# Trignometrical Equations Test 6

Total Questions:50 Total Time: 60 Min

Remaining:

## Questions 1 of 50

Question:If $$\frac{{1 - \cos 2\theta }}{{1 + \cos 2\theta }} = 3$$, then the general value of$$\theta$$is

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

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

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

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

## Questions 2 of 50

Question:If $$3({\sec ^2}\theta + {\tan ^2}\theta ) = 5$$, then the general value of $$\theta$$ is

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

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

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

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

## Questions 3 of 50

Question:If $$\cos 7\theta = \cos \theta - \sin 4\theta$$, then the general value of $$\theta$$is

$$\frac{{n\pi }}{4},\frac{{n\pi }}{3} + \frac{\pi }{{18}}$$

$$\frac{{n\pi }}{3},\frac{{n\pi }}{3} + {( - 1)^n}\frac{\pi }{{18}}$$

$$\frac{{n\pi }}{4},\frac{{n\pi }}{3} + {( - 1)^n}\frac{\pi }{{18}}$$

$$\frac{{n\pi }}{6},\frac{{n\pi }}{3} + {( - 1)^n}\frac{\pi }{{18}}$$

## Questions 4 of 50

Question:The general value of $$\theta$$ satisfying the equation $$\tan \theta + \tan \left( {\frac{\pi }{2} - \theta } \right) = 2$$, is

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

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

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

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

## Questions 5 of 50

Question:If $$\cos 2\theta = (\sqrt 2 + 1)\,\,\left( {\cos \theta - \frac{1}{{\sqrt 2 }}} \right)$$, then the value of $$\theta$$is

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

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

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

None of these

## Questions 6 of 50

Question:General solution of $$\tan 5\theta = \cot 2\theta$$ is

$$\theta = \frac{{n\pi }}{7} + \frac{\pi }{{14}}$$

$$\theta = \frac{{n\pi }}{7} + \frac{\pi }{5}$$

$$\theta = \frac{{n\pi }}{7} + \frac{\pi }{2}$$

$$\theta = \frac{{n\pi }}{7} + \frac{\pi }{3},n \in Z$$

## Questions 7 of 50

Question:The equation $$3{\sin ^2}x + 10\cos x - 6 = 0$$ is satisfied, if

$$x = n\pi \pm {\cos ^{ - 1}}(1/3)$$

$$x = 2n\pi \pm {\cos ^{ - 1}}(1/3)$$

$$x = n\pi \pm {\cos ^{ - 1}}(1/6)$$

$$x = 2n\pi \pm {\cos ^{ - 1}}(1/6)$$

## Questions 8 of 50

Question:The equation $$\sin x\cos x = 2$$has

One solution

Two solutions

Infinite solutions

No solutions

## Questions 9 of 50

Question:The number of solutions of the given equation $$\tan \theta + \sec \theta = \sqrt 3 ,$$ where $$0 < \theta < 2\pi$$is

0

1

2

3

## Questions 10 of 50

Question:The values of $$\theta$$ satisfying $$\sin 7\theta = \sin 4\theta - \sin \theta$$ and $$0 < \theta < \frac{\pi }{2}$$ are

$$\frac{\pi }{9},\frac{\pi }{4}$$

$$\frac{\pi }{3},\frac{\pi }{9}$$

$$\frac{\pi }{6},\frac{\pi }{9}$$

$$\frac{\pi }{3},\frac{\pi }{4}$$

## Questions 11 of 50

Question:The expression $$(1 + \tan x + {\tan ^2}x)$$ $$(1 - \cot x + {\cot ^2}x)$$ has the positive values for x, given by

$$0 \le x \le \frac{\pi }{2}$$

$$0 \le x \le \pi$$

For all $$x \in R$$

$$x \ge 0$$

## Questions 12 of 50

Question:The smallest positive angle which satisfies the equation $$2{\sin ^2}\theta + \sqrt 3 \cos \theta + 1 = 0$$, is

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

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

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

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

## Questions 13 of 50

Question:$$\cot \theta = \sin 2\theta (\theta \ne n\pi$$, n is integer), if $$\theta =$$

$${45^o}$$ and $${60^o}$$

$${45^o}$$and $${90^o}$$

$${45^o}$$only

$${90^o}$$only

## Questions 14 of 50

Question:Period of $$\frac{{\sin \theta + \sin 2\theta }}{{\cos \theta + \cos 2\theta }}$$is

$$2\pi$$

$$\pi$$

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

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

## Questions 15 of 50

Question:Period of $$\cos (7x - 5)$$is

$$\frac{{2\pi - 5}}{7}$$

$$2\pi - 5$$

$$\frac{{2\pi }}{7}$$

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

## Questions 16 of 50

Question:The period of the function |sinp x| is

$${\pi ^2}$$

$$2\pi$$

2

1

## Questions 17 of 50

Question:The period of $$f(x) = \sin \left( {\frac{{\pi x}}{{n - 1}}} \right) + \cos \,\left( {\frac{{\pi x}}{n}} \right)\,\,,\,n \in Z$$, $$n > 2$$ is

$$2\pi n(n - 1)$$

$$4n\,(n - 1)$$

$$2n\,(n - 1)$$

None of these

## Questions 18 of 50

Question:In $$\Delta ABC,$$$$1 - \tan \frac{A}{2}\tan \frac{B}{2} =$$

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

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

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

$$\frac{{4a}}{{a + b + c}}$$

## Questions 19 of 50

Question:In $$\Delta ABC,$$$${b^2}\cos 2A - {a^2}\cos 2B =$$

$${b^2} - {a^2}$$

$${b^2} - {c^2}$$

$${c^2} - {a^2}$$

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

## Questions 20 of 50

Question:If the lengths of the sides of a triangle be $$7,4\sqrt 3$$ and $$\sqrt {13}$$cm, then the smallest angle is

$${15^o}$$

$${30^o}$$

$$60^\circ$$

$${45^o}$$

## Questions 21 of 50

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

01:02:03

2:3:4

3:4:5

4:5:6

## Questions 22 of 50

Question:If in a triangle $$ABC$$, $$b = \sqrt 3$$, $$c = 1$$ and $$B - C = {90^o}$$then $$\angle A$$ is

$${30^o}$$

$${45^o}$$

$${75^o}$$

$${15^o}$$

## Questions 23 of 50

Question: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

A. P.

G. P.

H. P.

A. G.

## Questions 24 of 50

Question:If $$b = 3,c = 4$$and $$B = \frac{\pi }{3}$$, then the number of triangle that can be constructed is

Infinite

Two

One

Nil

## Questions 25 of 50

Question:If $${a^2},{b^2},{c^2}$$are in A. P. then which of the following are also in A.P.

$$\sin A,\sin B,\sin C$$

$$\tan A,\tan B,\tan C$$

$$\cot A,\cot B,\cot C$$

None of these

## Questions 26 of 50

Question:The smallest angle of the triangle whose sides are $$6 + \sqrt {12} ,\sqrt {48} ,\sqrt {24}$$is

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

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

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

None of these

## Questions 27 of 50

Question:If $$A = {30^o},c = 7\sqrt 3$$and $$C = {90^o}$$in $$\Delta ABC$$, then a =

$$7\sqrt 3$$

$$\frac{{7\sqrt 3 }}{2}$$

$$\frac{7}{2}$$

None of these

## Questions 28 of 50

Question:In any triangle $$AB = 2,BC = 4,CA = 3$$and D is mid point of BC, then

$$\cos B = \frac{{11}}{6}$$

$$\cos B = \frac{7}{8}$$

$$AD = 2.4$$

$$A{D^2} = 2.5$$

## Questions 29 of 50

Question:If the angles of a triangle are in the ratio 4:1:1, then the ratio of the longest side to the perimeter is

$$\sqrt 3 :(2 + \sqrt 3 )$$

$$1:6$$

$$1:(2 + \sqrt 3 )$$

$$2:3$$

## Questions 30 of 50

Question:If in triangle $$ABC,\frac{{{a^2} - {b^2}}}{{{a^2} + {b^2}}} = \frac{{\sin (A - B)}}{{\sin (A + B)}}$$, then the triangle is

Right angled

Isosceles

Right angled or isosceles

Right angled isosecles

## Questions 31 of 50

Question:In a triangle $$ABC$$, sin$$A:\sin B$$: $$\sin C = 1:2:3$$. If $$b = 4$$ cm, the perimeter of the triangle is

$$6cm$$

$$24cm$$

$$12cm$$

$$8cm$$

## Questions 32 of 50

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

$${30^o}$$

$${60^o}$$

$${90^o}$$

None of these

## Questions 33 of 50

Question:If $$\Delta = {a^2} - {(b - c)^2}$$, where $$\Delta$$is the area of triangle $$ABC$$, then tan A is equal to

$$\frac{{15}}{{16}}$$

$$\frac{8}{{15}}$$

$$\frac{8}{{17}}$$

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

## Questions 34 of 50

Question:If a triangle $$PQR$$, $$\sin P,\;\sin Q,\;\sin R$$are in A.P., then

The altitudes are in A.P.

The altitudes are in H.P.

The medians are in G.P.

The medians are in A.P.

## Questions 35 of 50

Question: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

A.P.

G.P.

H.P.

None of these

## Questions 36 of 50

Question:If the sides of a triangle are in ratio 3 : 7 : 8, then R : r is equal to

02:07

07:02

03:07

07:03

## Questions 37 of 50

Question:If the sides of triangle are 13, 14, 15, then the radius of its incircle is

$$\frac{{67}}{8}$$

$$\frac{{65}}{4}$$

4

24

## Questions 38 of 50

Question:$$\frac{{a\cos A + b\cos B + c\cos C}}{{a + b + c}} =$$

1/r

r/R

R/r

1/R

## Questions 39 of 50

Question:In a triangle $$ABC$$, if $$b = 2,\,B = 30^\circ$$then the area of circumcircle of triangle ABC in square units is

$$\pi$$

$$2\pi$$

$$4\pi$$

$$6\pi$$

## Questions 40 of 50

Question:From the top of a light house 60 meters high with its base at the sea level, the angle of depression of a boat is $$15^\circ$$. The distance of the boat from the foot of light house is

$$\left( {\frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}} \right)\,60\,\,m$$

$$\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)\,60\,\,m$$

$$\left( {\frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}} \right)\,\,\,m$$

None of these

## Questions 41 of 50

Question:An observer in a boat finds that the angle of elevation of a tower standing on the top of a cliff is $$60^\circ$$ and that of the top of cliff is $$30^\circ$$. If the height of the tower be 60 meters, then the height of the cliff is

30 m

$$60\sqrt 3 \,\,m$$

$$20\sqrt 3 \,\,m$$

None of these

## Questions 42 of 50

Question:The base of a cliff is circular. From the extremities of a diameter of the base the angles of elevation of the top of the cliff are $$30^\circ$$ and $$60^\circ$$. If the height of the cliff be 500 metres, then the diameter of the base of the cliff is

$$1000\,\sqrt 3 \,m$$

$$2000/\sqrt 3 \,m$$

$$1000/\sqrt 3 \,m$$

$$2000\sqrt 2 \,m$$

## Questions 43 of 50

Question:The angle of elevation of the top of a tower from the top of a house is $${60^0}$$ and the angle of depression of its base is $$30^\circ$$. If the horizontal distance between the house and the tower be 12 m, then the height of the tower is

$$48\,\sqrt 3 \,\,m$$

$$16\,\sqrt 3 \,\,m$$

$$24\,\sqrt 3 \,\,m$$

$$16/\,\sqrt 3 \,\,m$$

## Questions 44 of 50

Question:A man whose eye level is 1.5 metres above the ground observes the angle of elevation of a tower to be $${60^0}$$. If the distance of the man from the tower be 10 meters, the height of the tower is

$$(1.5 + 10\sqrt 3 )\,m$$

$$10\,\sqrt 3 \,m$$

$$\left( {1.5 + \frac{{10}}{{\sqrt 3 }}} \right)\,m$$

None of these

## Questions 45 of 50

Question:If a flagstaff of 6 metres high placed on the top of a tower throws a shadow of $$2\sqrt 3 \,metres$$ along the ground, then the angle (in degrees) that the sun makes with the ground is [EAMCET 1990]

60o

80o

75o

None of these

## Questions 46 of 50

Question:The angle of elevation of a cliff at a point A on the ground and a point B, 100 m vertically at A are a and b respectively. The height of the cliff is [EAMCET 1986]

$$\frac{{100\,\,\cot \,\alpha }}{{\cot \alpha - \cot \beta }}$$

$$\frac{{100\,\,\cot \beta }}{{\cot \,\alpha - \cot \,\beta }}$$

$$\frac{{100\,\,\cot \beta }}{{\cot \beta - \cot \alpha }}$$

$$\frac{{100\,\,\cot \beta }}{{\cot \beta + \cot \alpha }}$$

## Questions 47 of 50

Question:A flag-staff of 5 m high stands on a building of 25 m high. At an observer at a height of 30 m. The flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is

$$\frac{{5\sqrt 3 }}{2}$$

$$5\sqrt {\frac{3}{2}}$$

$$5\sqrt {\frac{2}{3}}$$

None of these

## Questions 48 of 50

Question:A tower subtends angles $$\alpha ,\,2\alpha ,\,3\alpha$$respectively at points A, B and $$C$$, all lying on a horizontal line through the foot of the tower. Then $$AB/BC =$$

$$\frac{{\sin 3\alpha }}{{\sin 2\alpha }}$$

$$1 + 2\cos 2\alpha$$

$$2 + \cos 3\alpha$$

$$\frac{{\sin 2\alpha }}{{\sin \alpha }}$$

## Questions 49 of 50

Question:Two pillars of equal height stand on either side of a roadway which is 60 metres wide. At a point in the roadway between the pillars, the elevation of the top of pillars are $${60^0}$$ and $${30^0}$$. The height of the pillars is

$$15\sqrt 3 m$$

$$\frac{{15}}{{\sqrt 3 }}m$$

$$15m$$

$$20m$$

## Questions 50 of 50

Question:A ladder rests against a wall making an angle $$\alpha$$with the horizontal. The foot of the ladder is pulled away from the wall through a distance x, so that it slides a distance y down the wall making an angle$$\beta$$with the horizontal. The correct relation is

$$x = y\tan \frac{{\alpha + \beta }}{2}$$
$$y = x\tan \frac{{\alpha + \beta }}{2}$$
$$x = y\tan (\alpha + \beta )$$
$$y = x\tan (\alpha + \beta )$$