Mathematics - Inverse Circular Functions
Exam Duration: 45 Mins Total Questions : 30
\(x\epsilon\left({3\pi\over 2}, 2\pi\right)\)then the value of the expression sin-1 [cos {cos-1 (cos x)} + sin-1 (sin x)], is
- (a)
\(-{\pi\over 2}\)
- (b)
\({\pi\over 2}\)
- (c)
0
- (d)
\(\pi\)
The sum of the infinite terms of the series \(tan^{-1}\left(1\over 3\right)+tan^{-1}\left(2\over 9\right)+tan^{-1}\left(4\over 33\right)+......\) is equal to
- (a)
\(\pi\over 6\)
- (b)
\(\pi\over 4\)
- (c)
\(\pi\over 3\)
- (d)
\(\pi\over 2\)
The greatest of tan1, tan-1 1, sin-11, sin1, cos 1, is
- (a)
sin 1
- (b)
tan 1
- (c)
tan-1 1
- (d)
none of these
The value of 'a' for which a x2 + sin-1 (x2 - 2x + 2) + cos-1 (x2 - 2x + 2) = 0 has a real solution, is
- (a)
\(-{2\over\pi}\)
- (b)
\({2\over\pi}\)
- (c)
\(-{\pi\over2}\)
- (d)
\({\pi\over2}\)
If the mapping f(x)=ax+b,a>0 maps [-1,1] onto [0, 2], then cot [cot-1 7 + cot-1 8 + cot-1 18] is equal to
- (a)
f( - 1)
- (b)
f(0)
- (c)
f(1)
- (d)
f(2)
If [sin-1 cos-1 sin-1 tan-1 x] = 1, where [.] denotes the greatest integer function, then x belongs to the interval
- (a)
[tan sin cos 1, tan sin cos sin 1]
- (b)
(tan sin cos 1, tan sin cos sin 1)
- (c)
[- 1, 1]
- (d)
[sin cos tan 1, sin cos sin tan 1]
If \(sin^{-1}x+sin^{-1}y={2\pi\over 3}\)then cos-1 x + cos-1 y is equal to
- (a)
\(2\pi\over 3\)
- (b)
\(\pi\over 3\)
- (c)
\(\pi\over 6\)
- (d)
\(\pi\)
If cos-1 x + cos-1 y + cos-1 z = 3rr, then x y + y z + z is equal to
- (a)
-3
- (b)
0
- (c)
3
- (d)
-1
The value of \(sin^{-1}\left[cot\left\{ sin^{-1}\sqrt{\left(2-\sqrt3\over 4\right)}+cos^{-1}\left(\sqrt{12}\over 4\right)+sec^{-1}(\sqrt2)\right\}\right]\)is equal to
- (a)
-3
- (b)
0
- (c)
3
- (d)
-1
If X2 + y2 + z2 =r2, then \(tan^{-1}\left(xy\over zr\right)+tan^{-1}\left(yz\over xr\right)+tan^{-1}\left(zx\over yr\right)\)is equal
- (a)
π
- (b)
\(π\over 2\)
- (c)
0
- (d)
none of these
If \(\frac{1}{2}\) < | x |< 1, then which of the following are real?
- (a)
sin-1x
- (b)
tan-1x
- (c)
sec-1x
- (d)
cos-1x
If 6 sin-1 (x2- 6x + 8.5) = \(\pi\), then
- (a)
x = 1
- (b)
x = 2
- (c)
x = 3
- (d)
x = 4
The value of \(tan^{-1}(1)+cos^{-1}\left(-{1\over 2}\right)+sin^{-1}\left(-{1\over2}\right)\)is equal to
- (a)
\(\pi\over 4\)
- (b)
\(5\pi\over12\)
- (c)
\(3\pi\over 4\)
- (d)
\(13\pi\over 12\)
If \(\sum_{i=i}^{2n}sin^{-1}x_i=n\pi\ then\ \sum_{i=1}^{2n}\)is equal to
- (a)
n
- (b)
2n
- (c)
\(n(n+1)\over 2\)
- (d)
none of these
3The equation sin-1 x = 2 sin-1 a has a solution for
- (a)
all real values of a
- (b)
a < 1
- (c)
\(-{1\over \sqrt2}\le a\le {1\over \sqrt2}\)
- (d)
-1< a< 1
The sum of the infinite' series cot-1 2 + cot-1I 8 + cot-1 18 + cot-1 32 + ... is equal to
- (a)
\(\pi\)
- (b)
\(\pi\over 2\)
- (c)
\(\pi\over x\)
- (d)
none of these
The number of positive integral solutions of tan-1 x + cot-1 Y = tan-1 3 is
- (a)
one
- (b)
two
- (c)
three
- (d)
four
\(\theta =\tan ^{ -1 }{ \left( 2\quad { tan }^{ 2 }\theta \right) } -{ tan }^{ -1 }\left\{ \left( \frac { 1 }{ 3 } \right) tan\quad \theta \right\} ,\)if
- (a)
\(tan\ \theta =-2\)
- (b)
\(tan\ \theta =0\)
- (c)
\(tan\ \theta =1\)
- (d)
\(tan\ \theta =2\)
If a, b are positive quantities and, if \(a_1={a+b\over 2},b_1=\sqrt{a_1,b}\) \(a_2={a_1+b_1\over1},b_2\sqrt{a_1b_1}\) and so on, then
- (a)
\(a_\infty={sqrt{(b^2-a^2)}\over cos^{-1}\left(a\over b\right)}\)
- (b)
\(b_\infty={sqrt{(b^2-a^2)}\over cos^{-1}\left(a\over b\right)}\)
- (c)
\(b_\infty={sqrt{(a^2-b^2)}\over cos^{-1}\left(b\over a\right)}\)
- (d)
none of these
The value(s) of x satisfying the equation \({ sin }^{ -1 }\left| sin\ x \right| =\sqrt { sin^{ -1 }\left| sin\ x \right| } \) is/are given by (n is any integer)
- (a)
\(n\pi -1\)
- (b)
\(n\pi\)
- (c)
\(n\pi +1\)
- (d)
\(2n\pi +1\)
The number of the positive integral solutions of \(tan^{-1}x+cos^{-1}\left(y\over \sqrt{(1+y^2)}\right)=sin^{-1}\left(3\over\sqrt{10}\right)\) is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Let f : A\(\rightarrow \)B be a function defined by y = f(x) such that f is both one-one (Injective) and onto (surjective)(ie, bijective), then there exists a unique function g: B\(\rightarrow \)A such that \(f\left( x \right) =y\Leftrightarrow g\left( y \right) =x,\ \forall x\epsilon A\ y\epsilon B\), then g is said to be inverse of f. Thus, g = f-1: B\(\rightarrow \)A = \(\left[ \left\{ f\left( x \right) ,x \right\} :\left\{ x,\ f(x) \right\} \epsilon { f }^{ -1 } \right] \).If no branch of an inverse trigonometric function is mentioned, then it means the principal value branch of that functon. On the basis of above information, answer the following question: If x takes negative permissible value, then sin-1 x is equal to
- (a)
\({ cos }^{ -1 }\sqrt { { \left( 1-{ x }^{ 2 } \right) } } \)
- (b)
\({ cos }^{ -1 }\left( \frac { \sqrt { { 1-{ x }^{ 2 } } } }{ x } \right) \)
- (c)
\(\pi -{ cos }^{ -1 }\sqrt { \left( 1-{ x }^{ 2 } \right) } \)
- (d)
\(-\pi +{ cot }^{ -1 }\left( \frac { \sqrt { { 1-{ x }^{ 2 } } } }{ x } \right) \)
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The sum to infinite terms of the series \({ tan }^{ -1 }\left( \frac { 1 }{ 3 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 9 } \right) +....+{ tan }^{ -1 }\left( \frac { { 2 }^{ n-1 } }{ 1+{ 2 }^{ 2n-1 } } \right) +......\)is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\pi\)
- (d)
none of these
\(\sum _{ r=1 }^{ n }{ { tan }^{ -1 } } \left( \frac { { x }_{ r }-{ x }_{ r-1 } }{ 1+{ x }_{ r-1 }{ x }_{ r } } \right) =\sum _{ r=1 }^{ n }{ \left( { tan }^{ -1 }{ x }_{ r }-{ tan }^{ -1 }{ x }_{ r-1 } \right) } ={ tan }^{ -1 }{ x }_{ n }-{ tan }^{ -1 }{ x }_{ 0 },\forall n\epsilon N\)
On the basis of above information, answer the following questions:
The sum to infinite terms of the series \({ tan }^{ -1 }\left( \frac { 2 }{ 1-{ 1 }^{ 2 }+{ 1 }^{ 4 } } \right) +{ tan }^{ -1 }\left( \frac { 4 }{ 1-{ 2 }^{ 2 }+{ 2 }^{ 4 } } \right) +{ tan }^{ -1 }\left( \frac { 6 }{ 1-{ 3 }^{ 2 }+{ 3 }^{ 4 } } \right) +\) ......is
- (a)
\(\frac { \pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 2 } \)
- (c)
\(\frac { 3\pi }{ 4 } \)
- (d)
none of these
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The principal value of \(\sin ^{ -1 }{ \left( sin\frac { 4\pi }{ 3 } \right) +\cos ^{ -1 }{ \left( cos\frac { 4\pi }{ 3 } \right) } } \) is
- (a)
\(\frac { 8\pi }{ 3 } \)
- (b)
\(\frac { 4\pi }{ 3 } \)
- (c)
\(\frac { 2\pi }{ 3 } \)
- (d)
\(\frac { \pi }{ 3 } \)
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The principal value of \({ tan }^{ -1 }\left( tan\left( -\frac { 3\pi }{ 4 } \right) \right) +{ cot }^{ -1 }cot\left( -\frac { 3\pi }{ 4 } \right) \) is
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(\pi\)
- (c)
\(\frac { -3\pi }{ 2 } \)
- (d)
0
Principal values for inverse circular functions:
x<0 | x\(\ge \)0 |
\(-\frac { \pi }{ 2 } \le \ { sin }^{ -1 }x<0\) | \(0\le { sin }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <\ cos^{ -1 }x\le \pi \) | \(0\le { cos }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } <{ tan }^{ -1 }x<0\) | \(0\le { tan }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ cot }^{ -1 }x<\pi \) | \(0\le { cot }^{ -1 }x\le \frac { \pi }{ 2 } \) |
\(\frac { \pi }{ 2 } <{ sec }^{ -1 }x\le \pi \) | \(0\le { sec }^{ -1 }x<\frac { \pi }{ 2 } \) |
\(-\frac { \pi }{ 2 } \le { cosec }^{ -1 }x<0\) | \(0<{ cosec }^{ -1 }x\le \frac { \pi }{ 2 } \) |
Ex. \({ sec }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) =\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } ,\ { tan }^{ -1 }\left( -\sqrt { 3 } \right) =-\frac { \pi }{ 3 } not\frac { 2\pi }{ 3 } \)
On the basis of above information, answer the following questions:
The value of sin-1 [cos{cos-1(cos x)+sin-1(sin x)}], where \(x\epsilon \left( \frac { \pi }{ 2 } ,\pi \right) \) is
- (a)
\(\frac { \pi }{ 2 } \)
- (b)
\(-\pi\)
- (c)
\(\pi\)
- (d)
\(-\frac { \pi }{ 2 } \)