JEE Main & Advanced Mathematics Trigonometric Equations Question Bank

Relation between sides and angles, Solutions of triangles Total Questions - 156

1)

In a triangle ABC, \[a=5,b=7\] and \[\sin A=\frac{3}{4}\] how many such triangles are possible [Roorkee 1990]

A)
1

B)
0

C)
2

D)
Infinite
View Solution
2)

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)
\[{{60}^{o}}\]
View Solution
3)

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
View Solution
4)

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)
\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}\]
View Solution
5)

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)
\[2(a+b+c)\]
View Solution
6)

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
View Solution
7)

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)
\[\frac{{{c}^{2}}}{{{a}^{2}}+{{b}^{2}}}\]
View Solution
8)

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)
\[\frac{ac}{4\Delta }\]
View Solution
9)

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
View Solution
10)

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)
\[\cot C\]

B)
\[c\cot C\]

C)
\[\cot \frac{C}{2}\]

D)
\[c\cot \frac{C}{2}\]
View Solution
11)

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.

B)
G. P.

C)
H. P.

D)
None of these
View Solution
12)

In \[\Delta ABC,{{(a-b)}^{2}}{{\cos }^{2}}\frac{C}{2}+{{(a+b)}^{2}}{{\sin }^{2}}\frac{C}{2}=\]

A)
\[{{a}^{2}}\]

B)
\[{{b}^{2}}\]

C)
\[{{c}^{2}}\]

D)
None of these
View Solution
13)

In \[\Delta ABC,\]if \[a=16,b=24\] and \[c=20,\]then \[\cos \frac{B}{2}=\] [MP PET 1988]

A)
3/4

B)
1/4

C)
1/2

D)
1/3
View Solution
14)

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
View Solution
15)

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)
\[\frac{4a}{a+b+c}\]
View Solution
16)

In \[\Delta ABC,\]\[{{b}^{2}}\cos 2A-{{a}^{2}}\cos 2B=\]

A)
\[{{b}^{2}}-{{a}^{2}}\]

B)
\[{{b}^{2}}-{{c}^{2}}\]

C)
\[{{c}^{2}}-{{a}^{2}}\]

D)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
View Solution
17)

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)
\[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]
View Solution
18)

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
View Solution
19)

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)
\[\angle A+\angle C={{90}^{o}}\]
View Solution
20)

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
View Solution
21)

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)
\[c/ab\]
View Solution
22)

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
View Solution
23)

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)
2 :1

C)
\[(\sqrt{5}+1):(\sqrt{5}-1)\]

D)
\[(\sqrt{5}-1):(\sqrt{5}+1)\]
View Solution
24)

If in a triangle \[ABC\], \[\angle C={{60}^{o}},\]then \[\frac{1}{a+c}+\frac{1}{b+c}=\] [IIT 1975]

A)
\[\frac{1}{a+b+c}\]

B)
\[\frac{2}{a+b+c}\]

C)
\[\frac{3}{a+b+c}\]

D)
None of these
View Solution
25)

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
View Solution
26)

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)
1/2

C)
2

D)
-1
View Solution
27)

If \[\tan \frac{B-C}{2}=x\cot \frac{A}{2},\]then \[x=\] [MP PET 1992, 2002]

A)
\[\frac{c-a}{c+a}\]

B)
\[\frac{a-b}{a+b}\]

C)
\[\frac{b-c}{b+c}\]

D)
None of these
View Solution
28)

In \[\Delta ABC\], if \[a=3,b=4,c=5\], then \[\sin 2B=\] [MP PET 1983]

A)
4/5

B)
3/20

C)
24/25

D)
1/50
View Solution
29)

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)
\[{{120}^{o}}\]
View Solution
30)

In a triangle ABC, \[{{a}^{3}}\cos (B-C)+{{b}^{3}}\cos (C-A)+{{c}^{3}}\cos (A-B)=\] [Kerala (Engg.) 2002]

A)
\[abc\]

B)
\[3abc\]

C)
\[a+b+c\]

D)
None of these
View Solution
31)

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)
\[{{45}^{o}}\]
View Solution
32)

If the sides of a right angled triangle be in A. P. , then their ratio will be

A)
1: 2: 3

B)
2 : 3 : 4

C)
3: 4: 5

D)
4 : 5 : 6
View Solution
33)

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

B)
Right angled isosceles

C)
Isosceles

D)
Obtuse angled
View Solution
34)

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
View Solution
35)

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)
\[8,\,\,8\sqrt{3}\]
View Solution
36)

In \[\Delta ABC\], \[c\cos (A-\alpha )+a\cos (C+\alpha )=\]

A)
\[a\cos \alpha \]

B)
\[b\cos \alpha \]

C)
\[c\cos \alpha \]

D)
\[2b\cos \alpha \]
View Solution
37)

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)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]
View Solution
38)

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

B)
1

C)
\[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\]

D)
\[2({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\]
View Solution
39)

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)
\[\frac{{{a}^{2}}+{{b}^{2}}}{{{a}^{2}}+{{c}^{2}}}\]
View Solution
40)

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)
\[\frac{a}{b+c}\]
View Solution
41)

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)
\[\frac{1}{2abc}\]
View Solution
42)

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)
\[{{\left( \frac{{{c}^{2}}-{{a}^{2}}}{2ca} \right)}^{2}}\]
View Solution
43)

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
View Solution
44)

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

B)
\[1:\sqrt{3}:2\]

C)
\[\sqrt{2}:\sqrt{3}:3\]

D)
\[1:\sqrt{3}:3\]
View Solution
45)

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)
\[{{60}^{o}}\]
View Solution
46)

If \[a=9,b=8\]and \[c=x\]satisfies \[3\cos C=2,\]then [MP PET 1984]

A)
\[x=5\]

B)
\[x=6\]

C)
\[x=4\]

D)
\[x=7\]
View Solution
47)

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)
\[{{15}^{o}}\]
View Solution
48)

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.
View Solution
49)

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)
\[{{75}^{o}}\]
View Solution
50)

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)
\[{{\cos }^{-1}}\left( -\frac{1}{4} \right)\]
View Solution
51)

\[\cot \frac{A+B}{2}\]. \[\tan \frac{A-B}{2}=\] [Roorkee 1975]

A)
\[\frac{a+b}{a-b}\]

B)
\[\frac{a-b}{a+b}\]

C)
\[\frac{a}{a+b}\]

D)
None of these
View Solution
52)

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
View Solution
53)

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)
\[5\pi /3\]
View Solution
54)

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)
\[\sin C,\sin \frac{A}{2}\]

B)
\[\cos C,\sin \frac{A}{2}\]

C)
\[\sin C,\cos \frac{A}{2}\]

D)
None of these
View Solution
55)

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
View Solution
56)

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
View Solution
57)

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
View Solution
58)

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
View Solution
59)

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

B)
3

C)
4

D)
5
View Solution
60)

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)
\[{{c}^{2}}+3c-7=0\]
View Solution
61)

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
View Solution
62)

If \[A={{30}^{o}},a=7,b=8\]in \[\Delta ABC\], then B has

A)
One solution

B)
Two solutions

C)
No solution

D)
None of these
View Solution
63)

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
View Solution
64)

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
View Solution
65)

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

B)
2, 3, 4

C)
3, 4, 5

D)
4, 5, 6
View Solution
66)

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
View Solution
67)

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

B)
2

C)
\[\frac{\sqrt{3}}{2}\]

D)
\[\sqrt{3}\]
View Solution
68)

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)
\[\pi \]
View Solution
69)

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
View Solution
70)

If in a \[\Delta ABC\], \[\cos A+2\cos B+\cos C=2\], then\[a,b,c\]are in

A)
A. P.

B)
H. P.

C)
G. P.

D)
None of these
View Solution
71)

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
View Solution
72)

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)
\[{{60}^{o}},{{60}^{o}},{{60}^{o}}\]
View Solution
73)

If in the \[\Delta ABC,AB=2BC\], then \[\tan \frac{B}{2}:\cot \left( \frac{C-A}{2} \right)\]

A)
3 :1

B)
2 : 1

C)
1 : 2

D)
1 : 3
View Solution
74)

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)
\[1+\sqrt{2}\]
View Solution
75)

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
View Solution
76)

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
View Solution
77)

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
View Solution
78)

In a \[\Delta ABC\], if \[A={{30}^{o}}\]\[b=2,c=\sqrt{3}+1\], then \[\frac{C-B}{2}=\]

A)
\[{{15}^{o}}\]

B)
\[{{30}^{o}}\]

C)
\[{{45}^{o}}\]

D)
None of these
View Solution
79)

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
View Solution
80)

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
View Solution
81)

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)
\[\sqrt{2}:2:(\sqrt{3}+1)\]

B)
\[2:\sqrt{2}:(\sqrt{3}+1)\]

C)
\[\sqrt{2}:(\sqrt{3}+1):2\]

D)
\[2:(\sqrt{3}+1):\sqrt{2}\]
View Solution
82)

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)
\[{{75}^{o}}\]
View Solution
83)

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)
\[\frac{c}{a+b}\]
View Solution
84)

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)
\[\frac{\sqrt{3}+1}{3\sqrt{2}}c{{m}^{2}}\]
View Solution
85)

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
View Solution
86)

If in a \[\Delta ABC,\,\angle A={{45}^{o}},\,\,\angle C={{60}^{o}}\], then \[a+c\sqrt{2}=\] [MP PET 1999]

A)
b

B)
2b

C)
\[\sqrt{2b}\]

D)
\[\sqrt{3}b\]
View Solution
87)

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)
\[3\pi /4\]
View Solution
88)

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)
\[1/\sqrt{2}\]
View Solution
89)

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
View Solution
90)

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)
\[{{c}^{2}}-{{a}^{2}}-{{b}^{2}}\]
View Solution
91)

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)
\[\frac{{{c}^{2}}}{ab}\]
View Solution
92)

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
View Solution
93)

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)
\[\frac{c+a}{b}\]
View Solution
94)

The angles of a triangle are in the ratio \[1:3:5\], then the greatest angle is [Kerala (Engg.) 2002]

A)
\[5\pi /9\]

B)
\[2\pi /9\]

C)
\[7\pi /9\]

D)
\[11\pi /9\]
View Solution
95)

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)
\[A{{D}^{2}}=2.5\]
View Solution
96)

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)
\[2:3\]
View Solution
97)

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)
\[\cot \frac{A}{2}\cot \frac{B}{2}=4\]

B)
\[\cot \frac{A}{2}\cot \frac{C}{2}=3\]

C)
\[\cot \frac{B}{2}\cot \frac{C}{2}=1\]

D)
\[\cot \frac{B}{2}\tan \frac{C}{2}=0\]
View Solution
98)

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
View Solution
99)

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)
\[\frac{16}{17}\]
View Solution
100)

In a\[\Delta ABC,\]if \[b=20,c=21\]and \[\sin A=3/5\], then \[a=\] [EAMCET 2003]

A)
12

B)
13

C)
14

D)
15
View Solution
101)

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)
\[1:4:3\]

B)
\[4:1:3\]

C)
\[4:3:1\]

D)
\[3:4:1\]
View Solution
102)

The ratio of the sides of triangle ABC is \[1:\sqrt{3}:2\]. The ratio of \[A:B:C\]is [IIT Screening 2004]

A)
\[3:5:2\]

B)
\[1:\sqrt{3}:2\]

C)
3 : 2 : 1

D)
1: 2 : 3
View Solution
103)

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)
\[{{135}^{o}}\]

B)
\[{{90}^{o}}\]

C)
\[{{60}^{o}}\]

D)
\[{{120}^{o}}\]
View Solution
104)

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)
\[\frac{3\pi }{4}\]

B)
\[\frac{\pi }{2}\]

C)
\[\frac{2\pi }{3}\]

D)
\[\frac{5\pi }{6}\]
View Solution
105)

The sides of a triangle are 4, 5 and 6cm. The area of the triangle is equal to [UPSEAT 2004]

A)
\[\frac{15}{4}c{{m}^{2}}\]

B)
\[\frac{15}{4}\sqrt{7}c{{m}^{2}}\]

C)
\[\frac{4}{15}\sqrt{7}c{{m}^{2}}\]

D)
None of these
View Solution
106)

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

B)
-1

C)
-2

D)
0
View Solution
107)

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

B)
8 square unit

C)
9 square unit

D)
None of these
View Solution
108)

In \[\Delta ABC\], if \[2s=a+b+c\], then the value of \[\frac{s(s-a)}{bc}-\frac{(s-b)(s-c)}{bc}=\]

A)
\[\sin A\]

B)
\[\cos A\]

C)
\[\tan A\]

D)
None of these
View Solution
109)

In \[\Delta ABC\] if \[a=2,b=4\]and \[\angle C={{60}^{o}}\], then \[\angle A\]and \[\angle B\] are equal to

A)
\[{{90}^{o}},{{30}^{o}}\]

B)
\[{{60}^{o}},{{60}^{o}}\]

C)
\[{{30}^{o}},{{90}^{o}}\]

D)
\[{{60}^{o}},{{45}^{o}}\]
View Solution
110)

The area of a \[\Delta ABC\]is equal to [MP PET 1984]

A)
\[\frac{1}{2}ab\sin A\]

B)
\[\frac{1}{2}bc\sin A\]

C)
\[\frac{1}{2}ca\sin A\]

D)
\[bc\sin A\]
View Solution
111)

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

B)
Isosceles

C)
Right angled or isosecles

D)
Right angled isosecles
View Solution
112)

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)
\[6cm\]

B)
\[24cm\]

C)
\[12cm\]

D)
\[8cm\]
View Solution
113)

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

B)
10, 24, 26

C)
15, 36, 39

D)
20, 48, 52
View Solution
114)

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

B)
Isosceles

C)
Right angled

D)
None of these
View Solution
115)

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)
\[{{60}^{o}}\]

B)
\[{{30}^{o}}\]

C)
\[{{90}^{o}}\]

D)
\[{{45}^{o}}\]
View Solution
116)

If the sides of triangle be 6, 10 and 14 then the triangle is [MP PET 1982]

A)
Obtuse angled

B)
Acute angled

C)
Right angled

D)
Equilateral
View Solution
117)

In any \[\Delta ABC\]if \[a\cos B=b\cos A\], then the triangle is [MP PET 1984]

A)
Equilateral triangle

B)
Isosceles triangle

C)
Scalene

D)
Right angled
View Solution
118)

In a triangle \[ABC,\]if \[a\sin A=b\sin B\], then the nature of the triangle [MP PET 1983]

A)
\[a>b\]

B)
\[a<b\]

C)
\[a=b\]

D)
\[a+b=c\]
View Solution
119)

If in a triangle \[ABC\], \[\cos A+\cos B+\cos C=\frac{3}{2}\], then the triangle is [IIT 1984]

A)
Isosceles

B)
Equilateral

C)
Right angled

D)
None of these
View Solution
120)

If in a triangle ABC the sides \[AB\]and AC are perpendicular, then the true equation is

A)
\[\tan A+\tan B=0\]

B)
\[\tan B+\tan C=0\]

C)
\[\tan A+2\tan C=0\]

D)
\[\tan B.\tan C=1\]
View Solution
121)

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)
\[\frac{3\sqrt{15}}{4}c{{m}^{2}}\]

B)
\[\frac{15\sqrt{3}}{4}c{{m}^{2}}\]

C)
\[\frac{15}{4}c{{m}^{2}}\]

D)
\[\frac{3\sqrt{3}}{4}c{{m}^{2}}\]
View Solution
122)

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)
\[3\Delta \]

B)
\[2\Delta \]

C)
\[4\Delta \]

D)
\[-4\Delta \]
View Solution
123)

In a right triangle \[AC=BC\] and D is the mid point of AC cotangent of angle \[DBC\] is equal to

A)
2

B)
3

C)
1/2

D)
1/3
View Solution
124)

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)
\[\sqrt{\frac{(s-c)(s-a)}{s(s-b)}}\]

B)
\[\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\]

C)
\[\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}\]

D)
\[\sqrt{\frac{(s-a)s}{(s-b)(s-c)}}\]
View Solution
125)

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)
\[3ab{{c}^{2}}\]

B)
\[3{{a}^{2}}bc\]

C)
\[3abc\]

D)
\[3a{{b}^{2}}c\]
View Solution
126)

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)
\[{{67}^{o}}\]

B)
\[{{44}^{o}}\]

C)
\[{{113}^{o}}\]

D)
None of these
View Solution
127)

If \[A={{60}^{o}}\], \[a=5,b=4\sqrt{3}\]in \[\Delta ABC\], then B =

A)
\[{{30}^{o}}\]

B)
\[{{60}^{o}}\]

C)
\[{{90}^{o}}\]

D)
None of these
View Solution
128)

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)
\[\frac{15}{16}\]

B)
\[\frac{8}{15}\]

C)
\[\frac{8}{17}\]

D)
\[\frac{1}{2}\]
View Solution
129)

If \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\], then \[4s(s-a)(s-b)(s-c)=\] [EAMCET 1986; Pb. CET 1990]

A)
\[{{s}^{4}}\]

B)
\[{{b}^{2}}{{c}^{2}}\]

C)
\[{{c}^{2}}{{a}^{2}}\]

D)
\[{{a}^{2}}{{b}^{2}}\]
View Solution
130)

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)
\[\frac{a+b+c}{\Delta }\]

B)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{4{{\Delta }^{2}}}\]

C)
\[\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{{{\Delta }^{2}}}\]

D)
None of these
View Solution
131)

If the median of \[\Delta ABC\]through A is perpendicular to \[AB\], then

A)
\[\tan A+\tan B=0\]

B)
\[2\tan A+\tan B=0\]

C)
\[\tan A+2\tan B=0\]

D)
None of these
View Solution
132)

If A is the area and 2s the sum of 3 sides of triangle, then

A)
\[A\le \frac{{{s}^{2}}}{3\sqrt{3}}\]

B)
\[A\le \frac{{{s}^{2}}}{2}\]

C)
\[A>\frac{{{s}^{2}}}{\sqrt{3}}\]

D)
None of these
View Solution
133)

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

B)
3, 4

C)
4, 3

D)
6, 8
View Solution
134)

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)
\[\frac{5\pi }{6}\]
View Solution
135)

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

B)
One triangle is possible

C)
Two triangles are possible

D)
A right angled triangle is possible
View Solution
136)

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

B)
Equilateral

C)
Obtuse angled

D)
None of these
View Solution
137)

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
View Solution
138)

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
View Solution
139)

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)
\[2a=bc\]
View Solution
140)

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
View Solution
141)

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)
\[\tan A\tan B+\tan B\tan C+\tan C\tan A\]
View Solution
142)

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)
\[b=c\]
View Solution
143)

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.
View Solution
144)

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
View Solution
145)

If in a triangle ABC, a, b, c and angle A is given and \[c\sin A<a<c,\] then [UPSEAT 1999]

A)
\[{{b}_{1}}+{{b}_{2}}=2c\cos A\]

B)
\[{{b}_{1}}+{{b}_{2}}=c\cos A\]

C)
\[{{b}_{1}}+{{b}_{2}}=3c\cos A\]

D)
\[{{b}_{1}}+{{b}_{2}}=4c\sin A\]
View Solution
146)

In a \[\Delta ABC,\,\,{{a}^{2}}\sin \,\,2C+{{c}^{2}}\sin 2A=\] [EAMCET 2001]

A)
\[\Delta \]

B)
\[2\Delta \]

C)
\[3\Delta \]

D)
\[4\Delta \]
View Solution
147)

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
View Solution
148)

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)
\[\frac{3}{2}\]
View Solution
149)

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
View Solution
150)

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.
View Solution
151)

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 \[120{}^\circ \]
View Solution
152)

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)
\[\sqrt{2}\]

B)
\[\frac{\sqrt{3}+1}{2}\]

C)
\[\frac{\sqrt{3}-1}{2}\]

D)
1
View Solution
153)

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)
\[(b-c)\sin \frac{B-C}{2}=2a\cos \frac{A}{2}\]
View Solution
154)

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
View Solution
155)

\[ABC\] 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)
\[\frac{35\sqrt{2}}{2}cm\]
View Solution
156)

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 \[2{{\tan }^{-1}}\frac{1}{40}\]
View Solution

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JEE Main & Advanced Mathematics Trigonometric Equations Question Bank

done Relation between sides and angles, Solutions of triangles Total Questions - 156

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Relation between sides and angles, Solutions of triangles Total Questions - 156