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

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

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

$$n\pi + \frac{\pi }{4} - \frac{\pi }{3}$$

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

Questions 2 of 50

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

$$n\pi \pm \frac{\pi }{3}$$

$$2n\pi \pm \frac{\pi }{3}$$

$$2n\pi \pm \frac{\pi }{6}$$

$$n\pi \pm \frac{\pi }{6}$$

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

$$2n\pi \pm \frac{\pi }{3}$$

$$2n\pi + \frac{\pi }{4}$$

$$n\pi \pm \frac{\pi }{3}$$

$$n\pi - \frac{\pi }{3}$$

Questions 4 of 50

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

$$2n\pi \pm \frac{\pi }{6}$$

$$2n\pi \pm \frac{\pi }{3}$$

$$n\pi \pm \frac{\pi }{3}$$

$$n\pi \pm \frac{\pi }{6}$$

Questions 5 of 50

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

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

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

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

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

Questions 6 of 50

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

$$2n\pi \pm \frac{\pi }{4} + \frac{\pi }{{12}}$$

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

$$2n\pi \pm \frac{\pi }{4} - \frac{\pi }{{12}}$$

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

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

$$\theta = n\pi$$

$$\theta = 2n\pi \pm \frac{\pi }{2}$$

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

$$\theta = 2n\pi \pm \frac{\pi }{4}$$

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

$$\varphi$$

$$\frac{\pi }{4}$$

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

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

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

0

5

6

10

Questions 10 of 50

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

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

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

$$\theta = {150^o}$$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

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

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

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

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

Questions 14 of 50

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

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 $$\cos \left( {\theta - \frac{\pi }{4}} \right)$$=

$$\frac{1}{{2\sqrt 2 }}$$

$$\frac{1}{{\sqrt 2 }}$$

$$\frac{1}{{3\sqrt 2 }}$$

$$\frac{1}{{4\sqrt 2 }}$$

Questions 16 of 50

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

$$\frac{1}{{\sqrt 2 }}$$

$$\frac{1}{2}$$

$$\frac{1}{{2\sqrt 2 }}$$

$$\frac{{\sqrt 3 }}{2}$$

Questions 17 of 50

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

$$\frac{{7\pi }}{{24}}$$ 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

$$3\pi$$

$$6\pi$$

$$9\pi$$

$$12\pi$$

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

1

4

8

2

Questions 20 of 50

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

$$\pi$$

$$2\pi$$

$$\frac{\pi }{2}$$

None of these

Questions 21 of 50

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

$$\frac{b}{{a + b}}$$

$$\frac{b}{c}$$

$$\frac{c}{b}$$

None of these

Questions 22 of 50

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

$$\frac{{{a^2} - {b^2}}}{{{c^2}}}$$

$$\frac{{{a^2} + {b^2}}}{{{c^2}}}$$

$$\frac{{{c^2}}}{{{a^2} - {b^2}}}$$

$$\frac{{{c^2}}}{{{a^2} + {b^2}}}$$

Questions 23 of 50

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

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

$$1:2$$

2:1

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

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

Questions 25 of 50

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

0

1

$${a^2} + {b^2} + {c^2}$$

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

Questions 26 of 50

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

$$\frac{{a - b}}{{a - c}}$$

$$\frac{{a + b}}{{a + c}}$$

$$\frac{{{a^2} - {b^2}}}{{{a^2} - {c^2}}}$$

$$\frac{{{a^2} + {b^2}}}{{{a^2} + {c^2}}}$$

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

$$\sin C,\sin \frac{A}{2}$$

$$\cos C,\sin \frac{A}{2}$$

$$\sin C,\cos \frac{A}{2}$$

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

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

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

$${90^o}$$

$${45^o}$$

$${120^o}$$

None of these

Questions 31 of 50

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

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

$$\pi /2$$

$$5\pi /6$$

$$2\pi /3$$

$$3\pi /4$$

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

$$3:5:2$$

$$1:\sqrt 3 :2$$

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

$${135^o}$$

$${90^o}$$

$${60^o}$$

$${120^o}$$

Questions 35 of 50

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

$$a > b$$

$$a < b$$

$$a = b$$

$$a + b = c$$

Questions 36 of 50

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

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

$$\frac{\pi }{3}$$

$$\frac{\pi }{2}$$

$$\frac{{2\pi }}{3}$$

$$\frac{{5\pi }}{6}$$

Questions 38 of 50

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

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

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

$$45^\circ$$ 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

$$\frac{{16}}{9}$$

$$\frac{{16}}{7}$$

$$\frac{{11}}{7}$$

$$\frac{7}{{16}}$$

Questions 42 of 50

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{{5h}}{3}$$

$$\frac{{4h}}{3}$$

$$\frac{{7h}}{5}$$

$$\frac{{3h}}{2}$$

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

$$\frac{{\tan \alpha \,.\,\tan \beta }}{{\cot \alpha + \cot \beta }}$$

$$\frac{{\tan \alpha + \tan \beta }}{{\tan \alpha \,.\,\tan \beta }}$$

$$\frac{{\cot \alpha + \cot \beta }}{{\tan \alpha \,.\,\tan \beta }}$$

$$\frac{{\tan \alpha \,.\,\tan \,\beta }}{{\tan \alpha + \tan \beta }}$$

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

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]

$$a\sqrt 3$$

$$\frac{{a\sqrt 3 }}{{\sqrt 3 - 1}}$$

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

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

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