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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.
| 1751. |
If \[R\subset A\times B\] and \[S\subset B\times C\,\] be two relations, then \[{{(SoR)}^{-1}}=\] |
| A. | \[{{S}^{-1}}o{{R}^{-1}}\] |
| B. | \[{{R}^{-1}}o{{S}^{-1}}\] |
| C. | \[SoR\] |
| D. | \[RoS\] |
| Answer» C. \[SoR\] | |
| 1752. |
Let X be a family of sets and R be a relation on X defined by 'A is disjoint from B'. Then R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Anti-symmetric |
| D. | Transitive |
| Answer» C. Anti-symmetric | |
| 1753. |
For real numbers x and y, we write \[x\,R\,y\Leftrightarrow \] \[x-y+\sqrt{2}\] is an irrational number. Then the relation R is |
| A. | Reflexive |
| B. | Symmetric |
| C. | Transitive |
| D. | None of these |
| Answer» B. Symmetric | |
| 1754. |
Let A = {1, 2, 3}. The total number of distinct relations that can be defined over A is |
| A. | \[{{2}^{9}}\] |
| B. | 6 |
| C. | 8 |
| D. | None of these |
| Answer» B. 6 | |
| 1755. |
In a \[\Delta ABC,\]if \[\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\], then \[\cos C=\] [Karnataka CET 2003] |
| A. | \[\frac{7}{5}\] |
| B. | \[\frac{5}{7}\] |
| C. | \[\frac{17}{36}\] |
| D. | \[\frac{16}{17}\] |
| Answer» C. \[\frac{17}{36}\] | |
| 1756. |
The smallest angle of the \[\Delta ABC\], when \[a=7,b=4\sqrt{3}\]and \[c=\sqrt{13},\] is [MP PET 2003] |
| A. | \[{{30}^{o}}\] |
| B. | \[{{15}^{o}}\] |
| C. | \[{{45}^{o}}\] |
| D. | None of these |
| Answer» B. \[{{15}^{o}}\] | |
| 1757. |
If the angles of a triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is [IIT Screening 2003] |
| A. | \[\sqrt{3}:(2+\sqrt{3})\] |
| B. | \[1:6\] |
| C. | \[1:(2+\sqrt{3})\] |
| D. | \[2:3\] |
| Answer» B. \[1:6\] | |
| 1758. |
In any triangle \[AB=2,BC=4,CA=3\]and D is mid point of BC, then [Roorkee 1995] |
| A. | \[\cos B=\frac{11}{6}\] |
| B. | \[\cos B=\frac{7}{8}\] |
| C. | \[AD=2.4\] |
| D. | \[A{{D}^{2}}=2.5\] |
| Answer» E. | |
| 1759. |
In a \[\Delta ABC,\,\,\frac{\cos C+\cos A}{c+a}+\frac{\cos B}{b}\]is equal to [EAMCET 2001] |
| A. | \[\frac{1}{a}\] |
| B. | \[\frac{1}{b}\] |
| C. | \[\frac{1}{c}\] |
| D. | \[\frac{c+a}{b}\] |
| Answer» C. \[\frac{1}{c}\] | |
| 1760. |
In a \[\Delta ABC,\] \[A:B:C\]. Then \[[a+b+c\sqrt{2}]\] is equal to [DCE 2001] |
| A. | 2b |
| B. | 2c |
| C. | 3b |
| D. | 3a |
| Answer» D. 3a | |
| 1761. |
In a triangle \[ABC\], right angled at C, the value of \[\tan A+\tan B\] is [Pb. CET 1990; Karnataka CET 1999; MP PET 2001] |
| A. | \[a+b\] |
| B. | \[\frac{{{a}^{2}}}{bc}\] |
| C. | \[\frac{{{b}^{2}}}{ac}\] |
| D. | \[\frac{{{c}^{2}}}{ab}\] |
| Answer» E. | |
| 1762. |
In a \[\Delta ABC,\]\[2a\sin \,\,\left( \frac{A-B+C}{2} \right)\] is equal to [IIT Screening 2000] |
| A. | \[{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\] |
| B. | \[{{c}^{2}}+{{a}^{2}}-{{b}^{2}}\] |
| C. | \[{{b}^{2}}-{{c}^{2}}-{{a}^{2}}\] |
| D. | \[{{c}^{2}}-{{a}^{2}}-{{b}^{2}}\] |
| Answer» C. \[{{b}^{2}}-{{c}^{2}}-{{a}^{2}}\] | |
| 1763. |
In a \[\Delta ABC\], if \[{{c}^{2}}+{{a}^{2}}-{{b}^{2}}=ac\], then \[\angle B=\] [MP PET 1983, 89, 90] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{3}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1764. |
The number of triangles ABC that can be formed with \[a=3,b=8\] and \[\sin A=\frac{5}{13}\]is [Roorkee Qualifying 1998] |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» B. 1 | |
| 1765. |
If in a triangle ABC, angle C is \[{{45}^{o}}\], then \[(1+\cot A)(1+\cot B)=\] [Kurukshetra CEE 1998] |
| A. | -1 |
| B. | 2 |
| C. | 3 |
| D. | \[1/\sqrt{2}\] |
| Answer» C. 3 | |
| 1766. |
If the lengths of the sides of a triangle are 3, 5, 7, then the largest angle of the triangle is [IIT Screening 1994; Kerala (Engg.) 2002] |
| A. | \[\pi /2\] |
| B. | \[5\pi /6\] |
| C. | \[2\pi /3\] |
| D. | \[3\pi /4\] |
| Answer» D. \[3\pi /4\] | |
| 1767. |
If in a right angled triangle the hypotenuse is four times as long as the perpendicular drawn to it from opposite vertex, then one of its acute angle is [MP PET 1998, 2004; UPSEAT 2002] |
| A. | \[{{15}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[{{45}^{o}}\] |
| D. | None of these |
| Answer» B. \[{{30}^{o}}\] | |
| 1768. |
If in a triangle ABC side \[a=(\sqrt{3}+1)\]cms and \[\angle B={{30}^{o}},\] \[\angle C={{45}^{o}}\], then the area of the triangle is [MP PET 1997] |
| A. | \[\frac{\sqrt{3}+1}{3}c{{m}^{2}}\] |
| B. | \[\frac{\sqrt{3}+1}{2}c{{m}^{2}}\] |
| C. | \[\frac{\sqrt{3}+1}{2\sqrt{2}}c{{m}^{2}}\] |
| D. | \[\frac{\sqrt{3}+1}{3\sqrt{2}}c{{m}^{2}}\] |
| Answer» C. \[\frac{\sqrt{3}+1}{2\sqrt{2}}c{{m}^{2}}\] | |
| 1769. |
In any triangle \[ABC,\frac{\tan \frac{A}{2}-\tan \frac{B}{2}}{\tan \frac{A}{2}+\tan \frac{B}{2}}=\] |
| A. | \[\frac{a-b}{a+b}\] |
| B. | \[\frac{a-b}{c}\] |
| C. | \[\frac{a-b}{a+b+c}\] |
| D. | \[\frac{c}{a+b}\] |
| Answer» C. \[\frac{a-b}{a+b+c}\] | |
| 1770. |
Sides of a triangle are \[2cm,\sqrt{6}\,cm\] and \[(\sqrt{3}+1)cm\]. The smallest angle of the triangle is |
| A. | \[{{30}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{60}^{o}}\] |
| D. | \[{{75}^{o}}\] |
| Answer» C. \[{{60}^{o}}\] | |
| 1771. |
If \[A={{30}^{o}},c=7\sqrt{3}\]and \[C={{90}^{o}}\]in \[\Delta ABC\], then a = |
| A. | \[7\sqrt{3}\] |
| B. | \[\frac{7\sqrt{3}}{2}\] |
| C. | \[\frac{7}{2}\] |
| D. | None of these |
| Answer» C. \[\frac{7}{2}\] | |
| 1772. |
In a \[\Delta ABC\], if \[{{b}^{2}}+{{c}^{2}}=3{{a}^{2}}\], then \[\cot B+\cot C-\cot A=\] [MP PET 1991] |
| A. | 1 |
| B. | \[\frac{ab}{4\Delta }\] |
| C. | 0 |
| D. | \[\frac{ac}{4\Delta }\] |
| Answer» D. \[\frac{ac}{4\Delta }\] | |
| 1773. |
The smallest angle of the triangle whose sides are \[6+\sqrt{12},\sqrt{48},\sqrt{24}\]is [EAMCET 1985] |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{4}\] |
| C. | \[\frac{\pi }{6}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1774. |
In a \[\Delta ABC\], \[a=5,b=4\]and \[\cos (A-B)=\frac{31}{32}\], then side c is equal to |
| A. | 6 |
| B. | 7 |
| C. | 9 |
| D. | None of these |
| Answer» B. 7 | |
| 1775. |
In a \[\Delta ABC\], \[b=2,C={{60}^{o}},c=\sqrt{6}\], then a = |
| A. | \[\sqrt{3}-1\] |
| B. | \[\sqrt{3}\] |
| C. | \[\sqrt{3}+1\] |
| D. | None of these |
| Answer» D. None of these | |
| 1776. |
In triangle ABC, \[A={{30}^{o}},b=8,a=6\], then \[B={{\sin }^{-1}}x\], where x = [Karnataka CET 1990] |
| A. | \[\frac{1}{2}\] |
| B. | \[\frac{1}{3}\] |
| C. | \[\frac{2}{3}\] |
| D. | 1 |
| Answer» D. 1 | |
| 1777. |
In a triangle \[ABC\], if \[a=2,B={{60}^{o}}\]and \[C={{75}^{o}}\], then b = [Karnataka CET 1992] |
| A. | \[\sqrt{3}\] |
| B. | \[\sqrt{6}\] |
| C. | \[\sqrt{9}\] |
| D. | \[1+\sqrt{2}\] |
| Answer» C. \[\sqrt{9}\] | |
| 1778. |
If in the \[\Delta ABC,AB=2BC\], then \[\tan \frac{B}{2}:\cot \left( \frac{C-A}{2} \right)\] |
| A. | 0.125694444444444 |
| B. | 0.0840277777777778 |
| C. | 0.0430555555555556 |
| D. | 0.04375 |
| Answer» E. | |
| 1779. |
ABC is a triangle such that \[\sin (2A+B)=\] \[\sin (C-A)=\] \[-\sin (B+2C)=\frac{1}{2}\]. If A, B and C are in A.P., then A, B and C are |
| A. | \[{{30}^{o}},{{60}^{o}},{{90}^{o}}\] |
| B. | \[{{45}^{o}},{{60}^{o}},{{75}^{o}}\] |
| C. | \[{{45}^{o}},{{45}^{o}},{{90}^{o}}\] |
| D. | \[{{60}^{o}},{{60}^{o}},{{60}^{o}}\] |
| Answer» C. \[{{45}^{o}},{{45}^{o}},{{90}^{o}}\] | |
| 1780. |
If in a \[\Delta ABC\], \[\cos 3A+\cos 3B+\cos 3C=1\], then one angle must be exactly equal to |
| A. | \[{{90}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{120}^{o}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1781. |
In \[\Delta ABC,\frac{\sin (A-B)}{\sin (A+B)}=\] [MP PET 1986] |
| A. | \[\frac{{{a}^{2}}-{{b}^{2}}}{{{c}^{2}}}\] |
| B. | \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}}\] |
| C. | \[\frac{{{c}^{2}}}{{{a}^{2}}-{{b}^{2}}}\] |
| D. | \[\frac{{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] |
| Answer» B. \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}}\] | |
| 1782. |
Point D, E are taken on the side BC of a triangle \[ABC\]such that \[BD=DE=EC\].If \[\angle BAD=x\], \[\angle DAE=y\], \[\angle EAC=z\], then the value of \[\frac{\sin (x+y)\sin (y+z)}{\sin x\sin z}=\] |
| A. | 1 |
| B. | 2 |
| C. | 4 |
| D. | None of these |
| Answer» D. None of these | |
| 1783. |
The perimeter of\[\Delta ABC\]is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is [IIT Screening 1992; DCE 1999] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\pi \] |
| Answer» B. \[\frac{\pi }{3}\] | |
| 1784. |
If in a triangle \[ABC,\] \[\cos A\cos B+\sin A\sin B\sin C=1,\] then the sides are proportional to |
| A. | 1: 1: \[\sqrt{2}\] |
| B. | \[1:\sqrt{2}:1\] |
| C. | \[\sqrt{2}:1:1\] |
| D. | None of these |
| Answer» B. \[1:\sqrt{2}:1\] | |
| 1785. |
If \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\]are in A. P. then which of the following are also in A.P. [ISM Dhandbad 1989] |
| A. | \[\sin A,\sin B,\sin C\] |
| B. | \[\tan A,\tan B,\tan C\] |
| C. | \[\cot A,\cot B,\cot C\] |
| D. | None of these |
| Answer» D. None of these | |
| 1786. |
If \[b=3,c=4\]and \[B=\frac{\pi }{3}\], then the number of triangle that can be constructed is [Roorkee 1992] |
| A. | Infinite |
| B. | Two |
| C. | One |
| D. | Nil |
| Answer» E. | |
| 1787. |
If \[a=2,b=3,c=5\]in \[\Delta ABC\], then C = [EAMCET 1984] |
| A. | \[\frac{\pi }{6}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{2}\] |
| D. | None of these |
| Answer» E. | |
| 1788. |
In a triangle\[ABC,\]\[a=4,b=3\], \[\angle A={{60}^{o}}\]. Then c is the root of the equation [Roorkee 1993] |
| A. | \[{{c}^{2}}-3c-7=0\] |
| B. | \[{{c}^{2}}+3c+7=0\] |
| C. | \[{{c}^{2}}-3c+7=0\] |
| D. | \[{{c}^{2}}+3c-7=0\] |
| Answer» B. \[{{c}^{2}}+3c+7=0\] | |
| 1789. |
In \[\Delta ABC,\frac{\sin B}{\sin (A+B)}=\] [MP PET 1989] |
| A. | \[\frac{b}{a+b}\] |
| B. | \[\frac{b}{c}\] |
| C. | \[\frac{c}{b}\] |
| D. | None of these |
| Answer» C. \[\frac{c}{b}\] | |
| 1790. |
In triangle\[ABC\]if \[A+C=2B\], then \[\frac{a+c}{\sqrt{{{a}^{2}}-ac+{{c}^{2}}}}\]is equal to [UPSEAT 1999] |
| A. | \[2\cos \frac{A-C}{2}\] |
| B. | \[\sin \frac{A+C}{2}\] |
| C. | \[\sin \frac{A}{2}\] |
| D. | None of these |
| Answer» B. \[\sin \frac{A+C}{2}\] | |
| 1791. |
Area of the triangle is \[10\sqrt{3}\]sq. cm, angle \[C={{60}^{o}}\]and its perimeter is 20 cm, then side c will be |
| A. | 5 |
| B. | 7 |
| C. | 8 |
| D. | 10 |
| Answer» C. 8 | |
| 1792. |
In a triangle \[ABC\] if \[2{{a}^{2}}{{b}^{2}}+2{{b}^{2}}{{c}^{2}}=\] \[{{a}^{4}}+{{b}^{4}}+{{c}^{4}}\], then angle B is equal to |
| A. | \[{{45}^{o}}\]or \[{{135}^{o}}\] |
| B. | \[{{135}^{o}}\]or \[{{120}^{o}}\] |
| C. | \[{{30}^{o}}\]or \[{{60}^{o}}\] |
| D. | None of these |
| Answer» B. \[{{135}^{o}}\]or \[{{120}^{o}}\] | |
| 1793. |
In \[\Delta ABC\], \[(b-c)\cot \frac{A}{2}+(c-a)\cot \frac{B}{2}+(a-b)\]\[\cot \frac{C}{2}\] is equal to [WB JEE 1989] |
| A. | 0 |
| B. | 1 |
| C. | \[\pm 1\] |
| D. | 2 |
| Answer» B. 1 | |
| 1794. |
If the sides of a triangle are p,q and\[\sqrt{{{p}^{2}}+pq+{{q}^{2}}}\], then the biggest angle is [Kerala (Engg.) 2005] |
| A. | \[\pi /2\] |
| B. | \[2\pi /3\] |
| C. | \[5\pi /4\] |
| D. | \[7\pi /4\] |
| E. | \[5\pi /3\] |
| Answer» C. \[5\pi /4\] | |
| 1795. |
If the angles \[A,B,C\]of a triangle are in A.P. and the sides \[a,b,c\] opposite to these angles are in G. P. then \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in [MP PET 1998] |
| A. | A. P. |
| B. | H. P. |
| C. | G. P. |
| D. | None of these |
| Answer» B. H. P. | |
| 1796. |
In \[\Delta \,ABC\], \[a=2cm,b=3cm\] and \[c=4cm\] , then angle A is [MNR 1973; MP PET 1984, 2002] |
| A. | \[{{\cos }^{-1}}\left( \frac{1}{24} \right)\] |
| B. | \[{{\cos }^{-1}}\left( \frac{11}{16} \right)\] |
| C. | \[{{\cos }^{-1}}\left( \frac{7}{8} \right)\] |
| D. | \[{{\cos }^{-1}}\left( -\frac{1}{4} \right)\] |
| Answer» C. \[{{\cos }^{-1}}\left( \frac{7}{8} \right)\] | |
| 1797. |
In triangle \[ABC\], \[(b+c)\cos A+(c+a)\cos B\] \[+(a+b)\cos C=\] [MP PET 1985] |
| A. | 0 |
| B. | 1 |
| C. | \[a+b+c\] |
| D. | \[2(a+b+c)\] |
| Answer» D. \[2(a+b+c)\] | |
| 1798. |
If in a triangle the angles are in A. P. and \[b:c=\sqrt{3}:\sqrt{2}\], then \[\angle A\]is equal to [IIT 1981; Kurukshetra CEE 1998; Pb. CET 1990] |
| A. | \[{{30}^{o}}\] |
| B. | \[{{60}^{o}}\] |
| C. | \[{{15}^{o}}\] |
| D. | \[{{75}^{o}}\] |
| Answer» E. | |
| 1799. |
If in a triangle, \[a{{\cos }^{2}}\frac{C}{2}+c{{\cos }^{2}}\frac{A}{2}=\frac{3b}{2},\]then its sides will be in [MP PET 1982; AMU 2000; AIEEE 2003] |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | A. G. |
| Answer» B. G. P. | |
| 1800. |
If in a triangle \[ABC\], \[b=\sqrt{3}\], \[c=1\] and \[B-C={{90}^{o}}\]then \[\angle A\] is [MP PET 1983] |
| A. | \[{{30}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{75}^{o}}\] |
| D. | \[{{15}^{o}}\] |
| Answer» B. \[{{45}^{o}}\] | |