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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.
| 1801. |
In a triangle \[ABC\], \[\frac{2\cos A}{a}+\frac{\cos B}{b}+\frac{2\cos C}{c}=\] \[\frac{a}{bc}+\frac{b}{ca}\], then the value of angle A is [IIT 1993] |
| A. | \[{{45}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[{{90}^{o}}\] |
| D. | \[{{60}^{o}}\] |
| Answer» D. \[{{60}^{o}}\] | |
| 1802. |
If the angles of a triangle are in the ratio 1: 2: 3, then their corresponding sides are in the ratio [MP PET 1993; BIT Ranchi 1992; Pb. CET 1990] |
| A. | 0.0430902777777778 |
| B. | \[1:\sqrt{3}:2\] |
| C. | \[\sqrt{2}:\sqrt{3}:3\] |
| D. | \[1:\sqrt{3}:3\] |
| Answer» C. \[\sqrt{2}:\sqrt{3}:3\] | |
| 1803. |
If the sides of a triangle are in A. P., then the cotangent of its half the angles will be in [MP PET 1993] |
| A. | H. P. |
| B. | G. P. |
| C. | A. P. |
| D. | No particular order |
| Answer» D. No particular order | |
| 1804. |
If in \[\Delta \,ABC\], \[2{{b}^{2}}={{a}^{2}}+{{c}^{2}},\]then \[\frac{\sin 3B}{\sin B}=\] [UPSEAT 1999] |
| A. | \[\frac{{{c}^{2}}-{{a}^{2}}}{2ca}\] |
| B. | \[\frac{{{c}^{2}}-{{a}^{2}}}{ca}\] |
| C. | \[{{\left( \frac{{{c}^{2}}-{{a}^{2}}}{ca} \right)}^{2}}\] |
| D. | \[{{\left( \frac{{{c}^{2}}-{{a}^{2}}}{2ca} \right)}^{2}}\] |
| Answer» E. | |
| 1805. |
In\[\Delta \,ABC\],\[({{b}^{2}}-{{c}^{2}})\cot A+({{c}^{2}}-{{a}^{2}})\cot B+({{a}^{2}}-{{b}^{2}})\cot C=\] |
| A. | 0 |
| B. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| C. | \[2\,({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] |
| D. | \[\frac{1}{2abc}\] |
| Answer» B. \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] | |
| 1806. |
In \[\Delta \,ABC\], \[\frac{\cos \frac{1}{2}(B-C)}{\sin \frac{1}{2}A}=\] [MP PET 1993; Roorkee 1973] |
| A. | \[\frac{b-c}{a}\] |
| B. | \[\frac{b+c}{a}\] |
| C. | \[\frac{a}{b-c}\] |
| D. | \[\frac{a}{b+c}\] |
| Answer» C. \[\frac{a}{b-c}\] | |
| 1807. |
If the angles of a triangle \[ABC\]be in A.P., then |
| A. | \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}-ab\] |
| B. | \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-ac\] |
| C. | \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}-ac\] |
| D. | \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] |
| Answer» C. \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}-ac\] | |
| 1808. |
In triangle \[ABC,\]\[\frac{1+\cos (A-B)\cos C}{1+\cos (A-C)\cos B}=\] |
| A. | \[\frac{a-b}{a-c}\] |
| B. | \[\frac{a+b}{a+c}\] |
| C. | \[\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}-{{c}^{2}}}\] |
| D. | \[\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}+{{c}^{2}}}\] |
| Answer» E. | |
| 1809. |
In \[\Delta ABC\], \[\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\] |
| A. | \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{abc}\] |
| B. | \[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{2abc}\] |
| C. | \[\frac{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{abc}\] |
| D. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| Answer» C. \[\frac{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{abc}\] | |
| 1810. |
In \[\Delta ABC\], \[c\cos (A-\alpha )+a\cos (C+\alpha )=\] |
| A. | \[a\cos \alpha \] |
| B. | \[b\cos \alpha \] |
| C. | \[c\cos \alpha \] |
| D. | \[2b\cos \alpha \] |
| Answer» C. \[c\cos \alpha \] | |
| 1811. |
In \[\Delta ABC\], if \[\angle C={{90}^{o}}\],\[\angle A={{30}^{o}}\], \[c=20\], then the values of a and b are |
| A. | 10, 10 |
| B. | \[10,\,10\sqrt{3}\] |
| C. | \[5,\,\,5\sqrt{3}\] |
| D. | \[8,\,\,8\sqrt{3}\] |
| Answer» C. \[5,\,\,5\sqrt{3}\] | |
| 1812. |
In a \[\Delta ABC\],side b is equal to [MP PET 1984, 92] |
| A. | \[c\cos A+a\cos C\] |
| B. | \[a\cos B+b\cos A\] |
| C. | \[b\cos C+c\cos B\] |
| D. | None of these |
| Answer» B. \[a\cos B+b\cos A\] | |
| 1813. |
If the sides of a right angled triangle be in A. P. , then their ratio will be |
| A. | 0.0430902777777778 |
| B. | 0.085462962962963 |
| C. | 0.127835648148148 |
| D. | 0.170208333333333 |
| Answer» D. 0.170208333333333 | |
| 1814. |
If the lengths of the sides of a triangle be \[7,4\sqrt{3}\] and \[\sqrt{13}\]cm, then the smallest angle is [MNR 1985] |
| A. | \[{{15}^{o}}\] |
| B. | \[{{30}^{o}}\] |
| C. | \[60{}^\circ \] |
| D. | \[{{45}^{o}}\] |
| Answer» C. \[60{}^\circ \] | |
| 1815. |
In a \[\Delta ABC\], if \[2s=a+b+c\]and \[(s-b)(s-c)=\] \[x{{\sin }^{2}}\frac{A}{2},\] then x = [MP PET 1992] |
| A. | bc |
| B. | ca |
| C. | ab |
| D. | abc |
| Answer» B. ca | |
| 1816. |
If the sides of a triangle are in the ratio \[2:\sqrt{6}:(\sqrt{3}+1)\], then the largest angle of the triangle will be [MP PET 1990] |
| A. | \[{{60}^{o}}\] |
| B. | \[{{75}^{o}}\] |
| C. | \[{{90}^{o}}\] |
| D. | \[{{120}^{o}}\] |
| Answer» C. \[{{90}^{o}}\] | |
| 1817. |
In \[\Delta ABC\], if \[a=3,b=4,c=5\], then \[\sin 2B=\] [MP PET 1983] |
| A. | 44320 |
| B. | 43891 |
| C. | 24/25 |
| D. | 18264 |
| Answer» D. 18264 | |
| 1818. |
In triangle \[ABC\]if \[a,b,c\]are in A. P., then the value of \[\frac{\sin \frac{A}{2}\sin \frac{C}{2}}{\sin \frac{B}{2}}=\] [AMU 1995] |
| A. | 1 |
| B. | 44228 |
| C. | 2 |
| D. | -1 |
| Answer» C. 2 | |
| 1819. |
In \[\Delta ABC\], if \[\tan \frac{A}{2}\tan \frac{C}{2}=\frac{1}{2},\]then \[a,b,c\]are in |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | None of these |
| Answer» E. | |
| 1820. |
If the angles of a triangle be in the ratio 1 : 2 : 7, then the ratio of its greatest side to the least side is |
| A. | \[1:2\] |
| B. | 0.0840277777777778 |
| C. | \[(\sqrt{5}+1):(\sqrt{5}-1)\] |
| D. | \[(\sqrt{5}-1):(\sqrt{5}+1)\] |
| Answer» D. \[(\sqrt{5}-1):(\sqrt{5}+1)\] | |
| 1821. |
If \[{{\cos }^{2}}A+{{\cos }^{2}}C={{\sin }^{2}}B,\]then \[\Delta ABC\]is [MP PET 1991] |
| A. | Equilateral |
| B. | Right angled |
| C. | Isosceles |
| D. | None of these |
| Answer» C. Isosceles | |
| 1822. |
In \[\Delta ABC,\] \[\text{cosec }A(\sin B\cos C+\cos B\sin C)=\] [MP PET 1986, 1995; Pb. CET 1990, 94] |
| A. | \[c/a\] |
| B. | \[a/c\] |
| C. | 1 |
| D. | \[c/ab\] |
| Answer» D. \[c/ab\] | |
| 1823. |
In\[\Delta ABC,\] if \[2(bc\cos A+ca\cos B+ab\cos C)=\] |
| A. | 0 |
| B. | \[a+b+c\] |
| C. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1824. |
If in a triangle \[ABC,\]\[(s-a)(s-b)=s\,\,(s-c)\], then angle C is equal to [MP PET 1986] |
| A. | \[{{90}^{o}}\] |
| B. | \[{{45}^{o}}\] |
| C. | \[{{30}^{o}}\] |
| D. | \[{{60}^{o}}\] |
| Answer» B. \[{{45}^{o}}\] | |
| 1825. |
In \[\Delta ABC,\] if \[(a+b+c)(a-b+c)\]=3ac, then [AMU 1996] |
| A. | \[\angle B={{60}^{o}}\] |
| B. | \[\angle B={{30}^{o}}\] |
| C. | \[\angle C={{60}^{o}}\] |
| D. | \[\angle A+\angle C={{90}^{o}}\] |
| Answer» B. \[\angle B={{30}^{o}}\] | |
| 1826. |
In\[\Delta ABC,\]a\[\sin (B-C)+b\sin (C-A)+c\sin (A-B)=\] [ISM Dhanbad 1973] |
| A. | 0 |
| B. | \[a+b+c\] |
| C. | \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] |
| D. | \[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] |
| Answer» B. \[a+b+c\] | |
| 1827. |
In\[\Delta ABC,\]if \[\cot A,\cot B,\cot C\]be in A. P., then \[{{a}^{2}},\text{ }{{b}^{2}},\text{ }{{c}^{2}}\] are in [MP PET 1997] |
| A. | H. P. |
| B. | G. P. |
| C. | A. P. |
| D. | None of these |
| Answer» D. None of these | |
| 1828. |
If in a triangle \[ABC,a=5,b=4,A=\frac{\pi }{2}+B\], then C [Kerala (Engg.) 2005] |
| A. | Is \[{{\tan }^{-1}}\left( \frac{1}{9} \right)\] |
| B. | Is \[{{\tan }^{-1}}\frac{1}{40}\] |
| C. | Cannot be evaluated |
| D. | Is\[2{{\tan }^{-1}}\left( 1/9 \right)\] |
| E. | Is \[2{{\tan }^{-1}}\frac{1}{40}\] |
| Answer» E. Is \[2{{\tan }^{-1}}\frac{1}{40}\] | |
| 1829. |
\[ABC\] is a right angled isosceles triangle with \[\angle B={{90}^{o}}\]. If D is a point on \[AB\] so that \[\angle DCB={{15}^{o}}\] and if \[AD=35cm\], then \[CD=\] [Kerala (Engg.) 2005] |
| A. | \[35\sqrt{2}\]cm |
| B. | \[70\sqrt{2}cm\] |
| C. | \[\frac{35\sqrt{3}}{2}cm\] |
| D. | \[35\sqrt{6}\]cm |
| E. | \[\frac{35\sqrt{2}}{2}cm\] |
| Answer» B. \[70\sqrt{2}cm\] | |
| 1830. |
Which of the following is true in a triangle ABC [IIT Screening 2005] |
| A. | \[(b+c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\] |
| B. | \[(b+c)\cos \frac{A}{2}=2a\sin \frac{B-C}{2}\] |
| C. | \[(b-c)\cos \frac{A}{2}=a\sin \frac{B-C}{2}\] |
| D. | \[(b-c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\] |
| Answer» D. \[(b-c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\] | |
| 1831. |
If the line segment joining the points \[A(a,\,b)\] and \[B(c,\,d)\] subtends an angle \[\theta \] at the origin, then \[\cos \theta \] is equal to [IIT 1961] |
| A. | \[\frac{ab+cd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] |
| B. | \[\frac{ac+bd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] |
| C. | \[\frac{ac-bd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] |
| D. | None of these |
| Answer» C. \[\frac{ac-bd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] | |
| 1832. |
If a, b and c are the sides of a triangle such that \[{{a}^{4}}+{{b}^{4}}+{{c}^{4}}=2{{c}^{2}}({{a}^{2}}+{{b}^{2}})\] then the angles opposite to the side C is [J & K 2005] |
| A. | \[45{}^\circ \] or \[135{}^\circ \] |
| B. | \[30{}^\circ \] or \[100{}^\circ \] |
| C. | \[50{}^\circ \] or \[100{}^\circ \] |
| D. | \[60{}^\circ \] or \[120{}^\circ \] |
| Answer» B. \[30{}^\circ \] or \[100{}^\circ \] | |
| 1833. |
If in a \[\Delta ABC\], the altitudes from the vertices A, B, C on opposite sides are in H.P. then \[\sin A,\,\sin B,\sin C\] are in [AIEEE 2005] |
| A. | A.G.P. |
| B. | H.P. |
| C. | G.P. |
| D. | A.P. |
| Answer» E. | |
| 1834. |
In \[\Delta ABC,\]\[1-\tan \frac{A}{2}\tan \frac{B}{2}=\] [Roorkee 1973] |
| A. | \[\frac{2c}{a+b+c}\] |
| B. | \[\frac{a}{a+b+c}\] |
| C. | \[\frac{2}{a+b+c}\] |
| D. | \[\frac{4a}{a+b+c}\] |
| Answer» B. \[\frac{a}{a+b+c}\] | |
| 1835. |
The area of triangle \[ABC,\] in which \[a=1,\ b=2\], \[\angle C=60{}^\circ \]is [MP PET 2004] |
| A. | \[\frac{1}{2}\] |
| B. | \[\sqrt{3}\] |
| C. | \[\frac{\sqrt{3}}{2}\] |
| D. | \[\frac{3}{2}\] |
| Answer» D. \[\frac{3}{2}\] | |
| 1836. |
In a triangle \[ABC\], if \[b+c=2a\] and \[\angle A=60{}^\circ ,\] then \[\Delta ABC\] is [MP PET 2004] |
| A. | Scalene |
| B. | Equilateral |
| C. | Isosecles |
| D. | Right angled |
| Answer» C. Isosecles | |
| 1837. |
In \[\Delta ABC,\]\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=ac+ab\sqrt{3},\]then triangle is [MP PET 2004] |
| A. | Equilateral |
| B. | Isosceles |
| C. | Right angled |
| D. | None of these |
| Answer» D. None of these | |
| 1838. |
In a \[\Delta ABC,\,\,{{a}^{2}}\sin \,\,2C+{{c}^{2}}\sin 2A=\] [EAMCET 2001] |
| A. | \[\Delta \] |
| B. | \[2\Delta \] |
| C. | \[3\Delta \] |
| D. | \[4\Delta \] |
| Answer» E. | |
| 1839. |
If in a triangle ABC, a, b, c and angle A is given and \[c\sin A |
| A. | \[{{b}_{1}}+{{b}_{2}}=2c\cos A\] |
| B. | \[{{b}_{1}}+{{b}_{2}}=c\cos A\] |
| C. | \[{{b}_{1}}+{{b}_{2}}=3c\cos A\] |
| D. | \[{{b}_{1}}+{{b}_{2}}=4c\sin A\] |
| Answer» B. \[{{b}_{1}}+{{b}_{2}}=c\cos A\] | |
| 1840. |
If a triangle \[PQR\], \[\sin P,\ \sin Q,\ \sin R\]are in A.P., then [IIT 1998] |
| A. | The altitudes are in A.P. |
| B. | The altitudes are in H.P. |
| C. | The medians are in G.P. |
| D. | The medians are in A.P. |
| Answer» C. The medians are in G.P. | |
| 1841. |
In a \[\Delta ABC,\]if \[\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)},\]then \[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\] are in [Pb. CET 2001; Karnataka CET 1999] |
| A. | A.P. |
| B. | G.P. |
| C. | H.P. |
| D. | None of these |
| Answer» B. G.P. | |
| 1842. |
In a triangle \[PQR\], \[\angle R=\frac{\pi }{2}.\]If \[\tan \left( \frac{P}{2} \right)\]and \[\tan \left( \frac{Q}{2} \right)\]are the roots of the equation \[a{{x}^{2}}+bx+c=0(a\ne 0).\] then [IIT 1999; MP PET 2000; AIEEE 2005] |
| A. | \[a+b=c\] |
| B. | \[b+c=a\] |
| C. | \[a+c=b\] |
| D. | \[b=c\] |
| Answer» B. \[b+c=a\] | |
| 1843. |
If the sides of a triangle are \[3,\ 5,\ 7,\]then [MP PET 1996] |
| A. | All its angles are acute |
| B. | One angle is obtuse |
| C. | Triangle is right angled |
| D. | None of these |
| Answer» C. Triangle is right angled | |
| 1844. |
If \[y=x\tan \frac{\alpha +\beta }{2}\], then \[\tan A+\tan B+\tan C=\] |
| A. | \[\frac{a+b+c}{abc}\] |
| B. | \[0\] |
| C. | \[\tan A\tan B\tan C\] |
| D. | \[\tan A\tan B+\tan B\tan C+\tan C\tan A\] |
| Answer» D. \[\tan A\tan B+\tan B\tan C+\tan C\tan A\] | |
| 1845. |
In\[\Delta ABC,\] if \[\cos A+\cos C=4{{\sin }^{2}}\frac{1}{2}B,\] then \[a,b,c\] are in |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | None of these |
| Answer» B. G. P. | |
| 1846. |
If in a triangle \[ABC\], \[2\cos A=\sin B\,\text{cosec}\,C,\] then [MP PET 1996] |
| A. | \[a=b\] |
| B. | \[b=c\] |
| C. | \[c=a\] |
| D. | \[2a=bc\] |
| Answer» D. \[2a=bc\] | |
| 1847. |
In a \[\Delta ABC\], \[a,\ b,\ A\]are given and \[{{c}_{1}},\ {{c}_{2}}\]are two values of the third side c. The sum of the areas of two triangles with sides \[a,\ b,\ {{c}_{1}}\] and \[a,b,\ {{c}_{2}}\] is |
| A. | \[\frac{1}{2}{{b}^{2}}\sin 2A\] |
| B. | \[\frac{1}{2}{{a}^{2}}\sin 2A\] |
| C. | \[{{b}^{2}}\sin 2A\] |
| D. | None of these |
| Answer» B. \[\frac{1}{2}{{a}^{2}}\sin 2A\] | |
| 1848. |
In a \[\Delta ABC\] \[a,\ c,A\]are given and \[{{b}_{1}},\ {{b}_{2}}\]are two values of the third side b such that \[{{b}_{2}}=2{{b}_{1}}\]. Then \[\sin A=\] |
| A. | \[\sqrt{\frac{9{{a}^{2}}-{{c}^{2}}}{8{{a}^{2}}}}\] |
| B. | \[\sqrt{\frac{9{{a}^{2}}-{{c}^{2}}}{8{{c}^{2}}}}\] |
| C. | \[\sqrt{\frac{9{{a}^{2}}+{{c}^{2}}}{8{{a}^{2}}}}\] |
| D. | None of these |
| Answer» C. \[\sqrt{\frac{9{{a}^{2}}+{{c}^{2}}}{8{{a}^{2}}}}\] | |
| 1849. |
In triangle ABC and DEF, AB = DE, AC = EF and \[\angle A=2\angle E\]. Two triangles will have the same area, if angle A is equal to |
| A. | \[\frac{\pi }{3}\] |
| B. | \[\frac{\pi }{2}\] |
| C. | \[\frac{2\pi }{3}\] |
| D. | \[\frac{5\pi }{6}\] |
| Answer» D. \[\frac{5\pi }{6}\] | |
| 1850. |
We are given b, c and \[\sin B\] such that B is acute and \[b |
| A. | No triangle is possible |
| B. | One triangle is possible |
| C. | Two triangles are possible |
| D. | A right angled triangle is possible |
| Answer» B. One triangle is possible | |