\frac{1}{2}({{3}^{9}}-3)\] |
| C. | \[\frac{1}{2}(7\times {{3}^{8}}-3)\] |
| D. | \[\frac{1}{2}({{3}^{9}}-3+{{3}^{8}})\] |
| Answer» D. \[\frac{1}{2}({{3}^{9}}-3+{{3}^{8}})\] | |
| 2947. |
There are four letters and four directed envelopers. The number of ways in which all the letters can be put I the wrong envelope is |
| A. | 8 |
| B. | 9 |
| C. | 16 |
| D. | none of these |
| Answer» C. 16 | |
| 2948. |
In a room, there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is |
| A. | \[{{12}^{2}}-1\] |
| B. | \[{{2}^{12}}\] |
| C. | \[{{2}^{12}}-1\] |
| D. | \[{{12}^{2}}\] |
| Answer» D. \[{{12}^{2}}\] | |
| 2949. |
A man has three friends. The number of ways he can invite one friend every day for dinner on six successive nights so that no friend is invited more than three times is |
| A. | 640 |
| B. | 320 |
| C. | 420 |
| D. | 510 |
| Answer» E. | |
| 2950. |
The maximum number of points of intersection of five lines and four circles is |
| A. | 60 |
| B. | 72 |
| C. | 62 |
| D. | none of these |
| Answer» D. none of these | |