-
question_answer1)
In a triangle ABC, \[a=5,b=7\] and \[\sin A=\frac{3}{4}\] how many such triangles are possible [Roorkee 1990]
A)
1 done
clear
B)
0 done
clear
C)
2 done
clear
D)
Infinite done
clear
View Solution play_arrow
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question_answer2)
If in a triangle \[ABC,\]\[(s-a)(s-b)=s\,\,(s-c)\], then angle C is equal to [MP PET 1986]
A)
\[{{90}^{o}}\] done
clear
B)
\[{{45}^{o}}\] done
clear
C)
\[{{30}^{o}}\] done
clear
D)
\[{{60}^{o}}\] done
clear
View Solution play_arrow
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question_answer3)
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 done
clear
B)
ca done
clear
C)
ab done
clear
D)
abc done
clear
View Solution play_arrow
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question_answer4)
If the angles of a triangle \[ABC\]be in A.P., then
A)
\[{{c}^{2}}={{a}^{2}}+{{b}^{2}}-ab\] done
clear
B)
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-ac\] done
clear
C)
\[{{a}^{2}}={{b}^{2}}+{{c}^{2}}-ac\] done
clear
D)
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\] done
clear
View Solution play_arrow
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question_answer5)
In triangle \[ABC\], \[(b+c)\cos A+(c+a)\cos B\] \[+(a+b)\cos C=\] [MP PET 1985]
A)
0 done
clear
B)
1 done
clear
C)
\[a+b+c\] done
clear
D)
\[2(a+b+c)\] done
clear
View Solution play_arrow
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question_answer6)
In \[\Delta ABC,\frac{\sin B}{\sin (A+B)}=\] [MP PET 1989]
A)
\[\frac{b}{a+b}\] done
clear
B)
\[\frac{b}{c}\] done
clear
C)
\[\frac{c}{b}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer7)
In \[\Delta ABC,\frac{\sin (A-B)}{\sin (A+B)}=\] [MP PET 1986]
A)
\[\frac{{{a}^{2}}-{{b}^{2}}}{{{c}^{2}}}\] done
clear
B)
\[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}}\] done
clear
C)
\[\frac{{{c}^{2}}}{{{a}^{2}}-{{b}^{2}}}\] done
clear
D)
\[\frac{{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}\] done
clear
View Solution play_arrow
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question_answer8)
In a \[\Delta ABC\], if \[{{b}^{2}}+{{c}^{2}}=3{{a}^{2}}\], then \[\cot B+\cot C-\cot A=\] [MP PET 1991]
A)
1 done
clear
B)
\[\frac{ab}{4\Delta }\] done
clear
C)
0 done
clear
D)
\[\frac{ac}{4\Delta }\] done
clear
View Solution play_arrow
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question_answer9)
In a \[\Delta ABC\], if \[{{c}^{2}}+{{a}^{2}}-{{b}^{2}}=ac\], then \[\angle B=\] [MP PET 1983, 89, 90]
A)
\[\frac{\pi }{6}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{3}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer10)
In \[\Delta ABC,\left( \cot \frac{A}{2}+\cot \frac{B}{2} \right)\,\left( a{{\sin }^{2}}\frac{B}{2}+b{{\sin }^{2}}\frac{A}{2} \right)\]= [Roorkee 1988]
A)
\[\cot C\] done
clear
B)
\[c\cot C\] done
clear
C)
\[\cot \frac{C}{2}\] done
clear
D)
\[c\cot \frac{C}{2}\] done
clear
View Solution play_arrow
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question_answer11)
In \[\Delta ABC,\]if \[{{\sin }^{2}}\frac{A}{2},{{\sin }^{2}}\frac{B}{2},{{\sin }^{2}}\frac{C}{2}\] be in H. P. then a, b, c will be in
A)
A. P. done
clear
B)
G. P. done
clear
C)
H. P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer12)
In \[\Delta ABC,{{(a-b)}^{2}}{{\cos }^{2}}\frac{C}{2}+{{(a+b)}^{2}}{{\sin }^{2}}\frac{C}{2}=\]
A)
\[{{a}^{2}}\] done
clear
B)
\[{{b}^{2}}\] done
clear
C)
\[{{c}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
In \[\Delta ABC,\]if \[a=16,b=24\] and \[c=20,\]then \[\cos \frac{B}{2}=\] [MP PET 1988]
A)
3/4 done
clear
B)
1/4 done
clear
C)
1/2 done
clear
D)
1/3 done
clear
View Solution play_arrow
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question_answer14)
In\[\Delta ABC,\] if \[\cos A+\cos C=4{{\sin }^{2}}\frac{1}{2}B,\] then \[a,b,c\] are in
A)
A. P. done
clear
B)
G. P. done
clear
C)
H. P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer15)
In \[\Delta ABC,\]\[1-\tan \frac{A}{2}\tan \frac{B}{2}=\] [Roorkee 1973]
A)
\[\frac{2c}{a+b+c}\] done
clear
B)
\[\frac{a}{a+b+c}\] done
clear
C)
\[\frac{2}{a+b+c}\] done
clear
D)
\[\frac{4a}{a+b+c}\] done
clear
View Solution play_arrow
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question_answer16)
In \[\Delta ABC,\]\[{{b}^{2}}\cos 2A-{{a}^{2}}\cos 2B=\]
A)
\[{{b}^{2}}-{{a}^{2}}\] done
clear
B)
\[{{b}^{2}}-{{c}^{2}}\] done
clear
C)
\[{{c}^{2}}-{{a}^{2}}\] done
clear
D)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
View Solution play_arrow
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question_answer17)
In\[\Delta ABC,\]a\[\sin (B-C)+b\sin (C-A)+c\sin (A-B)=\] [ISM Dhanbad 1973]
A)
0 done
clear
B)
\[a+b+c\] done
clear
C)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
D)
\[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done
clear
View Solution play_arrow
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question_answer18)
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. done
clear
B)
G. P. done
clear
C)
A. P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer19)
In \[\Delta ABC,\] if \[(a+b+c)(a-b+c)\]=3ac, then [AMU 1996]
A)
\[\angle B={{60}^{o}}\] done
clear
B)
\[\angle B={{30}^{o}}\] done
clear
C)
\[\angle C={{60}^{o}}\] done
clear
D)
\[\angle A+\angle C={{90}^{o}}\] done
clear
View Solution play_arrow
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question_answer20)
In\[\Delta ABC,\] if \[2(bc\cos A+ca\cos B+ab\cos C)=\]
A)
0 done
clear
B)
\[a+b+c\] done
clear
C)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer21)
In \[\Delta ABC,\] \[\text{cosec }A(\sin B\cos C+\cos B\sin C)=\] [MP PET 1986, 1995; Pb. CET 1990, 94]
A)
\[c/a\] done
clear
B)
\[a/c\] done
clear
C)
1 done
clear
D)
\[c/ab\] done
clear
View Solution play_arrow
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question_answer22)
If \[{{\cos }^{2}}A+{{\cos }^{2}}C={{\sin }^{2}}B,\]then \[\Delta ABC\]is [MP PET 1991]
A)
Equilateral done
clear
B)
Right angled done
clear
C)
Isosceles done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer23)
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)
\[1:2\] done
clear
B)
2 :1 done
clear
C)
\[(\sqrt{5}+1):(\sqrt{5}-1)\] done
clear
D)
\[(\sqrt{5}-1):(\sqrt{5}+1)\] done
clear
View Solution play_arrow
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question_answer24)
If in a triangle \[ABC\], \[\angle C={{60}^{o}},\]then \[\frac{1}{a+c}+\frac{1}{b+c}=\] [IIT 1975]
A)
\[\frac{1}{a+b+c}\] done
clear
B)
\[\frac{2}{a+b+c}\] done
clear
C)
\[\frac{3}{a+b+c}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
In \[\Delta ABC\], if \[\tan \frac{A}{2}\tan \frac{C}{2}=\frac{1}{2},\]then \[a,b,c\]are in
A)
A. P. done
clear
B)
G. P. done
clear
C)
H. P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer26)
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 done
clear
B)
1/2 done
clear
C)
2 done
clear
D)
-1 done
clear
View Solution play_arrow
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question_answer27)
If \[\tan \frac{B-C}{2}=x\cot \frac{A}{2},\]then \[x=\] [MP PET 1992, 2002]
A)
\[\frac{c-a}{c+a}\] done
clear
B)
\[\frac{a-b}{a+b}\] done
clear
C)
\[\frac{b-c}{b+c}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer28)
In \[\Delta ABC\], if \[a=3,b=4,c=5\], then \[\sin 2B=\] [MP PET 1983]
A)
4/5 done
clear
B)
3/20 done
clear
C)
24/25 done
clear
D)
1/50 done
clear
View Solution play_arrow
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question_answer29)
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)
\[{{60}^{o}}\] done
clear
B)
\[{{75}^{o}}\] done
clear
C)
\[{{90}^{o}}\] done
clear
D)
\[{{120}^{o}}\] done
clear
View Solution play_arrow
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question_answer30)
In a triangle ABC, \[{{a}^{3}}\cos (B-C)+{{b}^{3}}\cos (C-A)+{{c}^{3}}\cos (A-B)=\] [Kerala (Engg.) 2002]
A)
\[abc\] done
clear
B)
\[3abc\] done
clear
C)
\[a+b+c\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer31)
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)
\[{{15}^{o}}\] done
clear
B)
\[{{30}^{o}}\] done
clear
C)
\[60{}^\circ \] done
clear
D)
\[{{45}^{o}}\] done
clear
View Solution play_arrow
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question_answer32)
If the sides of a right angled triangle be in A. P. , then their ratio will be
A)
1: 2: 3 done
clear
B)
2 : 3 : 4 done
clear
C)
3: 4: 5 done
clear
D)
4 : 5 : 6 done
clear
View Solution play_arrow
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question_answer33)
In a \[\Delta ABC\], if \[\angle C={{30}^{o}}\], \[a=47cm\]and \[b=94\]cm, then the triangle is [MP PET 1986]
A)
Right angled done
clear
B)
Right angled isosceles done
clear
C)
Isosceles done
clear
D)
Obtuse angled done
clear
View Solution play_arrow
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question_answer34)
In a \[\Delta ABC\],side b is equal to [MP PET 1984, 92]
A)
\[c\cos A+a\cos C\] done
clear
B)
\[a\cos B+b\cos A\] done
clear
C)
\[b\cos C+c\cos B\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer35)
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 done
clear
B)
\[10,\,10\sqrt{3}\] done
clear
C)
\[5,\,\,5\sqrt{3}\] done
clear
D)
\[8,\,\,8\sqrt{3}\] done
clear
View Solution play_arrow
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question_answer36)
In \[\Delta ABC\], \[c\cos (A-\alpha )+a\cos (C+\alpha )=\]
A)
\[a\cos \alpha \] done
clear
B)
\[b\cos \alpha \] done
clear
C)
\[c\cos \alpha \] done
clear
D)
\[2b\cos \alpha \] done
clear
View Solution play_arrow
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question_answer37)
In \[\Delta ABC\], \[\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\]
A)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{abc}\] done
clear
B)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{2abc}\] done
clear
C)
\[\frac{2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})}{abc}\] done
clear
D)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
View Solution play_arrow
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question_answer38)
In \[\Delta ABC\], \[{{a}^{2}}({{\cos }^{2}}B-{{\cos }^{2}}C)+\] \[{{b}^{2}}({{\cos }^{2}}C-{{\cos }^{2}}A)+\] \[{{c}^{2}}({{\cos }^{2}}A-{{\cos }^{2}}B)=\]
A)
0 done
clear
B)
1 done
clear
C)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
D)
\[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done
clear
View Solution play_arrow
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question_answer39)
In triangle \[ABC,\]\[\frac{1+\cos (A-B)\cos C}{1+\cos (A-C)\cos B}=\]
A)
\[\frac{a-b}{a-c}\] done
clear
B)
\[\frac{a+b}{a+c}\] done
clear
C)
\[\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}-{{c}^{2}}}\] done
clear
D)
\[\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}+{{c}^{2}}}\] done
clear
View Solution play_arrow
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question_answer40)
In \[\Delta \,ABC\], \[\frac{\cos \frac{1}{2}(B-C)}{\sin \frac{1}{2}A}=\] [MP PET 1993; Roorkee 1973]
A)
\[\frac{b-c}{a}\] done
clear
B)
\[\frac{b+c}{a}\] done
clear
C)
\[\frac{a}{b-c}\] done
clear
D)
\[\frac{a}{b+c}\] done
clear
View Solution play_arrow
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question_answer41)
In\[\Delta \,ABC\],\[({{b}^{2}}-{{c}^{2}})\cot A+({{c}^{2}}-{{a}^{2}})\cot B+({{a}^{2}}-{{b}^{2}})\cot C=\]
A)
0 done
clear
B)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\] done
clear
C)
\[2\,({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done
clear
D)
\[\frac{1}{2abc}\] done
clear
View Solution play_arrow
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question_answer42)
If in \[\Delta \,ABC\], \[2{{b}^{2}}={{a}^{2}}+{{c}^{2}},\]then \[\frac{\sin 3B}{\sin B}=\] [UPSEAT 1999]
A)
\[\frac{{{c}^{2}}-{{a}^{2}}}{2ca}\] done
clear
B)
\[\frac{{{c}^{2}}-{{a}^{2}}}{ca}\] done
clear
C)
\[{{\left( \frac{{{c}^{2}}-{{a}^{2}}}{ca} \right)}^{2}}\] done
clear
D)
\[{{\left( \frac{{{c}^{2}}-{{a}^{2}}}{2ca} \right)}^{2}}\] done
clear
View Solution play_arrow
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question_answer43)
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. done
clear
B)
G. P. done
clear
C)
A. P. done
clear
D)
No particular order done
clear
View Solution play_arrow
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question_answer44)
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)
1 : 2 : 3 done
clear
B)
\[1:\sqrt{3}:2\] done
clear
C)
\[\sqrt{2}:\sqrt{3}:3\] done
clear
D)
\[1:\sqrt{3}:3\] done
clear
View Solution play_arrow
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question_answer45)
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)
\[{{45}^{o}}\] done
clear
B)
\[{{30}^{o}}\] done
clear
C)
\[{{90}^{o}}\] done
clear
D)
\[{{60}^{o}}\] done
clear
View Solution play_arrow
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question_answer46)
If \[a=9,b=8\]and \[c=x\]satisfies \[3\cos C=2,\]then [MP PET 1984]
A)
\[x=5\] done
clear
B)
\[x=6\] done
clear
C)
\[x=4\] done
clear
D)
\[x=7\] done
clear
View Solution play_arrow
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question_answer47)
If in a triangle \[ABC\], \[b=\sqrt{3}\], \[c=1\] and \[B-C={{90}^{o}}\]then \[\angle A\] is [MP PET 1983]
A)
\[{{30}^{o}}\] done
clear
B)
\[{{45}^{o}}\] done
clear
C)
\[{{75}^{o}}\] done
clear
D)
\[{{15}^{o}}\] done
clear
View Solution play_arrow
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question_answer48)
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. done
clear
B)
G. P. done
clear
C)
H. P. done
clear
D)
A. G. done
clear
View Solution play_arrow
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question_answer49)
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)
\[{{30}^{o}}\] done
clear
B)
\[{{60}^{o}}\] done
clear
C)
\[{{15}^{o}}\] done
clear
D)
\[{{75}^{o}}\] done
clear
View Solution play_arrow
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question_answer50)
In \[\Delta \,ABC\], \[a=2cm,b=3cm\] and \[c=4cm\] , then angle A is [MNR 1973; MP PET 1984, 2002]
A)
\[{{\cos }^{-1}}\left( \frac{1}{24} \right)\] done
clear
B)
\[{{\cos }^{-1}}\left( \frac{11}{16} \right)\] done
clear
C)
\[{{\cos }^{-1}}\left( \frac{7}{8} \right)\] done
clear
D)
\[{{\cos }^{-1}}\left( -\frac{1}{4} \right)\] done
clear
View Solution play_arrow
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question_answer51)
\[\cot \frac{A+B}{2}\]. \[\tan \frac{A-B}{2}=\] [Roorkee 1975]
A)
\[\frac{a+b}{a-b}\] done
clear
B)
\[\frac{a-b}{a+b}\] done
clear
C)
\[\frac{a}{a+b}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer52)
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. done
clear
B)
H. P. done
clear
C)
G. P. done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer53)
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)
\[\pi /2\] done
clear
B)
\[2\pi /3\] done
clear
C)
\[5\pi /4\] done
clear
D)
\[7\pi /4\] done
clear
E)
\[5\pi /3\] done
clear
View Solution play_arrow
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question_answer54)
In a triangle \[ABC\], if \[B=3C\], then the values of \[\sqrt{\left( \frac{b+c}{4c} \right)}\] and \[\left( \frac{b-c}{2c} \right)\] are
A)
\[\sin C,\sin \frac{A}{2}\] done
clear
B)
\[\cos C,\sin \frac{A}{2}\] done
clear
C)
\[\sin C,\cos \frac{A}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer55)
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 done
clear
B)
1 done
clear
C)
\[\pm 1\] done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer56)
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 \[{{135}^{o}}\] done
clear
B)
\[{{135}^{o}}\]or \[{{120}^{o}}\] done
clear
C)
\[{{30}^{o}}\]or \[{{60}^{o}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer57)
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 done
clear
B)
7 done
clear
C)
8 done
clear
D)
10 done
clear
View Solution play_arrow
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question_answer58)
In triangle\[ABC\]if \[A+C=2B\], then \[\frac{a+c}{\sqrt{{{a}^{2}}-ac+{{c}^{2}}}}\]is equal to [UPSEAT 1999]
A)
\[2\cos \frac{A-C}{2}\] done
clear
B)
\[\sin \frac{A+C}{2}\] done
clear
C)
\[\sin \frac{A}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer59)
The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is \[{{60}^{o}}\]. If the third side is 3, the remaining fourth side is [MNR 1994]
A)
2 done
clear
B)
3 done
clear
C)
4 done
clear
D)
5 done
clear
View Solution play_arrow
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question_answer60)
In a triangle\[ABC,\]\[a=4,b=3\], \[\angle A={{60}^{o}}\]. Then c is the root of the equation [Roorkee 1993]
A)
\[{{c}^{2}}-3c-7=0\] done
clear
B)
\[{{c}^{2}}+3c+7=0\] done
clear
C)
\[{{c}^{2}}-3c+7=0\] done
clear
D)
\[{{c}^{2}}+3c-7=0\] done
clear
View Solution play_arrow
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question_answer61)
If \[a=2,b=3,c=5\]in \[\Delta ABC\], then C = [EAMCET 1984]
A)
\[\frac{\pi }{6}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer62)
If \[A={{30}^{o}},a=7,b=8\]in \[\Delta ABC\], then B has
A)
One solution done
clear
B)
Two solutions done
clear
C)
No solution done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer63)
If \[b=3,c=4\]and \[B=\frac{\pi }{3}\], then the number of triangle that can be constructed is [Roorkee 1992]
A)
Infinite done
clear
B)
Two done
clear
C)
One done
clear
D)
Nil done
clear
View Solution play_arrow
-
question_answer64)
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)
\[\sin A,\sin B,\sin C\] done
clear
B)
\[\tan A,\tan B,\tan C\] done
clear
C)
\[\cot A,\cot B,\cot C\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer65)
The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangle are
A)
1, 2, 3 done
clear
B)
2, 3, 4 done
clear
C)
3, 4, 5 done
clear
D)
4, 5, 6 done
clear
View Solution play_arrow
-
question_answer66)
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: \[\sqrt{2}\] done
clear
B)
\[1:\sqrt{2}:1\] done
clear
C)
\[\sqrt{2}:1:1\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer67)
In a \[\Delta ABC\], \[\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\]and the side \[a=2,\]then area of the triangle is [IIT Screening 1993; MP PET 2000]
A)
1 done
clear
B)
2 done
clear
C)
\[\frac{\sqrt{3}}{2}\] done
clear
D)
\[\sqrt{3}\] done
clear
View Solution play_arrow
-
question_answer68)
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)
\[\frac{\pi }{6}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\pi \] done
clear
View Solution play_arrow
-
question_answer69)
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 done
clear
B)
2 done
clear
C)
4 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer70)
If in a \[\Delta ABC\], \[\cos A+2\cos B+\cos C=2\], then\[a,b,c\]are in
A)
A. P. done
clear
B)
H. P. done
clear
C)
G. P. done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer71)
If in a \[\Delta ABC\], \[\cos 3A+\cos 3B+\cos 3C=1\], then one angle must be exactly equal to
A)
\[{{90}^{o}}\] done
clear
B)
\[{{45}^{o}}\] done
clear
C)
\[{{120}^{o}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer72)
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)
\[{{30}^{o}},{{60}^{o}},{{90}^{o}}\] done
clear
B)
\[{{45}^{o}},{{60}^{o}},{{75}^{o}}\] done
clear
C)
\[{{45}^{o}},{{45}^{o}},{{90}^{o}}\] done
clear
D)
\[{{60}^{o}},{{60}^{o}},{{60}^{o}}\] done
clear
View Solution play_arrow
-
question_answer73)
If in the \[\Delta ABC,AB=2BC\], then \[\tan \frac{B}{2}:\cot \left( \frac{C-A}{2} \right)\]
A)
3 :1 done
clear
B)
2 : 1 done
clear
C)
1 : 2 done
clear
D)
1 : 3 done
clear
View Solution play_arrow
-
question_answer74)
In a triangle \[ABC\], if \[a=2,B={{60}^{o}}\]and \[C={{75}^{o}}\], then b = [Karnataka CET 1992]
A)
\[\sqrt{3}\] done
clear
B)
\[\sqrt{6}\] done
clear
C)
\[\sqrt{9}\] done
clear
D)
\[1+\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer75)
In triangle ABC, \[A={{30}^{o}},b=8,a=6\], then \[B={{\sin }^{-1}}x\], where x = [Karnataka CET 1990]
A)
\[\frac{1}{2}\] done
clear
B)
\[\frac{1}{3}\] done
clear
C)
\[\frac{2}{3}\] done
clear
D)
1 done
clear
View Solution play_arrow
-
question_answer76)
In a \[\Delta ABC\], \[b=2,C={{60}^{o}},c=\sqrt{6}\], then a =
A)
\[\sqrt{3}-1\] done
clear
B)
\[\sqrt{3}\] done
clear
C)
\[\sqrt{3}+1\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer77)
In a \[\Delta ABC\], \[a=5,b=4\]and \[\cos (A-B)=\frac{31}{32}\], then side c is equal to
A)
6 done
clear
B)
7 done
clear
C)
9 done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer78)
In a \[\Delta ABC\], if \[A={{30}^{o}}\]\[b=2,c=\sqrt{3}+1\], then \[\frac{C-B}{2}=\]
A)
\[{{15}^{o}}\] done
clear
B)
\[{{30}^{o}}\] done
clear
C)
\[{{45}^{o}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer79)
The smallest angle of the triangle whose sides are \[6+\sqrt{12},\sqrt{48},\sqrt{24}\]is [EAMCET 1985]
A)
\[\frac{\pi }{3}\] done
clear
B)
\[\frac{\pi }{4}\] done
clear
C)
\[\frac{\pi }{6}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer80)
If \[A={{30}^{o}},c=7\sqrt{3}\]and \[C={{90}^{o}}\]in \[\Delta ABC\], then a =
A)
\[7\sqrt{3}\] done
clear
B)
\[\frac{7\sqrt{3}}{2}\] done
clear
C)
\[\frac{7}{2}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer81)
If angles of a triangle are in the ratio of 2 : 3: 7, then the sides are in the ratio of [MP PET 1996]
A)
\[\sqrt{2}:2:(\sqrt{3}+1)\] done
clear
B)
\[2:\sqrt{2}:(\sqrt{3}+1)\] done
clear
C)
\[\sqrt{2}:(\sqrt{3}+1):2\] done
clear
D)
\[2:(\sqrt{3}+1):\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer82)
Sides of a triangle are \[2cm,\sqrt{6}\,cm\] and \[(\sqrt{3}+1)cm\]. The smallest angle of the triangle is
A)
\[{{30}^{o}}\] done
clear
B)
\[{{45}^{o}}\] done
clear
C)
\[{{60}^{o}}\] done
clear
D)
\[{{75}^{o}}\] done
clear
View Solution play_arrow
-
question_answer83)
In any triangle \[ABC,\frac{\tan \frac{A}{2}-\tan \frac{B}{2}}{\tan \frac{A}{2}+\tan \frac{B}{2}}=\]
A)
\[\frac{a-b}{a+b}\] done
clear
B)
\[\frac{a-b}{c}\] done
clear
C)
\[\frac{a-b}{a+b+c}\] done
clear
D)
\[\frac{c}{a+b}\] done
clear
View Solution play_arrow
-
question_answer84)
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)
\[\frac{\sqrt{3}+1}{3}c{{m}^{2}}\] done
clear
B)
\[\frac{\sqrt{3}+1}{2}c{{m}^{2}}\] done
clear
C)
\[\frac{\sqrt{3}+1}{2\sqrt{2}}c{{m}^{2}}\] done
clear
D)
\[\frac{\sqrt{3}+1}{3\sqrt{2}}c{{m}^{2}}\] done
clear
View Solution play_arrow
-
question_answer85)
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)
\[{{15}^{o}}\] done
clear
B)
\[{{30}^{o}}\] done
clear
C)
\[{{45}^{o}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer86)
If in a \[\Delta ABC,\,\angle A={{45}^{o}},\,\,\angle C={{60}^{o}}\], then \[a+c\sqrt{2}=\] [MP PET 1999]
A)
b done
clear
B)
2b done
clear
C)
\[\sqrt{2b}\] done
clear
D)
\[\sqrt{3}b\] done
clear
View Solution play_arrow
-
question_answer87)
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)
\[\pi /2\] done
clear
B)
\[5\pi /6\] done
clear
C)
\[2\pi /3\] done
clear
D)
\[3\pi /4\] done
clear
View Solution play_arrow
-
question_answer88)
If in a triangle ABC, angle C is \[{{45}^{o}}\], then \[(1+\cot A)(1+\cot B)=\] [Kurukshetra CEE 1998]
A)
-1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
\[1/\sqrt{2}\] done
clear
View Solution play_arrow
-
question_answer89)
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 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
-
question_answer90)
In a \[\Delta ABC,\]\[2a\sin \,\,\left( \frac{A-B+C}{2} \right)\] is equal to [IIT Screening 2000]
A)
\[{{a}^{2}}+{{b}^{2}}-{{c}^{2}}\] done
clear
B)
\[{{c}^{2}}+{{a}^{2}}-{{b}^{2}}\] done
clear
C)
\[{{b}^{2}}-{{c}^{2}}-{{a}^{2}}\] done
clear
D)
\[{{c}^{2}}-{{a}^{2}}-{{b}^{2}}\] done
clear
View Solution play_arrow
-
question_answer91)
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)
\[a+b\] done
clear
B)
\[\frac{{{a}^{2}}}{bc}\] done
clear
C)
\[\frac{{{b}^{2}}}{ac}\] done
clear
D)
\[\frac{{{c}^{2}}}{ab}\] done
clear
View Solution play_arrow
-
question_answer92)
In a \[\Delta ABC,\] \[A:B:C\]. Then \[[a+b+c\sqrt{2}]\] is equal to [DCE 2001]
A)
2b done
clear
B)
2c done
clear
C)
3b done
clear
D)
3a done
clear
View Solution play_arrow
-
question_answer93)
In a \[\Delta ABC,\,\,\frac{\cos C+\cos A}{c+a}+\frac{\cos B}{b}\]is equal to [EAMCET 2001]
A)
\[\frac{1}{a}\] done
clear
B)
\[\frac{1}{b}\] done
clear
C)
\[\frac{1}{c}\] done
clear
D)
\[\frac{c+a}{b}\] done
clear
View Solution play_arrow
-
question_answer94)
The angles of a triangle are in the ratio \[1:3:5\], then the greatest angle is [Kerala (Engg.) 2002]
A)
\[5\pi /9\] done
clear
B)
\[2\pi /9\] done
clear
C)
\[7\pi /9\] done
clear
D)
\[11\pi /9\] done
clear
View Solution play_arrow
-
question_answer95)
In any triangle \[AB=2,BC=4,CA=3\]and D is mid point of BC, then [Roorkee 1995]
A)
\[\cos B=\frac{11}{6}\] done
clear
B)
\[\cos B=\frac{7}{8}\] done
clear
C)
\[AD=2.4\] done
clear
D)
\[A{{D}^{2}}=2.5\] done
clear
View Solution play_arrow
-
question_answer96)
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)
\[\sqrt{3}:(2+\sqrt{3})\] done
clear
B)
\[1:6\] done
clear
C)
\[1:(2+\sqrt{3})\] done
clear
D)
\[2:3\] done
clear
View Solution play_arrow
-
question_answer97)
If in any \[\Delta ABC\], \[\cot \frac{A}{2},\cot \frac{B}{2},\cos \frac{C}{2}\]are in A. P. then [MP PET 2003]
A)
\[\cot \frac{A}{2}\cot \frac{B}{2}=4\] done
clear
B)
\[\cot \frac{A}{2}\cot \frac{C}{2}=3\] done
clear
C)
\[\cot \frac{B}{2}\cot \frac{C}{2}=1\] done
clear
D)
\[\cot \frac{B}{2}\tan \frac{C}{2}=0\] done
clear
View Solution play_arrow
-
question_answer98)
The smallest angle of the \[\Delta ABC\], when \[a=7,b=4\sqrt{3}\]and \[c=\sqrt{13},\] is [MP PET 2003]
A)
\[{{30}^{o}}\] done
clear
B)
\[{{15}^{o}}\] done
clear
C)
\[{{45}^{o}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer99)
In a \[\Delta ABC,\]if \[\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\], then \[\cos C=\] [Karnataka CET 2003]
A)
\[\frac{7}{5}\] done
clear
B)
\[\frac{5}{7}\] done
clear
C)
\[\frac{17}{36}\] done
clear
D)
\[\frac{16}{17}\] done
clear
View Solution play_arrow
-
question_answer100)
In a\[\Delta ABC,\]if \[b=20,c=21\]and \[\sin A=3/5\], then \[a=\] [EAMCET 2003]
A)
12 done
clear
B)
13 done
clear
C)
14 done
clear
D)
15 done
clear
View Solution play_arrow
-
question_answer101)
Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then \[{{a}^{2}}:{{b}^{2}}:{{c}^{2}}\]is equal to [Pb. CET 2004]
A)
\[1:4:3\] done
clear
B)
\[4:1:3\] done
clear
C)
\[4:3:1\] done
clear
D)
\[3:4:1\] done
clear
View Solution play_arrow
-
question_answer102)
The ratio of the sides of triangle ABC is \[1:\sqrt{3}:2\]. The ratio of \[A:B:C\]is [IIT Screening 2004]
A)
\[3:5:2\] done
clear
B)
\[1:\sqrt{3}:2\] done
clear
C)
3 : 2 : 1 done
clear
D)
1: 2 : 3 done
clear
View Solution play_arrow
-
question_answer103)
In a triangle \[ABC,\,\,b=\sqrt{3}\], \[c=1\]and \[\angle A={{30}^{o}}\], then the largest angle of the triangle is [MP PET 2004]
A)
\[{{135}^{o}}\] done
clear
B)
\[{{90}^{o}}\] done
clear
C)
\[{{60}^{o}}\] done
clear
D)
\[{{120}^{o}}\] done
clear
View Solution play_arrow
-
question_answer104)
The lengths of the sides of a triangle are \[\alpha -\beta ,\alpha +\beta \]and \[\sqrt{3{{\alpha }^{2}}+{{\beta }^{2}}},\] \[(\alpha >\beta >0)\]. Its largest angle is [Roorkee 1999]
A)
\[\frac{3\pi }{4}\] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{2\pi }{3}\] done
clear
D)
\[\frac{5\pi }{6}\] done
clear
View Solution play_arrow
-
question_answer105)
The sides of a triangle are 4, 5 and 6cm. The area of the triangle is equal to [UPSEAT 2004]
A)
\[\frac{15}{4}c{{m}^{2}}\] done
clear
B)
\[\frac{15}{4}\sqrt{7}c{{m}^{2}}\] done
clear
C)
\[\frac{4}{15}\sqrt{7}c{{m}^{2}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer106)
If \[\alpha ,\beta ,\gamma \] are angles of a triangle, then \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma -2\cos \alpha \cos \beta \cos \gamma \]is [Orissa JEE 2004]
A)
2 done
clear
B)
-1 done
clear
C)
-2 done
clear
D)
0 done
clear
View Solution play_arrow
-
question_answer107)
If in \[\Delta ABC,\]\[a=6,b=3\]and \[\cos (A-B)=\frac{4}{5}\], then its area will be [MP PET 2004]
A)
7 square unit done
clear
B)
8 square unit done
clear
C)
9 square unit done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer108)
In \[\Delta ABC\], if \[2s=a+b+c\], then the value of \[\frac{s(s-a)}{bc}-\frac{(s-b)(s-c)}{bc}=\]
A)
\[\sin A\] done
clear
B)
\[\cos A\] done
clear
C)
\[\tan A\] done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer109)
In \[\Delta ABC\] if \[a=2,b=4\]and \[\angle C={{60}^{o}}\], then \[\angle A\]and \[\angle B\] are equal to
A)
\[{{90}^{o}},{{30}^{o}}\] done
clear
B)
\[{{60}^{o}},{{60}^{o}}\] done
clear
C)
\[{{30}^{o}},{{90}^{o}}\] done
clear
D)
\[{{60}^{o}},{{45}^{o}}\] done
clear
View Solution play_arrow
-
question_answer110)
The area of a \[\Delta ABC\]is equal to [MP PET 1984]
A)
\[\frac{1}{2}ab\sin A\] done
clear
B)
\[\frac{1}{2}bc\sin A\] done
clear
C)
\[\frac{1}{2}ca\sin A\] done
clear
D)
\[bc\sin A\] done
clear
View Solution play_arrow
-
question_answer111)
If in triangle \[ABC,\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}=\frac{\sin (A-B)}{\sin (A+B)}\], then the triangle is [Roorkee 1987]
A)
Right angled done
clear
B)
Isosceles done
clear
C)
Right angled or isosecles done
clear
D)
Right angled isosecles done
clear
View Solution play_arrow
-
question_answer112)
In a triangle\[ABC\], sin\[A:\sin B\]: \[\sin C=1:2:3\]. If \[b=4\] cm, the perimeter of the triangle is [MP PET 1986]
A)
\[6cm\] done
clear
B)
\[24cm\] done
clear
C)
\[12cm\] done
clear
D)
\[8cm\] done
clear
View Solution play_arrow
-
question_answer113)
The ratios of the sides in a triangle are 5: 12: 13 and its area is 270 square cm. The sides of the triangle in cm are [MP PET 1989]
A)
5, 12, 13 done
clear
B)
10, 24, 26 done
clear
C)
15, 36, 39 done
clear
D)
20, 48, 52 done
clear
View Solution play_arrow
-
question_answer114)
If in triangle \[ABC,\cos A=\frac{\sin B}{2\sin C}\], then the triangle is [Orissa JEE 2002, 04; MP PET 2004]
A)
Equilateral done
clear
B)
Isosceles done
clear
C)
Right angled done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer115)
The area of an isosceles triangle is \[9c{{m}^{2}}\]. If the equal sides are \[6cm\]in length, the angle between them is[MP PET 1986]
A)
\[{{60}^{o}}\] done
clear
B)
\[{{30}^{o}}\] done
clear
C)
\[{{90}^{o}}\] done
clear
D)
\[{{45}^{o}}\] done
clear
View Solution play_arrow
-
question_answer116)
If the sides of triangle be 6, 10 and 14 then the triangle is [MP PET 1982]
A)
Obtuse angled done
clear
B)
Acute angled done
clear
C)
Right angled done
clear
D)
Equilateral done
clear
View Solution play_arrow
-
question_answer117)
In any \[\Delta ABC\]if \[a\cos B=b\cos A\], then the triangle is [MP PET 1984]
A)
Equilateral triangle done
clear
B)
Isosceles triangle done
clear
C)
Scalene done
clear
D)
Right angled done
clear
View Solution play_arrow
-
question_answer118)
In a triangle \[ABC,\]if \[a\sin A=b\sin B\], then the nature of the triangle [MP PET 1983]
A)
\[a>b\] done
clear
B)
\[a<b\] done
clear
C)
\[a=b\] done
clear
D)
\[a+b=c\] done
clear
View Solution play_arrow
-
question_answer119)
If in a triangle \[ABC\], \[\cos A+\cos B+\cos C=\frac{3}{2}\], then the triangle is [IIT 1984]
A)
Isosceles done
clear
B)
Equilateral done
clear
C)
Right angled done
clear
D)
None of these done
clear
View Solution play_arrow
-
question_answer120)
If in a triangle ABC the sides \[AB\]and AC are perpendicular, then the true equation is
A)
\[\tan A+\tan B=0\] done
clear
B)
\[\tan B+\tan C=0\] done
clear
C)
\[\tan A+2\tan C=0\] done
clear
D)
\[\tan B.\tan C=1\] done
clear
View Solution play_arrow
-
question_answer121)
In a triangle with one angle of \[{{120}^{o}}\]the lengths of the sides form an A. P. If the length of the greatest side is \[7cm\], the area of triangle is
A)
\[\frac{3\sqrt{15}}{4}c{{m}^{2}}\] done
clear
B)
\[\frac{15\sqrt{3}}{4}c{{m}^{2}}\] done
clear
C)
\[\frac{15}{4}c{{m}^{2}}\] done
clear
D)
\[\frac{3\sqrt{3}}{4}c{{m}^{2}}\] done
clear
View Solution play_arrow
-
question_answer122)
If the area of a triangle ABC is D, then \[{{a}^{2}}\sin 2B+{{b}^{2}}\sin 2A\] is equal to [WB JEE 1988]
A)
\[3\Delta \] done
clear
B)
\[2\Delta \] done
clear
C)
\[4\Delta \] done
clear
D)
\[-4\Delta \] done
clear
View Solution play_arrow
-
question_answer123)
In a right triangle \[AC=BC\] and D is the mid point of AC cotangent of angle \[DBC\] is equal to
A)
2 done
clear
B)
3 done
clear
C)
1/2 done
clear
D)
1/3 done
clear
View Solution play_arrow
-
question_answer124)
If a, b, c are the sides and A, B, C are the angles of a triangle \[ABC\], then \[\tan \left( \frac{A}{2} \right)\]is equal to [MP PET 1994]
A)
\[\sqrt{\frac{(s-c)(s-a)}{s(s-b)}}\] done
clear
B)
\[\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\] done
clear
C)
\[\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}\] done
clear
D)
\[\sqrt{\frac{(s-a)s}{(s-b)(s-c)}}\] done
clear
View Solution play_arrow
-
question_answer125)
In any triangle ABC, the value of \[a({{b}^{2}}+{{c}^{2}})\cos A+b({{c}^{2}}+{{a}^{2}})\cos B+c({{a}^{2}}+{{b}^{2}})\cos C\]is [MP PET 1994]
A)
\[3ab{{c}^{2}}\] done
clear
B)
\[3{{a}^{2}}bc\] done
clear
C)
\[3abc\] done
clear
D)
\[3a{{b}^{2}}c\] done
clear
View Solution play_arrow
-
question_answer126)
In a triangle \[ABC\], \[AD\] is altitude from A. Given \[b>c,\] \[\angle C={{23}^{o}}\]and \[AD=\frac{abc}{{{b}^{2}}-{{c}^{2}}},\]then \[\angle B=\] [IIT 1994]
A)
\[{{67}^{o}}\] done
clear
B)
\[{{44}^{o}}\] done
clear
C)
\[{{113}^{o}}\] done
clear
D)
None of these done
clear
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question_answer127)
If \[A={{60}^{o}}\], \[a=5,b=4\sqrt{3}\]in \[\Delta ABC\], then B =
A)
\[{{30}^{o}}\] done
clear
B)
\[{{60}^{o}}\] done
clear
C)
\[{{90}^{o}}\] done
clear
D)
None of these done
clear
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question_answer128)
If \[\Delta ={{a}^{2}}-{{(b-c)}^{2}}\], where \[\Delta \]is the area of triangle \[ABC\], then tan A is equal to [Pb. CET 1990; Kerala (Engg.) 2005]
A)
\[\frac{15}{16}\] done
clear
B)
\[\frac{8}{15}\] done
clear
C)
\[\frac{8}{17}\] done
clear
D)
\[\frac{1}{2}\] done
clear
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question_answer129)
If \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\], then \[4s(s-a)(s-b)(s-c)=\] [EAMCET 1986; Pb. CET 1990]
A)
\[{{s}^{4}}\] done
clear
B)
\[{{b}^{2}}{{c}^{2}}\] done
clear
C)
\[{{c}^{2}}{{a}^{2}}\] done
clear
D)
\[{{a}^{2}}{{b}^{2}}\] done
clear
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question_answer130)
If \[{{p}_{1}},{{p}_{2}},{{p}_{3}}\] are altitudes of a triangle \[ABC\]from the vertices \[A,B,C\] and \[\Delta \] the area of the triangle, then \[p_{1}^{-2}+p_{2}^{-2}+p_{3}^{-2}\] is equal to
A)
\[\frac{a+b+c}{\Delta }\] done
clear
B)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{4{{\Delta }^{2}}}\] done
clear
C)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\] done
clear
D)
None of these done
clear
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question_answer131)
If the median of \[\Delta ABC\]through A is perpendicular to \[AB\], then
A)
\[\tan A+\tan B=0\] done
clear
B)
\[2\tan A+\tan B=0\] done
clear
C)
\[\tan A+2\tan B=0\] done
clear
D)
None of these done
clear
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question_answer132)
If A is the area and 2s the sum of 3 sides of triangle, then
A)
\[A\le \frac{{{s}^{2}}}{3\sqrt{3}}\] done
clear
B)
\[A\le \frac{{{s}^{2}}}{2}\] done
clear
C)
\[A>\frac{{{s}^{2}}}{\sqrt{3}}\] done
clear
D)
None of these done
clear
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question_answer133)
If in a triangle \[ABC\]right angled at \[B,s-a=3\], \[s-c=2\], then the values of a and c are respectively
A)
2, 3 done
clear
B)
3, 4 done
clear
C)
4, 3 done
clear
D)
6, 8 done
clear
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question_answer134)
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)
\[\frac{\pi }{3}\] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\frac{2\pi }{3}\] done
clear
D)
\[\frac{5\pi }{6}\] done
clear
View Solution play_arrow
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question_answer135)
We are given b, c and \[\sin B\] such that B is acute and \[b<c\sin B\]. Then [Karnataka CET 1993]
A)
No triangle is possible done
clear
B)
One triangle is possible done
clear
C)
Two triangles are possible done
clear
D)
A right angled triangle is possible done
clear
View Solution play_arrow
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question_answer136)
The sides of triangle are \[3x+4y,\ 4x+3y\]and \[5x+5y\]units, where \[x,\ y>0.\]The triangle is [AIEEE 2002]
A)
Right angled done
clear
B)
Equilateral done
clear
C)
Obtuse angled done
clear
D)
None of these done
clear
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question_answer137)
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)
\[\sqrt{\frac{9{{a}^{2}}-{{c}^{2}}}{8{{a}^{2}}}}\] done
clear
B)
\[\sqrt{\frac{9{{a}^{2}}-{{c}^{2}}}{8{{c}^{2}}}}\] done
clear
C)
\[\sqrt{\frac{9{{a}^{2}}+{{c}^{2}}}{8{{a}^{2}}}}\] done
clear
D)
None of these done
clear
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question_answer138)
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)
\[\frac{1}{2}{{b}^{2}}\sin 2A\] done
clear
B)
\[\frac{1}{2}{{a}^{2}}\sin 2A\] done
clear
C)
\[{{b}^{2}}\sin 2A\] done
clear
D)
None of these done
clear
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question_answer139)
If in a triangle \[ABC\], \[2\cos A=\sin B\,\text{cosec}\,C,\] then [MP PET 1996]
A)
\[a=b\] done
clear
B)
\[b=c\] done
clear
C)
\[c=a\] done
clear
D)
\[2a=bc\] done
clear
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question_answer140)
If the sides of a triangle are \[3,\ 5,\ 7,\]then [MP PET 1996]
A)
All its angles are acute done
clear
B)
One angle is obtuse done
clear
C)
Triangle is right angled done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer141)
If \[y=x\tan \frac{\alpha +\beta }{2}\], then \[\tan A+\tan B+\tan C=\]
A)
\[\frac{a+b+c}{abc}\] done
clear
B)
\[0\] done
clear
C)
\[\tan A\tan B\tan C\] done
clear
D)
\[\tan A\tan B+\tan B\tan C+\tan C\tan A\] done
clear
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question_answer142)
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)
\[a+b=c\] done
clear
B)
\[b+c=a\] done
clear
C)
\[a+c=b\] done
clear
D)
\[b=c\] done
clear
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question_answer143)
If a triangle \[PQR\], \[\sin P,\ \sin Q,\ \sin R\]are in A.P., then [IIT 1998]
A)
The altitudes are in A.P. done
clear
B)
The altitudes are in H.P. done
clear
C)
The medians are in G.P. done
clear
D)
The medians are in A.P. done
clear
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question_answer144)
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. done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer145)
If in a triangle ABC, a, b, c and angle A is given and \[c\sin A<a<c,\] then [UPSEAT 1999]
A)
\[{{b}_{1}}+{{b}_{2}}=2c\cos A\] done
clear
B)
\[{{b}_{1}}+{{b}_{2}}=c\cos A\] done
clear
C)
\[{{b}_{1}}+{{b}_{2}}=3c\cos A\] done
clear
D)
\[{{b}_{1}}+{{b}_{2}}=4c\sin A\] done
clear
View Solution play_arrow
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question_answer146)
In a \[\Delta ABC,\,\,{{a}^{2}}\sin \,\,2C+{{c}^{2}}\sin 2A=\] [EAMCET 2001]
A)
\[\Delta \] done
clear
B)
\[2\Delta \] done
clear
C)
\[3\Delta \] done
clear
D)
\[4\Delta \] done
clear
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question_answer147)
In \[\Delta ABC,\]\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=ac+ab\sqrt{3},\]then triangle is [MP PET 2004]
A)
Equilateral done
clear
B)
Isosceles done
clear
C)
Right angled done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer148)
The area of triangle \[ABC,\] in which \[a=1,\ b=2\], \[\angle C=60{}^\circ \]is [MP PET 2004]
A)
\[\frac{1}{2}\] done
clear
B)
\[\sqrt{3}\] done
clear
C)
\[\frac{\sqrt{3}}{2}\] done
clear
D)
\[\frac{3}{2}\] done
clear
View Solution play_arrow
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question_answer149)
In a triangle \[ABC\], if \[b+c=2a\] and \[\angle A=60{}^\circ ,\] then \[\Delta ABC\] is [MP PET 2004]
A)
Scalene done
clear
B)
Equilateral done
clear
C)
Isosecles done
clear
D)
Right angled done
clear
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question_answer150)
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. done
clear
B)
H.P. done
clear
C)
G.P. done
clear
D)
A.P. done
clear
View Solution play_arrow
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question_answer151)
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 \[135{}^\circ \] done
clear
B)
\[30{}^\circ \] or \[100{}^\circ \] done
clear
C)
\[50{}^\circ \] or \[100{}^\circ \] done
clear
D)
\[60{}^\circ \] or \[120{}^\circ \] done
clear
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question_answer152)
In a \[\Delta ABC\] if the sides are \[a=3,\,b=5\] and \[c=4\], then \[\sin \frac{B}{2}+\cos \frac{B}{2}\] is equal to [Karnataka CET 2005]
A)
\[\sqrt{2}\] done
clear
B)
\[\frac{\sqrt{3}+1}{2}\] done
clear
C)
\[\frac{\sqrt{3}-1}{2}\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer153)
Which of the following is true in a triangle ABC [IIT Screening 2005]
A)
\[(b+c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\] done
clear
B)
\[(b+c)\cos \frac{A}{2}=2a\sin \frac{B-C}{2}\] done
clear
C)
\[(b-c)\cos \frac{A}{2}=a\sin \frac{B-C}{2}\] done
clear
D)
\[(b-c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\] done
clear
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question_answer154)
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)
\[\frac{ab+cd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] done
clear
B)
\[\frac{ac+bd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] done
clear
C)
\[\frac{ac-bd}{\sqrt{({{a}^{2}}+{{b}^{2}})\,({{c}^{2}}+{{d}^{2}})}}\] done
clear
D)
None of these done
clear
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question_answer155)
\[ABC\] 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 done
clear
B)
\[70\sqrt{2}cm\] done
clear
C)
\[\frac{35\sqrt{3}}{2}cm\] done
clear
D)
\[35\sqrt{6}\]cm done
clear
E)
\[\frac{35\sqrt{2}}{2}cm\] done
clear
View Solution play_arrow
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question_answer156)
If in a triangle \[ABC,a=5,b=4,A=\frac{\pi }{2}+B\], then C [Kerala (Engg.) 2005]
A)
Is \[{{\tan }^{-1}}\left( \frac{1}{9} \right)\] done
clear
B)
Is \[{{\tan }^{-1}}\frac{1}{40}\] done
clear
C)
Cannot be evaluated done
clear
D)
Is\[2{{\tan }^{-1}}\left( 1/9 \right)\] done
clear
E)
Is \[2{{\tan }^{-1}}\frac{1}{40}\] done
clear
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