-
question_answer1)
The general solution of \[\sin x-3\sin 2x+\sin 3x=\] \[\cos x-3\cos 2x+\cos 3x\] is [IIT 1989]
A)
\[n\pi +\frac{\pi }{8}\] done
clear
B)
\[\frac{n\pi }{2}+\frac{\pi }{8}\] done
clear
C)
\[{{(-1)}^{n}}\frac{n\pi }{2}+\frac{\pi }{8}\] done
clear
D)
\[2n\pi +{{\cos }^{-1}}\frac{3}{2}\] done
clear
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question_answer2)
If \[5{{\cos }^{2}}\theta +7{{\sin }^{2}}\theta -6=0\], then the general value of \[\theta \]is
A)
\[2n\pi \pm \frac{\pi }{4}\] done
clear
B)
\[n\pi \pm \frac{\pi }{4}\] done
clear
C)
\[n\pi +{{(-1)}^{n}}\frac{\pi }{4}\] done
clear
D)
None of these done
clear
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question_answer3)
If \[\cos \theta +\cos 7\theta +\cos 3\theta +\cos 5\theta =0\], then \[\theta \] [Dhanbad Engg. 1972]
A)
\[\frac{n\pi }{4}\] done
clear
B)
\[\frac{n\pi }{2}\] done
clear
C)
\[\frac{n\pi }{8}\] done
clear
D)
None of these done
clear
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question_answer4)
If \[\tan \theta =-\frac{1}{\sqrt{3}}\]and \[\sin \theta =\frac{1}{2}\], \[\cos \theta =-\frac{\sqrt{3}}{2}\], then the principal value of \[\theta \] will be [MP PET 1983, 84]
A)
\[\frac{\pi }{6}\] done
clear
B)
\[\frac{5\pi }{6}\] done
clear
C)
\[\frac{7\pi }{6}\] done
clear
D)
\[-\frac{\pi }{6}\] done
clear
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question_answer5)
The general solution of \[a\cos x+b\sin x=c,\] where \[a,\,\,b,\,\,c\] are constants
A)
\[x=n\pi +{{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] done
clear
B)
\[x=2n\pi -{{\tan }^{-1}}\left( \frac{b}{a} \right)\] done
clear
C)
\[x=2n\pi -{{\tan }^{-1}}\left( \frac{b}{a} \right)\pm {{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] done
clear
D)
\[x=2n\pi +{{\tan }^{-1}}\left( \frac{b}{a} \right)\pm {{\cos }^{-1}}\left( \frac{c}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)\] done
clear
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question_answer6)
If \[\sec 4\theta -\sec 2\theta =2\], then the general value of \[\theta \] is [IIT 1963]
A)
\[(2n+1)\frac{\pi }{4}\] done
clear
B)
\[(2n+1)\frac{\pi }{10}\] done
clear
C)
\[n\pi +\frac{\pi }{2}\]or \[\frac{n\pi }{5}+\frac{\pi }{10}\] done
clear
D)
None of these done
clear
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question_answer7)
If \[\sin 2x+\sin 4x=2\sin 3x,\]then \[x\]= [EAMCET 1989]
A)
\[\frac{n\pi }{3}\] done
clear
B)
\[n\pi +\frac{\pi }{3}\] done
clear
C)
\[2n\pi \pm \frac{\pi }{3}\] done
clear
D)
None of these done
clear
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question_answer8)
If \[\tan (\cot x)=\cot (\tan x),\]then \[\sin 2x\]= [MP PET 1999; Pb. CET 2001]
A)
\[(2n+1)\frac{\pi }{4}\] done
clear
B)
\[\frac{4}{(2n+1)\pi }\] done
clear
C)
\[4\pi (2n+1)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer9)
The solution set of \[(5+4\cos \theta )(2\cos \theta +1)=0\]in the interval \[[0,\,\,2\pi ]\] is [EAMCET 2003]
A)
\[\left\{ \frac{\pi }{3},\,\frac{2\pi }{3} \right\}\] done
clear
B)
\[\left\{ \frac{\pi }{3},\,\pi \right\}\] done
clear
C)
\[\left\{ \frac{2\pi }{3},\frac{4\pi }{3} \right\}\] done
clear
D)
\[\left\{ \frac{2\pi }{3},\frac{5\pi }{3} \right\}\] done
clear
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question_answer10)
If \[1+\sin x+{{\sin }^{2}}x+.....\]to \[\infty =4+2\sqrt{3},\,0<x<\pi ,\] then [DCE 2001]
A)
\[x=\frac{\pi }{6}\] done
clear
B)
\[x=\frac{\pi }{3}\] done
clear
C)
\[x=\frac{\pi }{3}\]or \[\frac{\pi }{6}\] done
clear
D)
\[x=\frac{\pi }{3}\]or \[\frac{2\pi }{3}\] done
clear
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question_answer11)
If the solution for \[\theta \]of \[\cos p\theta +\cos q\theta =0,\ p>0,\ q>0\]are in A.P., then the numerically smallest common difference of A.P. is [Kerala (Engg.) 2001]
A)
\[\frac{\pi }{p+q}\] done
clear
B)
\[\frac{2\pi }{p+q}\] done
clear
C)
\[\frac{\pi }{2(p+q)}\] done
clear
D)
\[\frac{1}{p+q}\] done
clear
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question_answer12)
The number of solution of the given equation \[a\sin x+b\cos x=c\] , where \[|c|\,>\,\sqrt{{{a}^{2}}+{{b}^{2}}},\]is [DCE 1998]
A)
1 done
clear
B)
2 done
clear
C)
Infinite done
clear
D)
None of these done
clear
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question_answer13)
The number of pairs (x, y) satisfying the equations \[\sin x+\sin y=\sin (x+y)\] and \[|x|+|y|=1\]is
A)
2 done
clear
B)
4 done
clear
C)
6 done
clear
D)
\[\infty \] done
clear
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question_answer14)
If \[a,\ b,\ c\]are the sides of a triangle ABC, then which of the following inequalities is not true [Kurukshetra CEE 1996]
A)
\[8abc\le (a+b)(b+c)(c+a)\] done
clear
B)
\[3abc\le {{a}^{3}}+{{b}^{3}}+{{c}^{3}}\] done
clear
C)
\[6abc\le bc(b+c)+ca(c+a)+ab(a+b)\] done
clear
D)
\[abc\le (a+b-c)(b+c-a)(c+a-b)\] done
clear
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question_answer15)
In \[\Delta ABC,\ a(b\cos C-c\cos B)=\] [EAMCET 1981]
A)
\[{{a}^{2}}-{{b}^{2}}\] 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
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question_answer16)
AB is a vertical tower. The point A is on the ground and C is the middle point of AB. The part CB subtend an angle \[\alpha \]at a point P on the ground. If \[AP=n\,AB,\]then the correct relation is [MNR 1989; IIT 1980]
A)
\[n=({{n}^{2}}+1)\tan \alpha \] done
clear
B)
\[n=(2{{n}^{2}}-1)\tan \alpha \] done
clear
C)
\[{{n}^{2}}=(2{{n}^{2}}+1)\tan \alpha \] done
clear
D)
\[n=(2{{n}^{2}}+1)\tan \alpha \] done
clear
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question_answer17)
In \[\Delta ABC,\ \frac{1}{a}{{\cos }^{2}}\frac{A}{2}+\frac{1}{b}{{\cos }^{2}}\frac{B}{2}+\frac{1}{c}{{\cos }^{2}}\frac{C}{2}=\]
A)
\[s\] done
clear
B)
\[\frac{s}{abc}\] done
clear
C)
\[\frac{{{s}^{2}}}{abc}\] done
clear
D)
\[\frac{{{s}^{3}}}{abc}\] done
clear
View Solution play_arrow
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question_answer18)
\[\Delta ABC,\] if \[\cos \frac{A}{2}=\sqrt{\frac{b+c}{2c}}\], then [MP PET 1990]
A)
\[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] done
clear
B)
\[{{b}^{2}}+{{c}^{2}}={{a}^{2}}\] done
clear
C)
\[{{c}^{2}}+{{a}^{2}}={{b}^{2}}\] done
clear
D)
\[b-c=c-a\] done
clear
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question_answer19)
In a triangle \[ABC\], \[\tan \frac{A}{2}=\frac{5}{6}\] and \[\tan \frac{C}{2}=\frac{2}{5},\] then [EAMCET 1994]
A)
\[a,\ b,\ c\]are in A.P. done
clear
B)
\[\cos A,\ \cos B,\ \cos C\]are in A.P. done
clear
C)
\[\sin A,\ \sin B,\ \sin C\]are in A.P. done
clear
D)
(a) and (c) both done
clear
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question_answer20)
In a \[\Delta ABC,\] if \[(\sin A+\sin B+\sin C)\] \[(\sin A+\sin B-\sin C)\] = \[3\sin A\sin B,\] then the angle C is equal to [AMU 1999]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{6}\] done
clear
View Solution play_arrow
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question_answer21)
If the two angle on the base of a triangle are \[{{\left( 22\frac{1}{2} \right)}^{o}}\] and \[{{\left( 112\frac{1}{2} \right)}^{o}}\], then the ratio of the height of the triangle to the length of the base is [MP PET 1993; Pb CET 2002]
A)
1 : 2 done
clear
B)
2 : 1 done
clear
C)
2 : 3 done
clear
D)
1: 1 done
clear
View Solution play_arrow
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question_answer22)
If a,b are different values of \[x\] satisfying \[a\cos x+b\sin x=c,\] then \[\tan \text{ }\left( \frac{\alpha +\beta }{2} \right)=\] [EAMCET 1986; Orissa JEE 2003]
A)
\[a+b\] done
clear
B)
\[a-b\] done
clear
C)
\[\frac{b}{a}\] done
clear
D)
\[\frac{a}{b}\] done
clear
View Solution play_arrow
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question_answer23)
In triangle \[ABC\]if \[\frac{\cos A}{a}=\frac{\cos B}{b}=\frac{\cos C}{c}\], then the triangle is [Karnataka 1991; Pb. CET 1989]
A)
Right angled done
clear
B)
Obtuse angled done
clear
C)
Equilateral done
clear
D)
Isosceles done
clear
View Solution play_arrow
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question_answer24)
If the perpendicular AD divides the base of the triangle ABC such that BD, CD and AD are in the ratio 2, 3 and 6, then angle A is equal to [MP PET 1993]
A)
\[\frac{\pi }{2}\] done
clear
B)
\[\frac{\pi }{3}\] done
clear
C)
\[\frac{\pi }{4}\] done
clear
D)
\[\frac{\pi }{6}\] done
clear
View Solution play_arrow
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question_answer25)
In a triangle, the length of the two larger sides are 10 cm and 9 cm respectively. If the angles of the triangle are in A.P., then the length of the third side in cm can be [MP PET 1990, 2001; IIT 1987; DCE 2001]
A)
\[5-\sqrt{6}\]only done
clear
B)
\[5+\sqrt{6}\]only done
clear
C)
\[5-\sqrt{6}\]or \[5+\sqrt{6}\] done
clear
D)
Neither\[5-\sqrt{6}\]nor \[5+\sqrt{6}\] done
clear
View Solution play_arrow
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question_answer26)
In triangle \[ABC\], if \[\angle A=45{}^\circ ,\] \[\angle B=75{}^\circ \], then \[a+c\sqrt{2}\] = [IIT 1988]
A)
0 done
clear
B)
1 done
clear
C)
b done
clear
D)
2b done
clear
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question_answer27)
In \[\Delta ABC,\]if\[\sin A:\sin C=\sin (A-B):\sin (B-C),\] then
A)
\[a,\ b,\ c\]are in A.P. done
clear
B)
\[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\]are in A.P. done
clear
C)
\[{{a}^{2}},\ {{b}^{2}},\ {{c}^{2}}\]are in G. P. done
clear
D)
None of these done
clear
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question_answer28)
If in a triangle \[ABC\], \[\frac{\sin A}{4}=\frac{\sin B}{5}=\frac{\sin C}{6}\], then the value of \[\cos A+\cos B+\cos C\]is equal to
A)
\[\frac{69}{48}\] done
clear
B)
\[\frac{96}{48}\] done
clear
C)
\[\frac{48}{69}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer29)
In a \[\Delta ABC,\]if \[a=2x,\]\[b=2y\]and \[\angle C=120{}^\circ \], then the area of the triangle is [MP PET 1986, 2002]
A)
\[xy\] done
clear
B)
\[xy\sqrt{3}\] done
clear
C)
\[3xy\] done
clear
D)
\[2xy\] done
clear
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question_answer30)
If the sides of a \[\Delta \]be\[({{x}^{2}}+x+1),\,(2x+1)\] and \[({{x}^{2}}-1),\]then the greatest angle is [EAMCET 1987; Kerala (Engg.) 2001]
A)
\[{{105}^{o}}\] done
clear
B)
\[{{120}^{o}}\] done
clear
C)
\[{{135}^{o}}\] done
clear
D)
None done
clear
View Solution play_arrow
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question_answer31)
In a \[\Delta ABC,\]if \[3a=b+c,\]then the value of \[\cot \frac{B}{2}\cot \frac{C}{2}\] is [Pb. CET 2003; Roorkee 1986; MP PET 1990, 97, 98, 2003; EAMCET 2003; Orissa JEE 2005]
A)
1 done
clear
B)
2 done
clear
C)
\[\sqrt{3}\] done
clear
D)
\[\sqrt{2}\] done
clear
View Solution play_arrow
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question_answer32)
In \[\Delta ABC,\]if \[8{{R}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}},\]then the triangle is
A)
Right angled done
clear
B)
Equilateral done
clear
C)
Acute angled done
clear
D)
Obtuse angled done
clear
View Solution play_arrow
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question_answer33)
In a \[\Delta ABC,\]let \[\angle C=\frac{\pi }{2}.\]If \[r\]and \[R\]are in radius and the circum-radius respectively of the triangle, then \[2(r+R)\] is equal to [IIT Screening 2000; AIEEE 2005]
A)
\[a+b\] done
clear
B)
\[b+c\] done
clear
C)
\[c+a\] done
clear
D)
\[a+b+c\] done
clear
View Solution play_arrow
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question_answer34)
A tower is situated on horizontal plane. From two points, the line joining three points passes through the base and which are \[a\]and \[b\]distance from the base. The angle of elevation of the top are \[\alpha \]and \[90{}^\circ -\alpha \]and \[\theta \]is that angle which two points joining the line makes at the top, the height of tower will be [UPSEAT 1999]
A)
\[\frac{a+b}{a-b}\] done
clear
B)
\[\frac{a-b}{a+b}\] done
clear
C)
\[\sqrt{ab}\] done
clear
D)
\[{{(ab)}^{1/3}}\] done
clear
View Solution play_arrow
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question_answer35)
ABC is triangular park with AB = AC = 100 m. A clock tower is situated at the mid-point of BC. The angles of elevation of the top of the tower at\[A\]and\[B\]are \[{{\cot }^{-1}}3.2\] and \[\text{cose}{{\text{c}}^{-1}}2.6\]respectively. The height of the tower is[EAMCET 1992]
A)
50 m done
clear
B)
25 m done
clear
C)
40 m done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer36)
From the bottom of a pole of height h, the angle of elevation of the top of a tower is \[\alpha \]and the pole subtends angle \[\beta \]at the top of the tower. The height of the tower is [Roorkee 1988]
A)
\[\frac{h\tan (\alpha -\beta )}{\tan (\alpha -\beta )-\tan \alpha }\] done
clear
B)
\[\frac{h\cot (\alpha -\beta )}{\cot (\alpha -\beta )-\cot \alpha }\] done
clear
C)
\[\frac{\cot (\alpha -\beta )}{\cot (\alpha -\beta )-\cot \alpha }\] done
clear
D)
None of these done
clear
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question_answer37)
A person observes the angle of elevation of a building as 30°. The person proceeds towards the building with a speed of \[25(\sqrt{3}-1)m/hour.\]After \[2\,hours\], he observes the angle of elevation as 45°. The height of the building (in meter) is [UPSEAT 2003]
A)
100 done
clear
B)
50 done
clear
C)
\[50(\sqrt{3}+1)\] done
clear
D)
\[50(\sqrt{3}-1)\] done
clear
View Solution play_arrow
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question_answer38)
A man from the top of a 100 meters high tower sees a car moving towards the tower at an angle of depression of 30 °. After some time, the angle of depression becomes\[{{60}^{o}}\]. The distance (in meters) travelled by the car during the time is [IIT Screening 2001]
A)
\[100\sqrt{3}\] done
clear
B)
\[\frac{200\sqrt{3}}{3}\] done
clear
C)
\[\frac{100\sqrt{3}}{3}\] done
clear
D)
\[200\sqrt{3}\] done
clear
View Solution play_arrow
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question_answer39)
The angular depressions of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first are \[\theta \] and \[\varphi \] respectively, then the distance between their tops when \[\tan \theta =\frac{4}{3}\]and \[\tan \varphi =\frac{5}{2}\], is [Pb. CET 2004; IIT 1965]
A)
\[\frac{150}{\sqrt{3}}metres\] done
clear
B)
\[100\sqrt{3}metres\] done
clear
C)
\[150metres\] done
clear
D)
\[100metres\] done
clear
View Solution play_arrow
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question_answer40)
The value of \[n\in Z\] for which the function \[f(x)=\frac{\sin nx}{\sin (x/n)}\] has \[4\pi \] as its period, is
A)
2 done
clear
B)
3 done
clear
C)
4 done
clear
D)
5 done
clear
View Solution play_arrow