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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.
| 2601. |
The probability of happening at least one of the events A and B is 0.6. If the events A and B happens simultaneously with the probability 0.2, then \[P\,(\bar{A})+P\,(\bar{B})=\] [Roorkee 1989; IIT 1987; MP PET 1997; DCE 2001; J & K 2005] |
| A. | 0.4 |
| B. | 0.8 |
| C. | 1.2 |
| D. | 1.4 |
| Answer» D. 1.4 | |
| 2602. |
If A and B are two mutually exclusive events, then \[P\,(A+B)=\] [MNR 1978; MP PET 1991, 92] |
| A. | \[P\,(A)+P\,(B)-P\,(AB)\] |
| B. | \[P\,(A)-P\,(B)\] |
| C. | \[P\,(A)+P\,(B)\] |
| D. | \[P\,(A)+P\,(B)+P\,(AB)\] |
| Answer» D. \[P\,(A)+P\,(B)+P\,(AB)\] | |
| 2603. |
A coin is tossed twice. If events A and B are defined as : A = head on first toss, \[B=\] head on second toss. Then the probability of \[A\cup B=\] |
| A. | \[\frac{1}{4}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{1}{8}\] |
| D. | \[\frac{3}{4}\] |
| Answer» E. | |
| 2604. |
If A and B are any two events, then the probability that exactly one of them occur is [BIT Ranchi 1990; IIT 1984; RPET 1995, 2002; MP PET 2004] |
| A. | \[P\,(A)+P\,(B)-P\,(A\cap B)\] |
| B. | \[P\,(A)+P\,(B)-2P\,(A\cap B)\] |
| C. | \[P\,(A)+P\,(B)-P\,(A\cup B)\] |
| D. | \[P\,(A)+P\,(B)-2P\,(A\cup B)\] |
| Answer» C. \[P\,(A)+P\,(B)-P\,(A\cup B)\] | |
| 2605. |
If the odds in favour of an event be 3 : 5, then the probability of non-occurrence of the event is |
| A. | \[\frac{3}{5}\] |
| B. | \[\frac{5}{3}\] |
| C. | \[\frac{3}{8}\] |
| D. | \[\frac{5}{8}\] |
| Answer» E. | |
| 2606. |
If \[P\,(A)=0.4,\,\,P\,(B)=x,\,\,P\,(A\cup B)=0.7\] and the events A and B are mutually exclusive, then \[x=\] [MP PET 1992] |
| A. | \[\frac{3}{10}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{2}{5}\] |
| D. | \[\frac{1}{5}\] |
| Answer» B. \[\frac{1}{2}\] | |
| 2607. |
If A and B are two events of a random experiment, \[P\,(A)=0.25\] , \[P\,(B)=0.5\] and \[P\,(A\cap B)=0.15,\] then \[P\,(A\cap \bar{B})=\] [MP PET 1987] |
| A. | 0.1 |
| B. | 0.35 |
| C. | 0.15 |
| D. | 0.6 |
| Answer» B. 0.35 | |
| 2608. |
If \[P\,(A)=0.4,\,\,P\,(B)=x,\,\,P\,(A\cup B)=0.7\] and the events A and B are independent, then x = [CEE 1993] |
| A. | \[\frac{1}{3}\] |
| B. | \[\frac{1}{2}\] |
| C. | \[\frac{2}{3}\] |
| D. | None of these |
| Answer» C. \[\frac{2}{3}\] | |
| 2609. |
A card is drawn at random from a pack of cards. The probability of this card being a red or a queen is [MP PET 1989] |
| A. | \[\frac{1}{13}\] |
| B. | \[\frac{1}{26}\] |
| C. | \[\frac{1}{2}\] |
| D. | \[\frac{7}{13}\] |
| Answer» E. | |
| 2610. |
Suppose that A, B, C are events such that \[P\,(A)=P\,(B)=P\,(C)=\frac{1}{4},\,P\,(AB)=P\,(CB)=0,\,P\,(AC)=\frac{1}{8},\] then \[P\,(A+B)=\] [MP PET 1992] |
| A. | 0.125 |
| B. | 0.25 |
| C. | 0.375 |
| D. | 0.5 |
| Answer» E. | |
| 2611. |
If A and B are two events such that \[P(A)=0.4\] , \[P\,(A+B)=0.7\] and \[P\,(AB)=0.2,\] then \[P\,(B)=\] [MP PET 1992] |
| A. | 0.1 |
| B. | 0.3 |
| C. | 0.5 |
| D. | None of these |
| Answer» D. None of these | |
| 2612. |
The probabilities of three mutually exclusive events are 2/3, 1/4 and 1/6. The statement is [MNR 1987; UPSEAT 2000] |
| A. | True |
| B. | Wrong |
| C. | Could be either |
| D. | Do not know |
| Answer» C. Could be either | |
| 2613. |
A party of 23 persons take their seats at a round table. The odds against two persons sitting together are [RPET 1999] |
| A. | 10 : 1 |
| B. | 1 : 11 |
| C. | 9 : 10 |
| D. | None of these |
| Answer» B. 1 : 11 | |
| 2614. |
\[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+........+.......\frac{1}{n.(n+1)}\]equals [AMU 1995; RPET 1996; UPSEAT 1999, 2001] |
| A. | \[\frac{1}{n(n+1)}\] |
| B. | \[\frac{n}{n+1}\] |
| C. | \[\frac{2n}{n+1}\] |
| D. | \[\frac{2}{n(n+1)}\] |
| Answer» C. \[\frac{2n}{n+1}\] | |
| 2615. |
The sum 1(1!) + 2(2!) + 3(3!) + ....+n (n!) equals [AMU 1999; DCE 2005] |
| A. | \[3\,(n\,!)\,+\,n-3\] |
| B. | \[(n+1)!\,-\,(n-1)!\] |
| C. | \[(n+1)\,!\,-1\] |
| D. | \[2\,(n\,!)-2n-1\] |
| Answer» D. \[2\,(n\,!)-2n-1\] | |
| 2616. |
The sum of the series \[1+(1+2)+(1+2+3)+............\]upto \[n\] terms, will be [MP PET 1986] |
| A. | \[{{n}^{2}}-2n+6\] |
| B. | \[\frac{n(n+1)(2n-1)}{6}\] |
| C. | \[{{n}^{2}}+2n+6\] |
| D. | \[\frac{n(n+1)(n+2)}{6}\] |
| Answer» E. | |
| 2617. |
The sum to \[n\] terms of the series \[{{2}^{2}}+{{4}^{2}}+{{6}^{2}}+...........\] is [MP PET 1994] |
| A. | \[\frac{n(n+1)(2n+1)}{3}\] |
| B. | \[\frac{2n(n+1)(2n+1)}{3}\] |
| C. | \[\frac{n(n+1)(2n+1)}{6}\] |
| D. | \[\frac{n(n+1)(2n+1)}{9}\] |
| Answer» C. \[\frac{n(n+1)(2n+1)}{6}\] | |
| 2618. |
If \[\frac{1}{{{1}^{4}}}+\frac{1}{{{2}^{4}}}+\frac{1}{{{3}^{4}}}+.....+\infty =\frac{{{\pi }^{4}}}{90}\], then the value of \[\frac{1}{{{1}^{4}}}+\frac{1}{{{3}^{4}}}+\frac{1}{{{5}^{4}}}+.....\infty \]is [AMU 2005] |
| A. | \[\frac{{{\pi }^{4}}}{96}\] |
| B. | \[\frac{{{\pi }^{4}}}{45}\] |
| C. | \[\frac{89}{90}{{\pi }^{4}}\] |
| D. | None of these |
| Answer» B. \[\frac{{{\pi }^{4}}}{45}\] | |
| 2619. |
If the sum of \[1+\frac{1+2}{2}+\frac{1+2+3}{3}+.....\] to n terms is S, then S is equal to [Kerala (Engg.) 2002] |
| A. | \[\frac{n(n+3)}{4}\] |
| B. | \[\frac{n(n+2)}{4}\] |
| C. | \[\frac{n(n+1)\,(n+2)}{6}\] |
| D. | \[{{n}^{2}}\] |
| Answer» B. \[\frac{n(n+2)}{4}\] | |
| 2620. |
If \[{{t}_{n}}=\frac{1}{4}(n+2)\,(n+3)\] for n = 1, 2, 3,?? then \[\frac{1}{{{t}_{1}}}+\frac{1}{{{t}_{2}}}+\frac{1}{{{t}_{3}}}+....+\frac{1}{{{t}_{2003}}}=\] [EAMCET 2003] |
| A. | \[\frac{4006}{3006}\] |
| B. | \[\frac{4003}{3007}\] |
| C. | \[\frac{4006}{3008}\] |
| D. | \[\frac{4006}{3009}\] |
| Answer» E. | |
| 2621. |
\[\frac{\frac{1}{2}.\frac{2}{2}}{{{1}^{3}}}+\frac{\frac{2}{2}.\frac{3}{2}}{{{1}^{3}}+{{2}^{3}}}+\frac{\frac{3}{2}.\frac{4}{2}}{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}}+.....n\] terms = [EAMCET 2000] |
| A. | \[{{\left( \frac{n}{n+1} \right)}^{2}}\] |
| B. | \[{{\left( \frac{n}{n+1} \right)}^{3}}\] |
| C. | \[\left( \frac{n}{n+1} \right)\] |
| D. | \[\left( \frac{1}{n+1} \right)\] |
| Answer» D. \[\left( \frac{1}{n+1} \right)\] | |
| 2622. |
The sum to infinity of the following series \[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...........\] shall be [MNR 1982] |
| A. | \[\infty \] |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 2623. |
First term of the \[{{11}^{th}}\]group in the following groups (1), (2, 3, 4), (5, 6, 7, 8, 9),???.is |
| A. | 89 |
| B. | 97 |
| C. | 101 |
| D. | 123 |
| Answer» D. 123 | |
| 2624. |
\[{{11}^{2}}+{{12}^{2}}+{{13}^{2}}+{{.......20}^{2}}=\] [MP PET 1995] |
| A. | 2481 |
| B. | 2483 |
| C. | 2485 |
| D. | 2487 |
| Answer» D. 2487 | |
| 2625. |
The sum of first \[n\] terms of the given series \[{{1}^{2}}+{{2.2}^{2}}+{{3}^{2}}+{{2.4}^{2}}+{{5}^{2}}+{{2.6}^{2}}+............\]is\[\frac{n{{(n+1)}^{2}}}{2}\], when \[n\] is even. When \[n\] is odd, the sum will be [IIT 1988] |
| A. | \[\frac{n{{(n+1)}^{2}}}{2}\] |
| B. | \[\frac{1}{2}{{n}^{2}}(n+1)\] |
| C. | \[n{{(n+1)}^{2}}\] |
| D. | None of these |
| Answer» C. \[n{{(n+1)}^{2}}\] | |
| 2626. |
The sum to \[n\] terms of\[(2n-1)+2\,(2n-3)\] \[+3\,(2n-5)+.....\] is [AMU 2001] |
| A. | \[(n+1)\,(n+2)\,(n+3)/6\] |
| B. | \[n\,(n+1)\,(n+2)/6\] |
| C. | \[n\,(n+1)\,(2n+3)\,\] |
| D. | \[n\,(n+1)\,(2n+1)/6\] |
| Answer» E. | |
| 2627. |
If the \[{{n}^{th}}\] term of a series be \[3+n\,(n-1)\], then the sum of \[n\] terms of the series is |
| A. | \[\frac{{{n}^{2}}+n}{3}\] |
| B. | \[\frac{{{n}^{3}}+8n}{3}\] |
| C. | \[\frac{{{n}^{2}}+8n}{5}\] |
| D. | \[\frac{{{n}^{2}}-8n}{3}\] |
| Answer» C. \[\frac{{{n}^{2}}+8n}{5}\] | |
| 2628. |
\[{{11}^{3}}+{{12}^{3}}+....+{{20}^{3}}\] [Pb. CET 1997; RPET 2002] |
| A. | Is divisible by 5 |
| B. | Is an odd integer divisible by 5 |
| C. | Is an even integer which is not divisible by 5 |
| D. | Is an odd integer which is not divisible by 5 |
| Answer» C. Is an even integer which is not divisible by 5 | |
| 2629. |
The sum of \[n\] terms of the following series \[1.2+2.3+3.4+4.5+.........\] shall be [MNR 1980] |
| A. | \[{{n}^{3}}\] |
| B. | \[\frac{1}{3}n\,(n+1)(n+2)\] |
| C. | \[\frac{1}{6}n\,(n+1)(n+2)\] |
| D. | \[\frac{1}{3}n\,(n+1)(2n+1)\] |
| Answer» C. \[\frac{1}{6}n\,(n+1)(n+2)\] | |
| 2630. |
\[\sum\limits_{m=1}^{n}{{{m}^{2}}}\] is equal to [RPET 1995] |
| A. | \[\frac{m(m+1)}{2}\] |
| B. | \[\frac{m(m+1)(2m+1)}{6}\] |
| C. | \[\frac{n(n+1)(2n+1)}{6}\] |
| D. | \[\frac{n(n+1)}{2}\] |
| Answer» D. \[\frac{n(n+1)}{2}\] | |
| 2631. |
Sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+.....\] to n terms is [Karnataka CET 2001] |
| A. | \[n-\frac{1}{2}({{3}^{n}}-1)\] |
| B. | \[n+\frac{1}{2}({{3}^{n}}-1)\] |
| C. | \[n+\frac{1}{2}(1-{{3}^{-n}})\] |
| D. | \[n+\frac{1}{2}({{3}^{-n}}-1)\] |
| Answer» E. | |
| 2632. |
Sum of the \[n\] terms of the series \[\frac{3}{{{1}^{2}}}+\frac{5}{{{1}^{2}}+{{2}^{2}}}+\frac{7}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}\,+...\,\,\text{is}\] [Pb. CET 1999; RPET 2001] |
| A. | \[\frac{2n}{n+1}\] |
| B. | \[\frac{4n}{n+1}\] |
| C. | \[\frac{6n}{n+1}\] |
| D. | \[\frac{9n}{n+1}\] |
| Answer» D. \[\frac{9n}{n+1}\] | |
| 2633. |
The sum of the series \[1.3.5+.2.5.8+3.7.11+.........\]upto \['n'\] terms is [Dhanbad Engg. 1972] |
| A. | \[\frac{n\,(n+1)(9{{n}^{2}}+23n+13)}{6}\] |
| B. | \[\frac{n\,(n-1)(9{{n}^{2}}+23n+12)}{6}\] |
| C. | \[\frac{(n+1)(9{{n}^{2}}+23n+13)}{6}\] |
| D. | \[\frac{n\,(9{{n}^{2}}+23n+13)}{6}\] |
| Answer» B. \[\frac{n\,(n-1)(9{{n}^{2}}+23n+12)}{6}\] | |
| 2634. |
\[\frac{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+{{4}^{3}}+{{........12}^{3}}}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}}+{{.........12}^{2}}}=\] [MP PET 1998] |
| A. | \[\frac{234}{25}\] |
| B. | \[\frac{243}{35}\] |
| C. | \[\frac{263}{27}\] |
| D. | None of these |
| Answer» B. \[\frac{243}{35}\] | |
| 2635. |
The sum of \[(n-1)\] terms of \[1+(1+3)+\] \[(1+3+5)+.......\] is [RPET 1999] |
| A. | \[\frac{n\,(n+1)\,(2n+1)}{6}\] |
| B. | \[\frac{{{n}^{2}}(n+1)}{4}\] |
| C. | \[\frac{n\,(n-1)\,(2n-1)}{6}\] |
| D. | \[{{n}^{2}}\] |
| Answer» D. \[{{n}^{2}}\] | |
| 2636. |
The sum of all the products of the first \[n\] natural numbers taken two at a time is |
| A. | \[\frac{1}{24}n(n-1)(n+1)(3n+2)\] |
| B. | \[\frac{{{n}^{2}}}{48}(n-1)(n-2)\] |
| C. | \[\frac{1}{6}n(n+1)(n+2)(n+5)\] |
| D. | None of these |
| Answer» B. \[\frac{{{n}^{2}}}{48}(n-1)(n-2)\] | |
| 2637. |
The \[{{n}^{th}}\] term of series \[\frac{1}{1}+\frac{1+2}{2}+\frac{1+2+3}{3}+.......\] will be [AMU 1982] |
| A. | \[\frac{n+1}{2}\] |
| B. | \[\frac{n-1}{2}\] |
| C. | \[\frac{{{n}^{2}}+1}{2}\] |
| D. | \[\frac{{{n}^{2}}-1}{2}\] |
| Answer» B. \[\frac{n-1}{2}\] | |
| 2638. |
Sum of first \[n\] terms in the following series \[{{\cot }^{-1}}3+{{\cot }^{-1}}7+{{\cot }^{-1}}13+{{\cot }^{-1}}21+.............\] is given by |
| A. | \[{{\tan }^{-1}}\left( \frac{n}{n+2} \right)\] |
| B. | \[{{\cot }^{-1}}\left( \frac{n+2}{n} \right)\] |
| C. | \[{{\tan }^{-1}}(n+1)-{{\tan }^{-1}}1\] |
| D. | All of these |
| Answer» E. | |
| 2639. |
Sum of the squares of first \[n\] natural numbers exceeds their sum by 330, then \[n=\] [Karnataka CET 1998] |
| A. | 8 |
| B. | 10 |
| C. | 15 |
| D. | 20 |
| Answer» C. 15 | |
| 2640. |
The sum of \[{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+{{4}^{3}}+.....+{{15}^{3}}\],is [MP PET 2003] |
| A. | 22000 |
| B. | 10000 |
| C. | 14400 |
| D. | 15000 |
| Answer» D. 15000 | |
| 2641. |
The sum of the series \[1.2.3+2.3.4+3.4.5+.......\] to n terms is [Kurukshetra CEE 1998] |
| A. | \[n(n+1)(n+2)\] |
| B. | \[(n+1)(n+2)(n+3)\] |
| C. | \[\frac{1}{4}n(n+1)(n+2)(n+3)\] |
| D. | \[\frac{1}{4}(n+1)(n+2)(n+3)\] |
| Answer» D. \[\frac{1}{4}(n+1)(n+2)(n+3)\] | |
| 2642. |
If \[\sum\limits_{i=1}^{n}{i=\frac{n(n+1)}{2}}\], then \[\sum\limits_{i=1}^{n}{(3i-2)=}\] |
| A. | \[\frac{n(3n-1)}{2}\] |
| B. | \[\frac{n(3n+1)}{2}\] |
| C. | \[n(3n+2)\] |
| D. | \[\frac{n(3n+1)}{4}\] |
| Answer» B. \[\frac{n(3n+1)}{2}\] | |
| 2643. |
The greatest integer less than or equal to \[{{(\sqrt{2}+1)}^{6}}\]is [RPET 2000] |
| A. | 196 |
| B. | 197 |
| C. | 198 |
| D. | 199 |
| Answer» C. 198 | |
| 2644. |
The number of terms in the expansion of \[{{(a+b+c)}^{n}}\] will be |
| A. | \[n+1\] |
| B. | \[n+3\] |
| C. | \[\frac{(n+1)(n+2)}{2}\] |
| D. | None of these |
| Answer» D. None of these | |
| 2645. |
Let \[R={{(5\sqrt{5}+11)}^{2n+1}}\] and \[f=R-[R]\], where [.] denotes the greatest integer function. The value of R.f is [IIT 1988] |
| A. | \[{{4}^{2n+1}}\] |
| B. | \[{{4}^{2n}}\] |
| C. | \[{{4}^{2n-1}}\] |
| D. | \[{{4}^{-2n}}\] |
| Answer» B. \[{{4}^{2n}}\] | |
| 2646. |
The number of terms which are free from radical signs in the expansion of \[{{({{y}^{1/5}}+{{x}^{1/10}})}^{55}}\]is |
| A. | 5 |
| B. | 6 |
| C. | 7 |
| D. | None of these |
| Answer» C. 7 | |
| 2647. |
In the expansion of \[{{({{5}^{1/2}}+{{7}^{1/8}})}^{1024}}\], the number of integral terms is |
| A. | 128 |
| B. | 129 |
| C. | 130 |
| D. | 131 |
| Answer» C. 130 | |
| 2648. |
If n is positive integer and three consecutive coefficients in the expansion of \[{{(1+x)}^{n}}\] are in the ratio 6 : 33 : 110, then n = |
| A. | 4 |
| B. | 6 |
| C. | 12 |
| D. | 16 |
| Answer» D. 16 | |
| 2649. |
If in the expansion of \[{{(1+x)}^{n}}\], a, b, c are three consecutive coefficients, then n= |
| A. | \[\frac{ac+ab+bc}{{{b}^{2}}+ac}\] |
| B. | \[\frac{2ac+ab+bc}{{{b}^{2}}-ac}\] |
| C. | \[\frac{ab+ac}{{{b}^{2}}-ac}\] |
| D. | None of these |
| Answer» C. \[\frac{ab+ac}{{{b}^{2}}-ac}\] | |
| 2650. |
If \[{{a}_{r}}\] is the coefficient of \[{{x}^{r}}\], in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\], then \[{{a}_{1}}-2{{a}_{2}}+3{{a}_{3}}-....-2n\,{{a}_{2n}}=\] [EAMCET 2003] |
| A. | 0 |
| B. | n |
| C. | ? n |
| D. | 2n |
| Answer» D. 2n | |