Inverse Trigonometric Functions - Trigonometry - Mathematics Previous Year Questions

Then \(\tan \theta =\) ____________

Let F(x) be an indefinite integral of \(\sin^2x\).

Statement 1 : The function F(x) satisfies F(\(x+\pi\)) = F(\(x\)) for all real x.

Statement 2 : \({\sin ^2}(x + \pi ) = {\sin ^2}x\) for all real x.

Let \((x,y)\) be such that \({\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}\).

Match the statements in Column I with the statements in Column II.

List \(I\)
\(P.\)\(\,\,\,\,\,\) \({\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }^{ - 1}}y} \right)} \over {\cot \left( {{{\sin }^{ - 1}}y} \right) + \tan \left( {{{\sin }^{ - 1}}y} \right)}}} \right)}^2} + {y^4}} \right)^{1/2}}\) takes value

\(Q.\) \(\,\,\,\,\) If \(\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z\) then
possible value of \(\cos {{x - y} \over 2}\) is

\(R.\) \(\,\,\,\,\,\) If \(\cos \left( {{\pi \over 4} - x} \right)\cos 2x + \sin x\sin 2\sec x = \cos x\sin 2x\sec x +\)
\(\cos \left( {{\pi \over 4} + x} \right)\cos 2x\) then possible value of \(\sec x\) is

\(S.\) \(\,\,\,\,\,\) If \(\cot \left( {{{\sin }^{ - 1}}\sqrt {1 - {x^2}} } \right) = \sin \left( {{{\tan }^{ - 1}}\left( {x\sqrt 6 } \right)} \right),\,\,x \ne 0,\)
Then possible value of \(x\) is

List \(II\)
\(1.\) \(\,\,\,\,\,\) \({1 \over 2}\sqrt {{5 \over 3}}\)

\(2.\) \(\,\,\,\,\,\) \(\sqrt 2\)

\(3.\) \(\,\,\,\,\,\) \({1 \over 2}\)

\(1.\) \(\,\,\,\,\) \(1\)

\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) List-\(I\)
(P.)\(\,\,\,\,\) Let \(y\left( x \right) = \cos \left( {3{{\cos }^{ - 1}}x} \right),x \in \left[ { - 1,1} \right],x \ne \pm {{\sqrt 3 } \over 2}.\) Then \({1 \over {y\left( x \right)}}\left\{ {\left( {{x^2} - 1} \right){{{d^2}y\left( x \right)} \over {d{x^2}}} + x{{dy\left( x \right)} \over {dx}}} \right\}\) equals
(Q.)\(\,\,\,\,\) Let \({A_1},{A_2},....,{A_n}\left( {n > 2} \right)\) be the vertices of a regular polygon of \(n\) sides with its centre at the origin. Let \({\overrightarrow {{a_k}} }\) be the position vector of the point \({A_k},k = 1,2,......,n.\) \(f\left| {\sum\nolimits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} \times \overrightarrow {{a_{k + 1}}} } \right)} } \right| = \left| {\sum\limits_{k = 1}^{n - 1} {\left( {\overrightarrow {{a_k}} .\,\overrightarrow {{a_{k + 1}}} } \right)} } \right|,\) then the minimum value of \(n\) is
(R.)\(\,\,\,\,\) If the normal from the point \(P(h, 1)\) on the ellipse \({{{x^2}} \over 6} + {{{y^2}} \over 3} = 1\) is perpendicular to the line \(x+y=8,\) then the value of \(h\) is
(S.)\(\,\,\,\,\) Number of positive solutions satisfying the equation \({\tan ^{ - 1}}\left( {{1 \over {2x + 1}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {4x + 1}}} \right) = {\tan ^{ - 1}}\left( {{2 \over {{x^2}}}} \right)\) is

\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\)List-\(II\)
(1.)\(\,\,\,\,\) \(1\)
(2.)\(\,\,\,\,\) \(2\)
(3.)\(\,\,\,\,\) \(8\)
(4.)\(\,\,\,\,\) \(9\)