Trignometrical Equations Test 4

Total Questions:50 Total Time: 60 Min

Remaining:

 

Questions 1 of 50

Question:If \(2\sin \theta + \tan \theta = 0\), then the general values of \(\theta \) are

Answers Choices:

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

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

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

\(n\pi ,\,\,n\pi + \frac{{2\pi }}{3}\)

Questions 2 of 50

Question:If \(\sqrt 3 \tan 2\theta + \sqrt 3 \tan 3\theta + \tan 2\theta \tan 3\theta = 1\), then the general value of\(\theta \)is

Answers Choices:

\(n\pi + \frac{\pi }{5}\)

\(\left( {n + \frac{1}{6}} \right)\frac{\pi }{5}\)

\(\left( {2n \pm \frac{1}{6}} \right)\frac{\pi }{5}\)

\(\left( {n + \frac{1}{3}} \right)\frac{\pi }{5}\)

Questions 3 of 50

Question:If \(\tan \theta - \sqrt 2 \sec \theta = \sqrt 3 \), then the general value of \(\theta \) is

Answers Choices:

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

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

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

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

Questions 4 of 50

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

Answers Choices:

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

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

\(n\pi - \frac{\pi }{4} \pm \alpha \)

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

Questions 5 of 50

Question:General value of \(\theta \) satisfying the equation \({\tan ^2}\theta + \sec 2\theta - = 1\) is

Answers Choices:

\(m\pi ,n\pi + \frac{\pi }{3}\)

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

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

None of these

Questions 6 of 50

Question:The general value \(\theta \) is obtained from the equation \(\cos 2\theta = \sin \alpha ,\) is

Answers Choices:

\(2\theta = \frac{\pi }{2} - \alpha \)

\(\theta = 2n\pi \pm \left( {\frac{\pi }{2} - \alpha } \right)\)

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

\(\theta = n\pi \pm \left( {\frac{\pi }{4} - \frac{\alpha }{2}} \right)\)

Questions 7 of 50

Question:If \(\frac{{\tan 3\theta - 1}}{{\tan 3\theta + 1}} = \sqrt 3 \), then the general value of \(\theta \)is

Answers Choices:

\(\frac{{n\pi }}{3} + \frac{\pi }{{12}}\)

\(\frac{{n\pi }}{3} + \frac{{7\pi }}{{36}}\)

\(n\pi + \frac{{7\pi }}{{12}}\)

\(n\pi + \frac{\pi }{{12}}\)

Questions 8 of 50

Question:If \(2{\cos ^2}x + 3\sin x - 3 = 0,\,\,0 \le x \le {180^o}\), then x =

Answers Choices:

\({30^o},{90^o},{150^o}\)

\({60^o},{120^o},{180^o}\)

\({0^o},{30^o},{150^o}\)

\({45^o},{90^o},{135^o}\)

Questions 9 of 50

Question:If \(2{\sin ^2}\theta = 3\cos \theta ,\)where \(0 \le \theta \le 2\pi \), then \(\theta = \)

Answers Choices:

\(\frac{\pi }{6},\frac{{7\pi }}{6}\)

\(\frac{\pi }{3},\frac{{5\pi }}{3}\)

\(\frac{\pi }{3},\frac{{7\pi }}{3}\)

None of these

Questions 10 of 50

Question:If\(\cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0\), where \(0 < \theta < {180^o}\), then \(\theta \) =

Answers Choices:

\({30^o},{45^o}\)

\({45^o},{90^o}\)

\({135^o},{150^o}\)

\({30^o},{45^o},{90^o},{135^o},{150^o}\)

Questions 11 of 50

Question:Common roots of the equations \(2{\sin ^2}x + {\sin ^2}2x = 2\) and \(\sin 2x + \cos 2x = \tan x,\) are

Answers Choices:

\(x = (2n - 1)\frac{\pi }{2}\)

\(x = (2n + 1)\frac{\pi }{4}\)

\(x = (2n + 1)\frac{\pi }{3}\)

None of these

Questions 12 of 50

Question:If \(r\,\sin \theta = 3,r = 4(1 + \sin \theta ),\,\,0 \le \theta \le 2\pi ,\)then \(\theta = \)

Answers Choices:

\(\frac{\pi }{6},\frac{\pi }{3}\)

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

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

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

Questions 13 of 50

Question:The general solution of \(\sin x - \cos x = \sqrt 2 \), for any integer n is

Answers Choices:

\(n\pi \)

\(2n\pi + \frac{{3\pi }}{4}\)

\(2n\pi \)

\((2n + 1)\,\pi \)

Questions 14 of 50

Question:If \(12{\cot ^2}\theta - 31\,{\rm{cosec }}\theta + {\rm{32}} = {\rm{0}}\), then the value of \(\sin \theta \) is

Answers Choices:

\(\frac{3}{5}\) or 1

\( - \sin (B + 2C) = \frac{1}{2}\) or \(\frac{{ - 2}}{3}\)

\(\frac{4}{5}\) or \(\frac{3}{4}\)

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

Questions 15 of 50

Question:The period of the function \(\sin \left( {\frac{{2x}}{3}} \right) + \sin \left( {\frac{{3x}}{2}} \right)\)is

Answers Choices:

\(2\pi \)

\(10\pi \)

\(6\pi \)

\(12\pi \)

Questions 16 of 50

Question:Let \(f(x) = \cos px + \sin x\) be periodic, then p must be

Answers Choices:

Irrational

Positive real number

None of these

Rational

Questions 17 of 50

Question:In \(\Delta ABC,\)if \({\sin ^2}\frac{A}{2},{\sin ^2}\frac{B}{2},{\sin ^2}\frac{C}{2}\) be in H. P. then a, b, c will be in

Answers Choices:

A. P.

G. P.

H. P.

None of these

Questions 18 of 50

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

Answers Choices:

\({a^2}\)

\({b^2}\)

\({c^2}\)

None of these

Questions 19 of 50

Question:If \(\tan \frac{{B - C}}{2} = x\cot \frac{A}{2},\)then \(x = \)

Answers Choices:

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

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

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

None of these

Questions 20 of 50

Question:In \(\Delta ABC\), if \(a = 3,b = 4,c = 5\), then \(\sin 2B = \)

Answers Choices:

05/Apr

20/Mar

24/25

01/50

Questions 21 of 50

Question:If the sides of a triangle are in A. P., then the cotangent of its half the angles will be in [MP PET 1993]

Answers Choices:

H. P.

G . P.

A. P.

No particular order

Questions 22 of 50

Question:If the angles of a triangle are in the ratio 1: 2: 3, then their corresponding sides are in the ratio

Answers Choices:

01:02:03

\(1:\sqrt 3 :2\)

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

\(1:\sqrt 3 :3\)

Questions 23 of 50

Question:The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is \({60^o}\). If the third side is 3, the remaining fourth side is

Answers Choices:

2

3

4

5

Questions 24 of 50

Question:In a triangle \(ABC,\)\(a = 4,b = 3\), \(\angle A = {60^o}\). Then c is the root of the equation

Answers Choices:

\({c^2} - 3c - 7 = 0\)

\({c^2} + 3c + 7 = 0\)

\({c^2} - 3c + 7 = 0\)

\({c^2} + 3c - 7 = 0\)

Questions 25 of 50

Question:In a triangle \(ABC\), if \(a = 2,B = {60^o}\)and \(C = {75^o}\), then b =

Answers Choices:

\(\sqrt 3 \)

\(\sqrt 6 \)

\(\sqrt 9 \)

\(1 + \sqrt 2 \)

Questions 26 of 50

Question:In triangle ABC, \(A = {30^o},b = 8,a = 6\), then \(B = {\sin ^{ - 1}}x\), where x =

Answers Choices:

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

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

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

1

Questions 27 of 50

Question:In a \(\Delta ABC\), \(b = 2,C = {60^o},c = \sqrt 6 \), then a =

Answers Choices:

\(\sqrt 3 - 1\)

\(\sqrt 3 \)

\(\sqrt 3 + 1\)

None of these

Questions 28 of 50

Question:In a \(\Delta ABC,\)\(2a\sin \,\,\left( {\frac{{A - B + C}}{2}} \right)\) is equal to

Answers Choices:

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

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

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

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

Questions 29 of 50

Question:In a triangle \(ABC\), right angled at C, the value of \(\tan A + \tan B\) is

Answers Choices:

\(a + b\)

\(\frac{{{a^2}}}{{bc}}\)

\(\frac{{{b^2}}}{{ac}}\)

\(\frac{{{c^2}}}{{ab}}\)

Questions 30 of 50

Question:In a \(\Delta ABC,\) \(A:B:C\). Then \([a + b + c\sqrt 2 ]\) is equal to

Answers Choices:

2b

2c

3b

3a

Questions 31 of 50

Question:If \(\alpha ,\beta ,\gamma \) are angles of a triangle, then \({\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma - 2\cos \alpha \cos \beta \cos \gamma \)is

Answers Choices:

2

1

2

0

Questions 32 of 50

Question:If in \(\Delta ABC,\)\(a = 6,b = 3\)and \(\cos (A - B) = \frac{4}{5}\), then its area will be

Answers Choices:

7 square unit

8 square unit

9 square unit

None of these

Questions 33 of 50

Question:In \(\Delta ABC\), if \(2s = a + b + c\), then the value of \(\frac{{s(s - a)}}{{bc}} - \frac{{(s - b)(s - c)}}{{bc}} = \)

Answers Choices:

\(\sin A\)

\(\cos A\)

\(\tan A\)

None of these

Questions 34 of 50

Question:If the area of a triangle ABC is D, then \({a^2}\sin 2B + {b^2}\sin 2A\) is equal to

Answers Choices:

\(3\Delta \)

\(2\Delta \)

\(4\Delta \)

\( - 4\Delta \)

Questions 35 of 50

Question:In a right triangle \(AC = BC\) and D is the mid point of AC cotangent of angle \(DBC\) is equal to

Answers Choices:

2

3

1/2

1/3

Questions 36 of 50

Question:If a, b, c are the sides and A, B, C are the angles of a triangle \(ABC\), then \(\tan \left( {\frac{A}{2}} \right)\)is equal to

Answers Choices:

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

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

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

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

Questions 37 of 50

Question:In a \(\Delta ABC\), \(a,\;b,\;A\)are given and \({c_1},\;{c_2}\)are two values of the third side c. The sum of the areas of two triangles with sides \(a,\;b,\;{c_1}\) and \(a,b,\;{c_2}\) is

Answers Choices:

\(\frac{1}{2}{b^2}\sin 2A\)

\(\frac{1}{2}{a^2}\sin 2A\)

\({b^2}\sin 2A\)

None of these

Questions 38 of 50

Question:If in a triangle \(ABC\), \(2\cos A = \sin B\,{\rm{cosec}}\,C,\) then

Answers Choices:

\(a = b\)

\(b = c\)

\(c = a\)

\(2a = bc\)

Questions 39 of 50

Question:If the line segment joining the points \(A(a,\,b)\) and \(B(c,\,d)\) subtends an angle \(\theta \) at the origin, then \(\cos \theta \) is equal to

Answers Choices:

\(\frac{{ab + cd}}{{\sqrt {({a^2} + {b^2})\,({c^2} + {d^2})} }}\)

\(\frac{{ac + bd}}{{\sqrt {({a^2} + {b^2})\,({c^2} + {d^2})} }}\)

\(\frac{{ac - bd}}{{\sqrt {({a^2} + {b^2})\,({c^2} + {d^2})} }}\)

None of these

Questions 40 of 50

Question:\(ABC\) is a right angled isosceles triangle with \(\angle B = {90^o}\). If D is a point on \(AB\) so that \(\angle DCB = {15^o}\) and if \(AD = 35cm\), then \(CD = \)

Answers Choices:

\(35\sqrt 2 \)cm\(70\sqrt 2 cm\)

\(\frac{{35\sqrt 3 }}{2}cm\)

\(35\sqrt 6 \)cm

\(\frac{{35\sqrt 2 }}{2}cm\)

Questions 41 of 50

Question:If R is the radius of the circumcircle of the \(\Delta ABC\)and \(\Delta \)is its area, then

Answers Choices:

\(R = \frac{{a + b + c}}{\Delta }\)

\(R = \frac{{a + b + c}}{{4\Delta }}\)

\(R = \frac{{abc}}{{4\Delta }}\)

\(R = \frac{{abc}}{\Delta }\)

Questions 42 of 50

Question:The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a, is

Answers Choices:

\(a\cot \left( {\frac{\pi }{n}} \right)\)

\(\frac{a}{2}\cot \left( {\frac{\pi }{{2n}}} \right)\)

\(a\cot \left( {\frac{\pi }{{2n}}} \right)\)

\(\frac{a}{2}\cot \left( {\frac{\pi }{{2n}}} \right)\)

Questions 43 of 50

Question:An observer on the top of a tree, finds the angle of depression of a car moving towards the tree to be \(30^\circ \)o. After 3 minutes this angle becomes 60o. After how much more time, the car will reach the tree

Answers Choices:

4 min.

4.5 min.

1.5 min.

2 min.

Questions 44 of 50

Question:A house of height 100 metres subtends a right angle at the window of an opposite house. If the height of the window be 64 metres, then the distance between the two houses is

Answers Choices:

48 m

36 m

54 m

72 m

Questions 45 of 50

Question:A ladder rests against a wall so that its top touches the roof of the house. If the ladder makes an angle of \({60^0}\) with the horizontal and height of the house be \(6\sqrt 3 \) meters, then the length of the ladder is

Answers Choices:

\(12\sqrt 3 \)

12 m

\(12/\sqrt 3 \,\,m\)

None of these

Questions 46 of 50

Question:If the angles of elevation of two towers from the middle point of the line joining their feet be \(60^\circ \) and \(30^\circ \) respectively, then the ratio of their heights is

Answers Choices:

02:01

\(1\,\,:\,\,\sqrt 2 \)

03:01

\(1\,\,:\,\,\sqrt 3 \)

Questions 47 of 50

Question:A balloon is coming down at the rate of 4 m/min. and its angle of elevation is \({45^0}\) from a point on the ground which has been reduced to 30o after 10 minutes. Balloon will be on the ground at a distance of how many meters from the observer

Answers Choices:

\(20\,\sqrt 3 \,m\)

\(20\,(3 + \sqrt 3 )\,m\)

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

None of these

Questions 48 of 50

Question:A person standing on the bank of a river finds that the angle of elevation of the top of a tower on the opposite bank is 45o. Then which of the following statements is correct

Answers Choices:

Breadth of the river is twice the height of the tower

Breadth of the river and the height of the tower are the same

Breadth of the river is half of the height of the tower

None of the above

Questions 49 of 50

Question:The shadow of a tower standing on a level ground is found to be 60 m longer when the sun's altitude is \({30^0}\) than when it is \({45^0}\). The height of the tower is

Answers Choices:

60 m

30 m

\(60\sqrt 3 m\)

\(30(\sqrt 3 + 1)m\)

Questions 50 of 50

Question:If the angle of elevation of the top of tower at a distance 500 m from its foot is \({30^0}\), then height of the tower is

Answers Choices:

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

\(\frac{{500}}{{\sqrt 3 }}\)

\(\sqrt 3 \)

\(\frac{1}{{500}}\)