Inverse trigonometrical functions Test 1

Total Questions:50 Total Time: 60 Min

Remaining:

 

Questions 1 of 50

Question:\(\tan \left[ {{{\sec }^{ - 1}}\sqrt {1 + {x^2}} } \right] = \)

Answers Choices:

\(\frac{1}{x}\)

\(x\)

\(\frac{1}{{\sqrt {1 + {x^2}} }}\)

\(\frac{x}{{\sqrt {1 + {x^2}} }}\)

Questions 2 of 50

Question:\({\sec ^{ - 1}}[\sec ( - {30^o})] = \)

Answers Choices:

\( - {60^o}\)

\( - {30^o}\)

\({30^o}\)

\({150^o}\)

Questions 3 of 50

Question:\({\tan ^{ - 1}}\left[ {\frac{{\cos x}}{{1 + \sin x}}} \right] = \)

Answers Choices:

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

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

\(\frac{x}{2}\)

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

Questions 4 of 50

Question:\({\tan ^{ - 1}}\frac{1}{{\sqrt {{x^2} - 1} }} = \)

Answers Choices:

\(\frac{\pi }{2} + {\rm{cose}}{{\rm{c}}^{ - 1}}x\)

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

\({\rm{cose}}{{\rm{c}}^{ - 1}}x\)

\({\sec ^{ - 1}}x\)

Questions 5 of 50

Question:The principal value of \({\sin ^{ - 1}}\left( { - \frac{1}{2}} \right)\)is

Answers Choices:

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

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

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

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

Questions 6 of 50

Question:\({\sec ^2}({\tan ^{ - 1}}2) + {\rm{cose}}{{\rm{c}}^2}({\cot ^{ - 1}}3) = \)

Answers Choices:

5

13

15

6

Questions 7 of 50

Question:\({\sin ^{ - 1}}\left[ {x\sqrt {1 - x} - \sqrt x \sqrt {1 - {x^2}} } \right] = \)

Answers Choices:

\({\sin ^{ - 1}}x + {\sin ^{ - 1}}\sqrt x \)

\({\sin ^{ - 1}}x - {\sin ^{ - 1}}\sqrt x \)

\({\sin ^{ - 1}}\sqrt x - {\sin ^{ - 1}}x\)

None of these

Questions 8 of 50

Question:The principal value of \({\sin ^{ - 1}}\left( { - \,\,\frac{{\sqrt 3 }}{2}} \right)\)is

Answers Choices:

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

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

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

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

Questions 9 of 50

Question:\(\cot \,\,\left[ {{{\cos }^{ - 1}}\left( {\frac{7}{{25}}} \right)} \right] = \)

Answers Choices:

\(\frac{{25}}{{24}}\)

\(\frac{{25}}{7}\)

\(\frac{{24}}{{25}}\)

None of these

Questions 10 of 50

Question:If \(\frac{\pi }{2} \le x \le \frac{{3\pi }}{2},\)then\({\sin ^{ - 1}}(\sin x)\)is equal to

Answers Choices:

x

\( - x\)

\(\pi + x\)

\(\pi - x\)

Questions 11 of 50

Question:If \(\pi \le x \le 2\pi \), then \({\cos ^{ - 1}}(\cos x)\)is equal to

Answers Choices:

x

\( - x\)

\(2\pi + x\)

\(2\pi - x\)

Questions 12 of 50

Question:The value of \({\sin ^{ - 1}}(\sin 10)\)is

Answers Choices:

10

\(10 - 3\pi \)

\(3\pi - 10\)

None of these

Questions 13 of 50

Question:The smallest and the largest values of \({\tan ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right){\rm{ }},\,\,0 \le x \le 1\)are

Answers Choices:

\(0,\,\,\pi \)

\(0,\,\frac{\pi }{4}\)

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

\(\frac{\pi }{4},\,\frac{\pi }{2}\)

Questions 14 of 50

Question:If \(x\)takes non-positive permissible value, then \({\sin ^{ - 1}}x\)=

Answers Choices:

\({\cos ^{ - 1}}\sqrt {1 - {x^2}} \)

\( - {\cos ^{ - 1}}\sqrt {1 - {x^2}} \)

\({\cos ^{ - 1}}\sqrt {{x^2} - 1} \)

\(\pi - {\cos ^{ - 1}}\sqrt {1 - {x^2}} \)

Questions 15 of 50

Question:\(\sin \left[ {\frac{\pi }{2} - {{\sin }^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right)} \right] = \)

Answers Choices:

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

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

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

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

Questions 16 of 50

Question:\(\sin [{\cot ^{ - 1}}(\cos {\tan ^{ - 1}}x)]\)=

Answers Choices:

\(\frac{x}{{\sqrt {{x^2} + 2} }}\)

\(\frac{x}{{\sqrt {{x^2} + 1} }}\)

\(\frac{1}{{\sqrt {{x^2} + 2} }}\)

\(\sqrt {\frac{{{x^2} + 1}}{{{x^2} + 2}}} \)

Questions 17 of 50

Question:If \(\sin ({\cot ^{ - 1}}(x + 1) = \cos ({\tan ^{ - 1}}x)\), then x =

Answers Choices:

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

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

0

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

Questions 18 of 50

Question:\({\cos ^{ - 1}}\frac{4}{5} + {\tan ^{ - 1}}\frac{3}{5} = \)

Answers Choices:

\({\tan ^{ - 1}}\frac{{27}}{{11}}\)

\({\sin ^{ - 1}}\frac{{11}}{{27}}\)

\({\cos ^{ - 1}}\frac{{11}}{{27}}\)

None of these

Questions 19 of 50

Question:\({\sin ^{ - 1}}x + {\sin ^{ - 1}}\frac{1}{x} + {\cos ^{ - 1}}x + {\cos ^{ - 1}}\frac{1}{x} = \)

Answers Choices:

\(\pi \)

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

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

None of these

Questions 20 of 50

Question:\(2{\tan ^{ - 1}}\frac{1}{3} + {\tan ^{ - 1}}\frac{1}{2} = \)

Answers Choices:

\({90^o}\)

\({60^o}\)

\({45^o}\)

\({\tan ^{ - 1}}2\)

Questions 21 of 50

Question:\({\tan ^{ - 1}}\frac{{a - b}}{{1 + ab}} + {\tan ^{ - 1}}\frac{{b - c}}{{1 + bc}} = \)

Answers Choices:

\({\tan ^{ - 1}}a - {\tan ^{ - 1}}b\)

\({\tan ^{ - 1}}a - {\tan ^{ - 1}}c\)

\({\tan ^{ - 1}}b - {\tan ^{ - 1}}c\)

\({\tan ^{ - 1}}c - {\tan ^{ - 1}}a\)

Questions 22 of 50

Question:If \({\tan ^{ - 1}}2x + {\tan ^{ - 1}}3x = \frac{\pi }{4}\), then x =

Answers Choices:

1

\(\frac{1}{6}\)

\( - 1,\,\frac{1}{6}\)

None of these

Questions 23 of 50

Question:If \({\cot ^{ - 1}}x + {\tan ^{ - 1}}3 = \frac{\pi }{2}\), then x =

Answers Choices:

1/3

1/4

3

4

Questions 24 of 50

Question:\(2{\sin ^{ - 1}}\frac{3}{5} + {\cos ^{ - 1}}\frac{{24}}{{25}} = \)

Answers Choices:

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

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

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

None of these

Questions 25 of 50

Question:\(\cos \left[ {{{\tan }^{ - 1}}\frac{1}{3} + {{\tan }^{ - 1}}\frac{1}{2}} \right] = \)

Answers Choices:

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

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

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

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

Questions 26 of 50

Question:\({\tan ^{ - 1}}x + {\cot ^{ - 1}}(x + 1) = \)

Answers Choices:

\({\tan ^{ - 1}}({x^2} + 1)\)

\({\tan ^{ - 1}}({x^2} + x)\)

\({\tan ^{ - 1}}(x + 1)\)

\({\tan ^{ - 1}}({x^2} + x + 1)\)

Questions 27 of 50

Question:\({\cos ^{ - 1}}\frac{1}{2} + 2{\sin ^{ - 1}}\frac{1}{2}\)is equal to

Answers Choices:

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

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

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

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

Questions 28 of 50

Question:\({\tan ^{ - 1}}\frac{3}{4} + {\tan ^{ - 1}}\frac{3}{5} - {\tan ^{ - 1}}\frac{8}{{19}} = \)

Answers Choices:

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

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

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

None of these

Questions 29 of 50

Question:\(4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}} = \)

Answers Choices:

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

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

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

None of these

Questions 30 of 50

Question:If \({\sin ^{ - 1}}x + {\sin ^{ - 1}}y = \frac{{2\pi }}{3},\)then \({\cos ^{ - 1}}x + {\cos ^{ - 1}}y = \)

Answers Choices:

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

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

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

\(\pi \)

Questions 31 of 50

Question:\({\tan ^{ - 1}}\left( {\frac{1}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{2}{9}} \right) = \)

Answers Choices:

\(\frac{1}{2}{\cos ^{ - 1}}\left( {\frac{3}{5}} \right)\)

\(\frac{1}{2}{\sin ^{ - 1}}\left( {\frac{3}{5}} \right)\)

\(\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{3}{5}} \right)\)

\({\tan ^{ - 1}}\left( {\frac{1}{2}} \right)\)

Questions 32 of 50

Question:\({\tan ^{ - 1}}\left( {\frac{x}{y}} \right) - {\tan ^{ - 1}}\,\left( {\frac{{x - y}}{{x + y}}} \right)\) is

Answers Choices:

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

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

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

\(\frac{\pi }{4}\)or \( - \frac{{3\pi }}{4}\)

Questions 33 of 50

Question:If \(\tan (x + y) = 33\)and \(x = {\tan ^{ - 1}}3,\)then y will be

Answers Choices:

\(0.3\)

\({\tan ^{ - 1}}(1.3)\)

\({\tan ^{ - 1}}(0.3)\)

\({\tan ^{ - 1}}\left( {\frac{1}{{18}}} \right)\)

Questions 34 of 50

Question:If \({\tan ^{ - 1}}\frac{{x - 1}}{{x + 1}} + {\tan ^{ - 1}}\frac{{2x - 1}}{{2x + 1}} = {\tan ^{ - 1}}\frac{{23}}{{36}},\)then x =

Answers Choices:

\(\frac{3}{4},\frac{{ - 3}}{8}\)

\(\frac{3}{4},\frac{3}{8}\)

\(\frac{4}{3},\frac{3}{8}\)

None of these

Questions 35 of 50

Question:\({\tan ^{ - 1}}\frac{{{c_1}x - y}}{{{c_1}y + x}} + {\tan ^{ - 1}}\frac{{{c_2} - {c_1}}}{{1 + {c_2}{c_1}}} + \) \({\tan ^{ - 1}}\frac{{{c_3} - {c_2}}}{{1 + {c_3}{c_2}}} + ... + {\tan ^{ - 1}}\frac{1}{{{c_n}}} = \)

Answers Choices:

\({\tan ^{ - 1}}\frac{y}{x}\)

\({\tan ^{ - 1}}yx\)

\({\tan ^{ - 1}}\frac{x}{y}\)

\({\tan ^{ - 1}}(x - y)\)

Questions 36 of 50

Question:\(\sin \left\{ {{{\sin }^{ - 1}}\frac{1}{2} + {{\cos }^{ - 1}}\frac{1}{2}} \right\} = \)

Answers Choices:

0

1

2

1

Questions 37 of 50

Question:\({\sin ^{ - 1}}\frac{4}{5} + 2{\tan ^{ - 1}}\frac{1}{3} = \)

Answers Choices:

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

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

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

None of these

Questions 38 of 50

Question:\({\sin ^{ - 1}}x + {\cos ^{ - 1}}x\) is equal to

Answers Choices:

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

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

1

1

Questions 39 of 50

Question:The value of \({\sin ^{ - 1}}\left( {\frac{{\sqrt 3 }}{2}} \right) - {\sin ^{ - 1}}\left( {\frac{1}{2}} \right)\)is

Answers Choices:

\({45^0}\)

\({90^0}\)

\({15^0}\)

\({30^0}\)

Questions 40 of 50

Question:If \({\cos ^{ - 1}}x + {\cos ^{ - 1}}y + {\cos ^{ - 1}}z = 3\pi ,\)then \(xy + yz + zx = \)

Answers Choices:

0

1

3

3

Questions 41 of 50

Question:\(\cos {\rm{ }}\left[ {{{\cos }^{ - 1}}{\rm{ }}\left( {\frac{{ - 1}}{7}} \right) + {{\sin }^{ - 1}}{\rm{ }}\left( {\frac{{ - 1}}{7}} \right)} \right] = \)

Answers Choices:

\( - 1/3\)

0

\(1/3\)

\(4/9\)

Questions 42 of 50

Question:The value of \(\tan \left[ {{{\sin }^{ - 1}}\left( {\frac{3}{5}} \right) + {{\cos }^{ - 1}}\left( {\frac{3}{{\sqrt {13} }}} \right)} \right]\)is

Answers Choices:

\(\frac{6}{{17}}\)

\(\frac{6}{{\sqrt {13} }}\)

\(\frac{{\sqrt {13} }}{5}\)

\(\frac{{17}}{6}\)

Questions 43 of 50

Question:The value of \(\tan \left( {{{\tan }^{ - 1}}\frac{1}{2} - {{\tan }^{ - 1}}\frac{1}{3}} \right)\)is

Answers Choices:

\(5/6\)

\(7/6\)

\(1/6\)

\(1/7\)

Questions 44 of 50

Question:If \({\cos ^{ - 1}}\sqrt p + {\cos ^{ - 1}}\sqrt {1 - p} + {\cos ^{ - 1}}\sqrt {1 - q} = \frac{{3\pi }}{4},\) then the value of q is

Answers Choices:

1

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

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

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

Questions 45 of 50

Question:\(\frac{1}{2}{\cos ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right) = \)

Answers Choices:

\({\cot ^{ - 1}}\sqrt x \)

\({\tan ^{ - 1}}\sqrt x \)

\({\tan ^{ - 1}}x\)

\({\cot ^{ - 1}}x\)

Questions 46 of 50

Question:\(\sin \left( {4{{\tan }^{ - 1}}\frac{1}{3}} \right) = \)

Answers Choices:

\(\frac{{12}}{{25}}\)

\(\frac{{24}}{{25}}\)

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

None of these

Questions 47 of 50

Question:\(3{\tan ^{ - 1}}a\)is equal to

Answers Choices:

\({\tan ^{ - 1}}\frac{{3a + {a^3}}}{{1 + 3{a^2}}}\)

\({\tan ^{ - 1}}\frac{{3a - {a^3}}}{{1 + 3{a^2}}}\)

\({\tan ^{ - 1}}\frac{{3a + {a^3}}}{{1 - 3{a^2}}}\)

\({\tan ^{ - 1}}\frac{{3a - {a^3}}}{{1 - 3{a^2}}}\)

Questions 48 of 50

Question:\(4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}\)is equal to

Answers Choices:

\(\pi \)

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

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

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

Questions 49 of 50

Question:If \(3{\sin ^{ - 1}}\frac{{2x}}{{1 - {x^2}}} - 4{\cos ^{ - 1}}\frac{{1 - {x^2}}}{{1 + {x^2}}} + 2{\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}} = \frac{\pi }{3}\) then \(x\) =

Answers Choices:

\(\sqrt 3 \)

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

1

None of these

Questions 50 of 50

Question:The value of \(\sin \left( {2{{\tan }^{ - 1}}\left( {\frac{1}{3}} \right)} \right) + \cos ({\tan ^{ - 1}}2\sqrt 2 ) = \)

Answers Choices:

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

\(\frac{{14}}{{15}}\)

\(\frac{{12}}{{15}}\)

\(\frac{{11}}{{15}}\)