Trignometrical Equations Test 2

Total Questions:50 Total Time: 60 Min

Remaining:

 

Questions 1 of 50

Question:If \(\sqrt 3 \cos \,\theta + \sin \theta = \sqrt 2 ,\)then the most general value of \(\theta \) is

Answers Choices:

\(n\pi + {( - 1)^n}\frac{\pi }{4}\)

\({( - 1)^n}\frac{\pi }{4} - \frac{\pi }{3}\)

\(n\pi + \frac{\pi }{4} - \frac{\pi }{3}\)

\(n\pi + {( - 1)^n}\frac{\pi }{4} - \frac{\pi }{3}\)

Questions 2 of 50

Question:If \({\sin ^2}\theta - 2\cos \theta + \frac{1}{4} = 0,\)then the general value of \(\theta \)is

Answers Choices:

\(n\pi \pm \frac{\pi }{3}\)

\(2n\pi \pm \frac{\pi }{3}\)

\(2n\pi \pm \frac{\pi }{6}\)

\(n\pi \pm \frac{\pi }{6}\)

Questions 3 of 50

Question:If \(4{\sin ^2}\theta + 2(\sqrt 3 + 1)\cos \theta = 4 + \sqrt 3 \), then the general value of \(\theta \)is

Answers Choices:

\(2n\pi \pm \frac{\pi }{3}\)

\(2n\pi + \frac{\pi }{4}\)

\(n\pi \pm \frac{\pi }{3}\)

\(n\pi - \frac{\pi }{3}\)

Questions 4 of 50

Question:If \(\cot \theta + \cot \left( {\frac{\pi }{4} + \theta } \right) = 2\), then the general value of \(\theta \) is

Answers Choices:

\(2n\pi \pm \frac{\pi }{6}\)

\(2n\pi \pm \frac{\pi }{3}\)

\(n\pi \pm \frac{\pi }{3}\)

\(n\pi \pm \frac{\pi }{6}\)

Questions 5 of 50

Question:The general value of \(\theta \)satisfying the equation \(2{\sin ^2}\theta - 3\sin \theta - 2 = 0\) is

Answers Choices:

\(n\pi + {( - 1)^n}\frac{\pi }{6}\)

\(n\pi + {( - 1)^n}\frac{\pi }{2}\)

\(n\pi + {( - 1)^n}\frac{{5\pi }}{6}\)

\(n\pi + {( - 1)^n}\frac{{7\pi }}{6}\)

Questions 6 of 50

Question:The general solution of the equation \((\sqrt 3 - 1)\sin \theta + (\sqrt 3 + 1)\cos \theta = 2\) is

Answers Choices:

\(2n\pi \pm \frac{\pi }{4} + \frac{\pi }{{12}}\)

\(n\pi + {( - 1)^n}\frac{\pi }{4} + \frac{\pi }{{12}}\)

\(2n\pi \pm \frac{\pi }{4} - \frac{\pi }{{12}}\)

\(n\pi + {( - 1)^n}\frac{\pi }{4} - \frac{\pi }{{12}}\)

Questions 7 of 50

Question:The solution of the equation \(\left| {\,\begin{array}{*{20}{c}} {\cos \theta } & {\sin \theta } & {\cos \theta } \\ { - \sin \theta } & {\cos \theta } & {\sin \theta } \\ { - \cos \theta } & { - \sin \theta } & {\cos \theta } \\\end{array}\,} \right| = 0\), is

Answers Choices:

\(\theta = n\pi \)

\(\theta = 2n\pi \pm \frac{\pi }{2}\)

\(\theta = n\pi \pm {( - 1)^n}\frac{\pi }{4}\)

\(\theta = 2n\pi \pm \frac{\pi }{4}\)

Questions 8 of 50

Question:The set of values of x for which the expression \(\frac{{\tan 3x - \tan 2x}}{{1 + \tan 3x\tan 2x}} = 1\), is

Answers Choices:

\(\varphi \)

\(\frac{\pi }{4}\)

\(\left\{ {n\pi + \frac{\pi }{4}:n = 1,\,2,\,3.....} \right\}\)

\(\left\{ {2n\pi + \frac{\pi }{4}:n = 1,\,2,\,3.....} \right\}\)

Questions 9 of 50

Question:The number of values of x in the interval [0, 5\(\pi \)] satisfying the equation \(3{\sin ^2}x - 7\sin x + 2 = 0\)is

Answers Choices:

0

5

6

10

Questions 10 of 50

Question:The equation \(\sqrt 3 \sin x + \cos x = 4\)has

Answers Choices:

Only one solution

Two solutions

Infinitely many solutions

No solution

Questions 11 of 50

Question:The equation \(3\cos x + 4\sin x = 6\)has

Answers Choices:

Finite solution

Infinite solution

One solution

No solution

Questions 12 of 50

Question:The value of \(\theta \) in between \({0^o}\)and \({360^o}\)and satisfying the equation \(\tan \theta + \frac{1}{{\sqrt 3 }} = 0\)is equal to

Answers Choices:

\(\theta = {150^o}\)and \({300^o}\)

\(\theta = {120^o}\)and \({300^o}\)

\(\theta = {60^o}\)and \({240^o}\)

\(\theta = {150^o}\)and \({330^o}\)

Questions 13 of 50

Question:The solution of equation \({\cos ^2}\theta + \sin \theta + 1 = 0\) lies in the interval

Answers Choices:

\(\left( { - \frac{\pi }{4},\frac{\pi }{4}} \right)\)

\(\left( {\frac{\pi }{4},\frac{{3\pi }}{4}} \right)\)

\(\left( {\frac{{3\pi }}{4},\frac{{5\pi }}{4}} \right)\)

\(\left( {\frac{{5\pi }}{4},\frac{{7\pi }}{4}} \right)\)

Questions 14 of 50

Question:The number of solution of the equation \(2\cos ({e^x}) = {5^x} + {5^{ - x}}\), are

Answers Choices:

No solution

One solution

Two solutions

Infinitely many solutions

Questions 15 of 50

Question:If \(\tan (\pi \cos \theta ) = \cot (\pi \sin \theta ),\)then the value of \(\cos \left( {\theta - \frac{\pi }{4}} \right)\)=

Answers Choices:

\(\frac{1}{{2\sqrt 2 }}\)

\(\frac{1}{{\sqrt 2 }}\)

\(\frac{1}{{3\sqrt 2 }}\)

\(\frac{1}{{4\sqrt 2 }}\)

Questions 16 of 50

Question:If \(\tan (\pi \cos \theta ) = \cot (\pi \sin \theta )\), then \(\sin \left( {\theta + \frac{\pi }{4}} \right)\) equals

Answers Choices:

\(\frac{1}{{\sqrt 2 }}\)

\(\frac{1}{2}\)

\(\frac{1}{{2\sqrt 2 }}\)

\(\frac{{\sqrt 3 }}{2}\)

Questions 17 of 50

Question:The value of \(\theta \)lying between 0 and \(\pi /2\)and satisfying the equation \(\left| {\,\begin{array}{*{20}{c}} {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \\ {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \\ {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \\\end{array}\,} \right| = 0\)

Answers Choices:

\(\frac{{7\pi }}{{24}}\) or \(\frac{{11\pi }}{{24}}\)

\(\frac{{5\pi }}{{24}}\)

\(\frac{\pi }{{24}}\)

None of these

Questions 18 of 50

Question:The period of the function \(f(\theta ) = \sin \frac{\theta }{3} + \cos \frac{\theta }{2}\)is

Answers Choices:

\(3\pi \)

\(6\pi \)

\(9\pi \)

\(12\pi \)

Questions 19 of 50

Question:If the period of the function \(f(x) = \sin \left( {\frac{x}{n}} \right)\)is \(4\pi \), then n is equal to

Answers Choices:

1

4

8

2

Questions 20 of 50

Question:Period of \({\sin ^2}x\)is

Answers Choices:

\(\pi \)

\(2\pi \)

\(\frac{\pi }{2}\)

None of these\(\)

Questions 21 of 50

Question:In \(\Delta ABC,\frac{{\sin B}}{{\sin (A + B)}} = \)

Answers Choices:

\(\frac{b}{{a + b}}\)

\(\frac{b}{c}\)

\(\frac{c}{b}\)

None of these

Questions 22 of 50

Question:In \(\Delta ABC,\frac{{\sin (A - B)}}{{\sin (A + B)}} = \)

Answers Choices:

\(\frac{{{a^2} - {b^2}}}{{{c^2}}}\)

\(\frac{{{a^2} + {b^2}}}{{{c^2}}}\)

\(\frac{{{c^2}}}{{{a^2} - {b^2}}}\)

\(\frac{{{c^2}}}{{{a^2} + {b^2}}}\)

Questions 23 of 50

Question:If \({\cos ^2}A + {\cos ^2}C = {\sin ^2}B,\)then \(\Delta ABC\)is

Answers Choices:

Equilateral

Right angled

Isosceles

None of these

Questions 24 of 50

Question: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

Answers Choices:

\(1:2\)

2:1

\((\sqrt 5 + 1):(\sqrt 5 - 1)\)

\((\sqrt 5 - 1):(\sqrt 5 + 1)\)

Questions 25 of 50

Question:In \(\Delta ABC\), \({a^2}({\cos ^2}B - {\cos ^2}C) + \) \({b^2}({\cos ^2}C - {\cos ^2}A) + \)\({c^2}({\cos ^2}A - {\cos ^2}B) = \)

Answers Choices:

0

1

\({a^2} + {b^2} + {c^2}\)

\(2({a^2} + {b^2} + {c^2})\)

Questions 26 of 50

Question:In triangle \(ABC,\)\(\frac{{1 + \cos (A - B)\cos C}}{{1 + \cos (A - C)\cos B}} = \)

Answers Choices:

\(\frac{{a - b}}{{a - c}}\)

\(\frac{{a + b}}{{a + c}}\)

\(\frac{{{a^2} - {b^2}}}{{{a^2} - {c^2}}}\)

\(\frac{{{a^2} + {b^2}}}{{{a^2} + {c^2}}}\)

Questions 27 of 50

Question: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

Answers Choices:

\(\sin C,\sin \frac{A}{2}\)

\(\cos C,\sin \frac{A}{2}\)

\(\sin C,\cos \frac{A}{2}\)

None of these

Questions 28 of 50

Question:In \(\Delta ABC\), \((b - c)\cot \frac{A}{2} + (c - a)\cot \frac{B}{2} + (a - b)\)\(\cot \frac{C}{2}\) is equal to

Answers Choices:

0

1

\( \pm 1\)

2

Questions 29 of 50

Question:If in a \(\Delta ABC\), \(\cos A + 2\cos B + \cos C = 2\), then\(a,b,c\)are in

Answers Choices:

A. P.

H. P.

G. P.

None of these

Questions 30 of 50

Question:If in a \(\Delta ABC\), \(\cos 3A + \cos 3B + \cos 3C = 1\), then one angle must be exactly equal to

Answers Choices:

\({90^o}\)

\({45^o}\)

\({120^o}\)

None of these

Questions 31 of 50

Question:If in a \(\Delta ABC,\,\angle A = {45^o},\,\,\angle C = {60^o}\), then \(a + c\sqrt 2 = \)

Answers Choices:

b

2b

\(\sqrt {2b} \)

\(\sqrt 3 b\)

Questions 32 of 50

Question:If the lengths of the sides of a triangle are 3, 5, 7, then the largest angle of the triangle is

Answers Choices:

\(\pi /2\)

\(5\pi /6\)

\(2\pi /3\)

\(3\pi /4\)

Questions 33 of 50

Question:The ratio of the sides of triangle ABC is \(1:\sqrt 3 :2\). The ratio of \(A:B:C\)is

Answers Choices:

\(3:5:2\)

\(1:\sqrt 3 :2\)

03:02:01

01:02:03

Questions 34 of 50

Question:In a triangle \(ABC,\,\,b = \sqrt 3 \), \(c = 1\)and \(\angle A = {30^o}\), then the largest angle of the triangle is

Answers Choices:

\({135^o}\)

\({90^o}\)

\({60^o}\)

\({120^o}\)

Questions 35 of 50

Question:In a triangle \(ABC,\)if \(a\sin A = b\sin B\), then the nature of the triangle

Answers Choices:

\(a > b\)

\(a < b\)

\(a = b\)

\(a + b = c\)

Questions 36 of 50

Question:If in a triangle \(ABC\), \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle is

Answers Choices:

Isosceles

Equilateral

Right angled

None of these

Questions 37 of 50

Question: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

Answers Choices:

\(\frac{\pi }{3}\)

\(\frac{\pi }{2}\)

\(\frac{{2\pi }}{3}\)

\(\frac{{5\pi }}{6}\)

Questions 38 of 50

Question:We are given b, c and \(\sin B\) such that B is acute and \(b < c\sin B\). Then

Answers Choices:

No triangle is possible

One triangle is possible

Two triangles are possible

A right angled triangle is possible

Questions 39 of 50

Question: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

Answers Choices:

A.G.P.

H.P.

G.P.

A.P.

Questions 40 of 50

Question: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

Answers Choices:

\(45^\circ \) or \(135^\circ \)

\(30^\circ \) or \(100^\circ \)

\(50^\circ \) or \(100^\circ \)

\(60^\circ \) or \(120^\circ \)

Questions 41 of 50

Question:In a triangle ABC, \(a:b:c = 4:5:6\). The ratio of the radius of the circumcircle to that of the incircle is

Answers Choices:

\(\frac{{16}}{9}\)

\(\frac{{16}}{7}\)

\(\frac{{11}}{7}\)

\(\frac{7}{{16}}\)

Questions 42 of 50

Question:Which is true in the following [UPSEAT 1999]

Answers Choices:

\(a\cos A + b\cos B + c\cos C = R\sin A\sin B\sin C\)

\(a\cos A + b\cos B + c\cos C = 2R\sin A\sin B\sin C\)

\(a\cos A + b\cos B + c\cos C = 4R\sin A\sin B\sin C\)

\(a\cos A + b\cos B + c\cos C = 8R\sin A\sin B\sin C\)

Questions 43 of 50

Question:The angle of elevation of the top of a tower from a point A due south of the tower is \(\alpha \)and from a point B due east of the tower is \(\beta \). If AB =d, then the height of the tower is

Answers Choices:

\(\frac{d}{{\sqrt {{{\tan }^2}\alpha - {{\tan }^2}\beta } }}\)

\(\frac{d}{{\sqrt {{{\tan }^2}\alpha + {{\tan }^2}\beta } }}\)

\(\frac{d}{{\sqrt {{{\cot }^2}\alpha + {{\cot }^2}\beta } }}\)

\(\frac{d}{{\sqrt {{{\cot }^2}\alpha - {{\cot }^2}\beta } }}\)

Questions 44 of 50

Question:A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \(60^\circ \). When he retires 40 meters from the bank, he finds the angle to be \(30^\circ \). The breadth of the river is

Answers Choices:

20 m

40 m

30 m

60 m

Questions 45 of 50

Question:Two vertical poles of equal heights are 120 m apart. On the line joining their bottoms, A and B are two points. Angle of elevation of the top of one pole from A is \(45^\circ \) and that of the other pole from B is also \(45^\circ \). If AB = 30 m, then the height of each pole is

Answers Choices:

40 m

45 m

50 m

42 m

Questions 46 of 50

Question:At a distance 2h from the foot of a tower of height h, the tower and a pole at the top of the tower subtend equal angles. Height of the pole should be

Answers Choices:

\(\frac{{5h}}{3}\)

\(\frac{{4h}}{3}\)

\(\frac{{7h}}{5}\)

\(\frac{{3h}}{2}\)

Questions 47 of 50

Question:From an aeroplane vertically over a straight horizontally road, the angles of depression of two consecutive mile stones on opposite sides of the aeroplane are observed to be a and b, then the height in miles of aeroplane above the road is

Answers Choices:

\(\frac{{\tan \alpha \,.\,\tan \beta }}{{\cot \alpha + \cot \beta }}\)

\(\frac{{\tan \alpha + \tan \beta }}{{\tan \alpha \,.\,\tan \beta }}\)

\(\frac{{\cot \alpha + \cot \beta }}{{\tan \alpha \,.\,\tan \beta }}\)

\(\frac{{\tan \alpha \,.\,\tan \,\beta }}{{\tan \alpha + \tan \beta }}\)

Questions 48 of 50

Question:A balloon is observed simultaneously from three points A, B and C on a straight road directly under it. The angular elevation at B is twice and at C is thrice that of A. If the distance between A and B is 200 metres and the distance between B and C is 100 metres, then the height of balloon is given by

Answers Choices:

50 metres

\(50\,\sqrt 3 \) metres

\(50\,\sqrt 2 \) Metres

None of these

Questions 49 of 50

Question:The angles of elevation of the top of a tower from the top and bottom at a building of height a are \({30^0}\) and \({45^0}\) respectively. If the tower and the building stand at the same level, then the height of the tower is [Karnataka CET 2000]

Answers Choices:

\(a\sqrt 3 \)

\(\frac{{a\sqrt 3 }}{{\sqrt 3 - 1}}\)

\(\frac{{a\,(3 + \sqrt 3 )}}{2}\)

\(a\,(\sqrt 3 - 1)\)

Questions 50 of 50

Question:A ladder 5 metre long leans against a vertical wall. The bottom of the ladder is 3 metre from the wall. If the bottom of the ladder is pulled 1 metre farther from the wall, how much does the top of the ladder slide down the wall

Answers Choices:

1 m

7 m

2 m

None of these