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

Answers Choices:

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

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

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

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

Questions 2 of 50

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

Answers Choices:

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

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

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

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

Questions 3 of 50

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

Answers Choices:

\(\frac{{n\pi }}{4},\frac{{n\pi }}{3} + \frac{\pi }{{18}}\)

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

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

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

Questions 4 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

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

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

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

None of these

Questions 6 of 50

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

Answers Choices:

\(\theta = \frac{{n\pi }}{7} + \frac{\pi }{{14}}\)

\(\theta = \frac{{n\pi }}{7} + \frac{\pi }{5}\)

\(\theta = \frac{{n\pi }}{7} + \frac{\pi }{2}\)

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

Questions 7 of 50

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

Answers Choices:

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

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

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

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

Questions 8 of 50

Question:The equation \(\sin x\cos x = 2\)has

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\(\frac{\pi }{9},\frac{\pi }{4}\)

\(\frac{\pi }{3},\frac{\pi }{9}\)

\(\frac{\pi }{6},\frac{\pi }{9}\)

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

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

Answers Choices:

\(0 \le x \le \frac{\pi }{2}\)

\(0 \le x \le \pi \)

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

Answers Choices:

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

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

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

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

Questions 13 of 50

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

Answers Choices:

\({45^o}\) 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

Answers Choices:

\(2\pi \)

\(\pi \)

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

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

Questions 15 of 50

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

Answers Choices:

\(\frac{{2\pi - 5}}{7}\)

\(2\pi - 5\)

\(\frac{{2\pi }}{7}\)

\(\frac{\pi }{7}\)

Questions 16 of 50

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

Answers Choices:

\({\pi ^2}\)

\(2\pi \)

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

Answers Choices:

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

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

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

None of these

Questions 18 of 50

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

Answers Choices:

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

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

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

\(\frac{{4a}}{{a + b + c}}\)

Questions 19 of 50

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

Answers Choices:

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

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

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

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

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

Answers Choices:

\({15^o}\)

\({30^o}\)

\(60^\circ \)

\({45^o}\)

Questions 21 of 50

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

Answers Choices:

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

Answers Choices:

\({30^o}\)

\({45^o}\)

\({75^o}\)

\({15^o}\)

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

Answers Choices:

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

Answers Choices:

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.

Answers Choices:

\(\sin A,\sin B,\sin C\)

\(\tan A,\tan B,\tan C\)

\(\cot A,\cot B,\cot C\)

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

Answers Choices:

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

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

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

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 =

Answers Choices:

\(7\sqrt 3 \)

\(\frac{{7\sqrt 3 }}{2}\)

\(\frac{7}{2}\)

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

Answers Choices:

\(\cos B = \frac{{11}}{6}\)

\(\cos B = \frac{7}{8}\)

\(AD = 2.4\)

\(A{D^2} = 2.5\)

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

Answers Choices:

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

\(1:6\)

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

\(2:3\)

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

Answers Choices:

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

Answers Choices:

\(6cm\)

\(24cm\)

\(12cm\)

\(8cm\)

Questions 32 of 50

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

Answers Choices:

\({30^o}\)

\({60^o}\)

\({90^o}\)

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

Answers Choices:

\(\frac{{15}}{{16}}\)

\(\frac{8}{{15}}\)

\(\frac{8}{{17}}\)

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

Questions 34 of 50

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

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

Answers Choices:

\(\frac{{67}}{8}\)

\(\frac{{65}}{4}\)

4

24

Questions 38 of 50

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

Answers Choices:

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

Answers Choices:

\(\pi \)

\(2\pi \)

\(4\pi \)

\(6\pi \)

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

Answers Choices:

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

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

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

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

Answers Choices:

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

Answers Choices:

\(1000\,\sqrt 3 \,m\)

\(2000/\sqrt 3 \,m\)

\(1000/\sqrt 3 \,m\)

\(2000\sqrt 2 \,m\)

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

Answers Choices:

\(48\,\sqrt 3 \,\,m\)

\(16\,\sqrt 3 \,\,m\)

\(24\,\sqrt 3 \,\,m\)

\(16/\,\sqrt 3 \,\,m\)

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

Answers Choices:

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

\(10\,\sqrt 3 \,m\)

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

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]

Answers Choices:

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]

Answers Choices:

\(\frac{{100\,\,\cot \,\alpha }}{{\cot \alpha - \cot \beta }}\)

\(\frac{{100\,\,\cot \beta }}{{\cot \,\alpha - \cot \,\beta }}\)

\(\frac{{100\,\,\cot \beta }}{{\cot \beta - \cot \alpha }}\)

\(\frac{{100\,\,\cot \beta }}{{\cot \beta + \cot \alpha }}\)

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

Answers Choices:

\(\frac{{5\sqrt 3 }}{2}\)

\(5\sqrt {\frac{3}{2}} \)

\(5\sqrt {\frac{2}{3}} \)

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 \(AB/BC = \)

Answers Choices:

\(\frac{{\sin 3\alpha }}{{\sin 2\alpha }}\)

\(1 + 2\cos 2\alpha \)

\(2 + \cos 3\alpha \)

\(\frac{{\sin 2\alpha }}{{\sin \alpha }}\)

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

Answers Choices:

\(15\sqrt 3 m\)

\(\frac{{15}}{{\sqrt 3 }}m\)

\(15m\)

\(20m\)

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

Answers Choices:

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

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

\(x = y\tan (\alpha + \beta )\)

\(y = x\tan (\alpha + \beta )\)